ADDED doc/grg32.lyx Index: doc/grg32.lyx ================================================================== --- /dev/null +++ doc/grg32.lyx @@ -0,0 +1,56729 @@ +#LyX file created by tex2lyx 2.3 +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin /home/jhj/src/grg/ +\textclass report +\begin_preamble +%==========================================================================% +% GRG 3.2 Manual (C) 1988-97 Vadim V. Zhytnikov % +%==========================================================================% +% LaTeX 2e and MakeIndex are required to pront this document: % +% % +% latex grg32 % +% latex grg32 % +% latex grg32 % +% makeindex grg32 % +% latex grg32 % +% % +% If you do not have MakeIndex just omit two last steps. % +% The document is intended for two-side printing. % +%==========================================================================% + + + +\oddsidemargin=1.5cm +\evensidemargin=1.3cm + +%%% This is for PS fonts and dvips driver +%\usepackage{mathptm} +%\usepackage{palatino} +%\renewcommand{\bfdefault}{b} +%\newcommand{\grgtt}{\bfseries\ttfamily} +%\usepackage[dvips]{color} +%\definecolor{shade}{gray}{.9} +%\newcommand{\shadedbox}[1]{\fcolorbox{black}{shade}{#1}} +%%% This is for CM fonts +\newcommand{\grgtt}{\ttfamily} +\newcommand{\shadedbox}[1]{\fbox{#1}} +%%% + + +%\usepackage{calrsfs} % rsfs for mathcal + +%%% + +\let\@afterindentfalse\@afterindenttrue +\@afterindenttrue + +%%% + +%%% +\usepackage{makeidx} +\newcommand{\cmdind}[1]{\index{Commands!\comm{#1}}\index{#1@\comm{#1} (command)}} +\newcommand{\cmdindx}[2]{\index{Commands!\comm{#1}}\index{#1@\comm{#1} (command)!\comm{#2}}} +\newcommand{\swind}[1]{\index{Switches!\comm{#1}}% +\index{#1@\comm{#1} (switch)}% +\label{#1}} +\newcommand{\swinda}[1]{\index{Switches!\comm{#1}}% +\index{#1@\comm{#1} (switch)}} +%%% + +%%% +\newcommand{\rim}[1]{\stackrel{\scriptscriptstyle\{\}}{#1}\!} +%%% + +%%% +\newcommand{\object}[2]{% +\begin{equation} +\mbox{\comm{#1}} =\ #2 +\end{equation}} +\newcommand{\tsst}{\longleftrightarrow} +\newcommand{\vv}{\vphantom{\rule{5mm}{5mm}}} +\newcommand{\RR}[1]{\stackrel{\rm #1}{R}\!{}} +\newcommand{\OO}[1]{\stackrel{\rm #1}{\Omega}\!{}} +%%% + +%%% +\newcommand{\ipr}{\rule{1.8mm}{.1mm}\rule{.1mm}{2.2mm}\,} % _| int. product +%%% + +%%% +\newcommand{\spref}[1]{section \ref{#1} on page \pageref{#1}} +\newcommand{\pref}[1]{page \pageref{#1}} +%%% + +%%% +\newcommand{\seethis}[1]{\marginpar{\footnotesize\it #1}} +\newcommand{\rseethis}[1]{ +\reversemarginpar +\marginpar{\footnotesize\it #1} +\normalmarginpar} +\newcommand{\important}[1]{\marginpar{\itshape\bfseries\fbox{\ !\ } #1}} +%%% + +%%% Footnotes simbol ... +\renewcommand{\thefootnote}{\fnsymbol{footnote}} % + ++ etc for footnotes + +\def\@fnsymbol#1{\ensuremath{\ifcase#1\or \dagger\or \ddagger\or + \mathchar "278\or \mathchar "27B\or \|\or *\or **\or \dagger\dagger + \or \ddagger\ddagger \else\@ctrerr\fi}} + +%%% + +%%% Page layout ... +\textheight=180mm +\textwidth=120mm +%\marginparsep=2mm +%\marginparwidth=28mm +\marginparsep=5mm +\marginparwidth=25mm +\parindent=6mm +\parskip=1.2mm plus 1mm minus 1mm +%%% +\newlength{\myparindent} +\myparindent=\parindent + +%%% My own \tt font ... + +\def\verbatim@font{\grgtt} + +\renewcommand{\tt}{\grgtt} +%%% + +%%% +%%% Special symbols ... +\def\^{{\tt \char'136}} %%% \^ is ^ +\def\_{{\tt \char'137}} %%% \_ is _ +\newcommand{\w}{{\tt \char'057 \char'134}} %%% \w is /\ +\newcommand{\bs}{{\tt \char'134}} %%% \bs is \ +\newcommand{\ul}{{\tt \char'137}} %%% \ul is _ +\newcommand{\dd}{{\tt \char'043}} %%% \dd is # +\newcommand{\cc}{{\tt \char'176}} %%% \cc is ~ +\newcommand{\ip}{{\tt \char'137 \char'174}} %%% \ip is _| +\newcommand{\ii}{{\tt \char'174}} %%% \ii is | +\newcommand{\udr}{\mbox{$\Updownarrow$}} +%%% + +%%% \grg GRG logo ... +\newcommand{\grg}{{\sc GRG}} +\newcommand{\reduce}{{\sc Reduce}} +\newcommand{\maple}{{\sc Maple}} +\newcommand{\macsyma}{{\sc Macsyma}} +\newcommand{\mathematica}{{\sc Mathematica}} + +%%% \marg ... +\newcommand{\marg}[1]{\marginpar{\tiny#1}} + +%%% \command{...} commands in (shaded) box +\def\mynewline{\ifvmode \relax \else + \unskip\nobreak\hfil\break\fi} +\newcommand{\command}[1]{\vspace{1.2mm}\mynewline\hspace*{6mm}% +\shadedbox{\begin{tabular}{l}\tt% +#1 \end{tabular}}\vspace{1.2mm}\newline} +%%% parts of the commands +\newcommand{\file}[1]{{\sf#1}} +\newcommand{\comm}[1]{{\upshape\tt#1}} % \comm short in-line command +\newcommand{\parm}[1]{{\sf\slshape#1\/}} % \parm command parameter +\newcommand{\opt}[1]{{\rm[}#1{\rm]}} % \opt optional part of command +\newcommand{\user}[1]{{\bfseries\ttfamily#1}} % \user user input +\newcommand{\rpt}[1]{#1{\rm[}{\tt,}#1{\rm\dots}{\rm]}} % \rpt repetition + + +\def\closerule{\rule{.1mm}{1mm}\rule{119.8mm}{.1mm}} +\def\openrule{\rule{.1mm}{1mm}\rule[1mm]{119.8mm}{.1mm}} + +%%% \begin{slisting} ... \end{slisting} small font listing with frame +%%% \begin{listing} ... \end{listing} normal font listing without frame +\newcommand{\etrivlistrule}{\vspace*{-3mm}\endtrivlist{\closerule}\newline} + +\newdimen\allttindent +\allttindent=0mm +\def\docspecials{\do\ \do\$\do\&% + \do\#\do\^\do\^^K\do\_\do\^^A\do\%\do\~} +\def\slisting{\vspace*{-2mm}% +\trivlist \item[]\if@minipage\else\relax\fi +\leftskip\@totalleftmargin \advance\leftskip\allttindent \rightskip\z@ +\parindent\z@\parfillskip\@flushglue\parskip\z@ +\@tempswafalse\openrule \def\par{\if@tempswa\hbox{}\fi\@tempswatrue\@@par} +\obeylines \small\grgtt% + \catcode``=13 \@noligs +\let\do\@makeother \docspecials + \frenchspacing\@vobeyspaces} +\def\listing{\trivlist \item[]\if@minipage\else\relax\fi +\leftskip\@totalleftmargin \advance\leftskip\allttindent \rightskip\z@ +\parindent\z@\parfillskip\@flushglue\parskip\z@ +\@tempswafalse \def\par{\if@tempswa\hbox{}\fi\@tempswatrue\@@par} +\obeylines \grgtt% + \catcode``=13 \@noligs +\let\do\@makeother \docspecials + \frenchspacing\@vobeyspaces} +\let\endslisting=\etrivlistrule +\let\endlisting=\endtrivlist + +%%% + +%%% Headings style ... +%\usepackage{fancyheadings} +%%% We just inserat the fancyheadings.sty here literally ... + +% fancyheadings.sty version 1.7 +% Fancy headers and footers. +% Piet van Oostrum, Dept of Computer Science, University of Utrecht +% Padualaan 14, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands +% Telephone: +31-30-531806. piet@cs.ruu.nl (mcvax!sun4nl!ruuinf!piet) +% Sep 16, 1994 +% version 1.4: Correction for use with \reversemargin +% Sep 29, 1994: +% version 1.5: Added the \iftopfloat, \ifbotfloat and \iffloatpage commands +% Oct 4, 1994: +% version 1.6: Reset single spacing in headers/footers for use with +% setspace.sty or doublespace.sty +% Oct 4, 1994: +% version 1.7: changed \let\@mkboth\markboth to +% \def\@mkboth{\protect\markboth} to make it more robust + +\def\lhead{\@ifnextchar[{\@xlhead}{\@ylhead}} +\def\@xlhead[#1]#2{\gdef\@elhead{#1}\gdef\@olhead{#2}} +\def\@ylhead#1{\gdef\@elhead{#1}\gdef\@olhead{#1}} + +\def\chead{\@ifnextchar[{\@xchead}{\@ychead}} +\def\@xchead[#1]#2{\gdef\@echead{#1}\gdef\@ochead{#2}} +\def\@ychead#1{\gdef\@echead{#1}\gdef\@ochead{#1}} + +\def\rhead{\@ifnextchar[{\@xrhead}{\@yrhead}} +\def\@xrhead[#1]#2{\gdef\@erhead{#1}\gdef\@orhead{#2}} +\def\@yrhead#1{\gdef\@erhead{#1}\gdef\@orhead{#1}} + +\def\lfoot{\@ifnextchar[{\@xlfoot}{\@ylfoot}} +\def\@xlfoot[#1]#2{\gdef\@elfoot{#1}\gdef\@olfoot{#2}} +\def\@ylfoot#1{\gdef\@elfoot{#1}\gdef\@olfoot{#1}} + +\def\cfoot{\@ifnextchar[{\@xcfoot}{\@ycfoot}} +\def\@xcfoot[#1]#2{\gdef\@ecfoot{#1}\gdef\@ocfoot{#2}} +\def\@ycfoot#1{\gdef\@ecfoot{#1}\gdef\@ocfoot{#1}} + +\def\rfoot{\@ifnextchar[{\@xrfoot}{\@yrfoot}} +\def\@xrfoot[#1]#2{\gdef\@erfoot{#1}\gdef\@orfoot{#2}} +\def\@yrfoot#1{\gdef\@erfoot{#1}\gdef\@orfoot{#1}} + +\newdimen\headrulewidth +\newdimen\footrulewidth +\newdimen\plainheadrulewidth +\newdimen\plainfootrulewidth +\newdimen\headwidth +\newif\if@fancyplain \@fancyplainfalse +\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi} + +% Command to reset various things in the headers: +% a.o. single spacing (taken from setspace.sty) +% and the catcode of ^^M (so that epsf files in the header work if a +% verbatim crosses a page boundary) +\def\fancy@reset{\restorecr + \def\baselinestretch{1}% + \ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e + \ifx\@currsize\normalsize\@normalsize\else\@currsize\fi% + \else% NFSS (2.09) present + \@newbaseline% + \fi} + +% Initialization of the head and foot text. + +\headrulewidth 0.4pt +\footrulewidth\z@ +\plainheadrulewidth\z@ +\plainfootrulewidth\z@ + +\lhead[\fancyplain{}{\sl\rightmark}]{\fancyplain{}{\sl\leftmark}} +% i.e. empty on ``plain'' pages \rightmark on even, \leftmark on odd pages +\chead{} +\rhead[\fancyplain{}{\sl\leftmark}]{\fancyplain{}{\sl\rightmark}} +% i.e. empty on ``plain'' pages \leftmark on even, \rightmark on odd pages +\lfoot{} +\cfoot{\rm\thepage} % page number +\rfoot{} + +% Put together a header or footer given the left, center and +% right text, fillers at left and right and a rule. +% The \lap commands put the text into an hbox of zero size, +% so overlapping text does not generate an errormessage. + +\def\@fancyhead#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset\vbox{\hbox +{\rlap{\parbox[b]{\headwidth}{\raggedright#2\strut}}\hfill +\parbox[b]{\headwidth}{\centering#3\strut}\hfill +\llap{\parbox[b]{\headwidth}{\raggedleft#4\strut}}}\headrule}}#5} + + +\def\@fancyfoot#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset\vbox{\footrule +\hbox{\rlap{\parbox[t]{\headwidth}{\raggedright#2\strut}}\hfill +\parbox[t]{\headwidth}{\centering#3\strut}\hfill +\llap{\parbox[t]{\headwidth}{\raggedleft#4\strut}}}}}#5} + +\def\headrule{{\if@fancyplain\headrulewidth\plainheadrulewidth\fi +\hrule\@height\headrulewidth\@width\headwidth \vskip-\headrulewidth}} + +\def\footrule{{\if@fancyplain\footrulewidth\plainfootrulewidth\fi +\vskip-0.3\normalbaselineskip\vskip-\footrulewidth +\hrule\@width\headwidth\@height\footrulewidth\vskip0.3\normalbaselineskip}} + +\def\ps@fancy{ +\def\@mkboth{\protect\markboth} +\@ifundefined{chapter}{\def\sectionmark##1{\markboth +{\uppercase{\ifnum \c@secnumdepth>\z@ + \thesection\hskip 1em\relax \fi ##1}}{}} +\def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne + \thesubsection\hskip 1em\relax \fi ##1}}} +{\def\chaptermark##1{\markboth {\uppercase{\ifnum \c@secnumdepth>\m@ne + \@chapapp\ \thechapter. \ \fi ##1}}{}} +\def\sectionmark##1{\markright{\uppercase{\ifnum \c@secnumdepth >\z@ + \thesection. \ \fi ##1}}}} +\ps@@fancy +\global\let\ps@fancy\ps@@fancy +\headwidth\textwidth} +\def\ps@fancyplain{\ps@fancy \let\ps@plain\ps@plain@fancy} +\def\ps@plain@fancy{\@fancyplaintrue\ps@@fancy} +\def\ps@@fancy{ +\def\@oddhead{\@fancyhead\@lodd\@olhead\@ochead\@orhead\@rodd} +\def\@oddfoot{\@fancyfoot\@lodd\@olfoot\@ocfoot\@orfoot\@rodd} +\def\@evenhead{\@fancyhead\@rodd\@elhead\@echead\@erhead\@lodd} +\def\@evenfoot{\@fancyfoot\@rodd\@elfoot\@ecfoot\@erfoot\@lodd} +} +\def\@lodd{\if@reversemargin\hss\else\relax\fi} +\def\@rodd{\if@reversemargin\relax\else\hss\fi} + +\let\latex@makecol\@makecol +\def\@makecol{\let\topfloat\@toplist\let\botfloat\@botlist\latex@makecol} +\def\iftopfloat#1#2{\ifx\topfloat\empty #2\else #1\fi} +\def\ifbotfloat#1#2{\ifx\botfloat\empty #2\else #1\fi} +\def\iffloatpage#1#2{\if@fcolmade #1\else #2\fi} + +%%% + +\addtolength{\headwidth}{\marginparsep} +\addtolength{\headwidth}{\marginparwidth} +\lhead[\bfseries\thepage]{\bfseries\slshape\rightmark} +\chead{} +\rhead[\bfseries\slshape\leftmark]{\bfseries\thepage} +\lfoot{} +\cfoot{} +\rfoot{} +\renewcommand{\uppercase}[1]{#1} +%%% + +%%% Chapter style ... + +\def\@makechapterhead#1{% + \noindent\grgrule\break% + { \hsize=150mm + \parindent \z@ \raggedleft \reset@font + \ifnum \c@secnumdepth >\m@ne + \Large\slshape \@chapapp{} \Huge\bfseries \thechapter + \par + \vskip 20\p@ + \fi + \Huge \bfseries\upshape #1\par + \nobreak + \vskip 40\p@ + }} +\def\@makeschapterhead#1{% + \noindent\grgrule\break% + { \hsize=150mm + \parindent \z@ \raggedleft + \reset@font + \Large\slshape #1\par + \nobreak + \vskip 20\p@ + }} +\renewcommand{\chapter}{\if@openright\cleardoublepage\else\clearpage\fi + \thispagestyle{empty}% + \global\@topnum\z@ + %\@afterindentfalse + \secdef\@chapter\@schapter} + +\renewcommand{\chaptername}{CHAPTER} +\renewcommand{\contentsname}{CONTENTS} +\renewcommand{\appendixname}{APPENDIX} +\newcommand{\grgrule}{\rule{150mm}{.3mm}\relax} +%%% + +%%% Sections ... +%\renewcommand{\thesection}{} +%\renewcommand{\thesubsection}{} +%\renewcommand{\thesubsubsection}{} + +%\renewcommand\section{\@startsection {section}{1}{\z@}% +% {-3.5ex \@plus -1ex \@minus -.2ex}% +% {2.3ex \@plus.2ex}% +% {\normalfont\Large\bfseries}} +\renewcommand{\subsection}{\@startsection{subsection}{2}{\z@}% + {-3.25ex\@plus -1ex \@minus -.2ex}% + {1.5ex \@plus .2ex}% + {\normalfont\large\slshape\bfseries}} +%\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% +% {-3.25ex\@plus -1ex \@minus -.2ex}% +% {1.5ex \@plus .2ex}% +% {\normalfont\normalsize\bfseries}} + +%%% + + + + +\end_preamble +\options openright +\use_default_options false +\maintain_unincluded_children false +\language english +\language_package none +\inputencoding auto +\fontencoding default +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "cmtt" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 0 +\use_package cancel 0 +\use_package esint 1 +\use_package mathdots 0 +\use_package mathtools 0 +\use_package mhchem 0 +\use_package stackrel 0 +\use_package stmaryrd 0 +\use_package undertilde 0 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 0 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 +\papercolumns 1 +\papersides 2 +\paperpagestyle fancy +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{titlepage} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +hsize +\end_layout + +\end_inset + +=150mm +\begin_inset space \hrulefill{} + +\end_inset + + +\begin_inset VSpace 20mm* +\end_inset + + +\end_layout + +\begin_layout Standard +\align center + +\size giant + +\series bold +GRG +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash + +\backslash +[1mm] +\end_layout + +\end_inset + + +\size normal +Version 3.2 +\end_layout + +\begin_layout Standard +\align center + +\series bold + +\size normal + +\size larger +Computer Algebra System for +\begin_inset Newline newline +\end_inset + + Differential Geometry, +\begin_inset Newline newline +\end_inset + + Gravitation and +\begin_inset Newline newline +\end_inset + + Field Theory +\begin_inset VSpace 25mm* +\end_inset + + +\begin_inset Newline newline +\end_inset + + +\shape italic +Vadim V. Zhytnikov +\size larger + +\begin_inset Newline newline +\end_inset + + +\begin_inset VSpace vfill +\end_inset + + +\size normal +Moscow, 1992–1997 +\begin_inset Formula $\bullet$ +\end_inset + + Chung-Li, 1994 +\size larger + +\end_layout + +\begin_layout Standard + +\series bold + +\size larger + +\begin_inset space \hrulefill{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{titlepage} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +setcounter{page}{0} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +thispagestyle{empty} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\begin_inset CommandInset toc +LatexCommand tableofcontents + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +thispagestyle{empty} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Chapter +Introduction +\end_layout + +\begin_layout Standard +Calculation of various geometrical and physical quantities and equations is the usual technical problem which permanently arises in geometry, field and gravity theory. Numerous indices, contractions and components make these calculations very tedious and error-prone. Since this calculus obeys the well defined rules the idea to automate this kind of problems using computer is quite natural. Now there are several computer algebra systems such as +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +maple +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mathematica +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +macsyma +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +which in principle allow one to do this and it is not so hard to write a program to calculate, for example, the curvature tensor or connection. But suppose that we want to make a non-trivial coordinate transformation or tetrad rotation, calculate covariant or Lie derivative, compute a complicated expression with numerous contraction or raise or lower some indices. All these operations are typical in differential geometry and field theory but their realization with the help of general purpose computer algebra systems requires hard programming since all these systems really know nothing about +\emph on +covariant properties +\emph default + of geometrical quantities. +\end_layout + +\begin_layout Standard +The computer algebra system +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is designed in such a way to make calculation in differential geometry and field theory as simple and natural as possible. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is based on the computer algebra system +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +but +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +has its own simple input language whose commands resembles English phrases. Working with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +no any knowledge of programming is required. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +understands tensors, spinors, vectors, differential forms and knows all standard operations with these quantities. Input form for mathematical expressions is very close to traditional mathematical notation including Einstein summation rule. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +knows the covariant properties of these objects, you can easily raise and lower indices, compute covariant and Lie derivatives, perform coordinate and frame transformations. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +works in any dimension and allows one to represent tensor quantities with respect to holonomic, orthogonal and even any other arbitrary frame. +\end_layout + +\begin_layout Standard +One of the useful features of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is that it has a large number of built-in standard field-theory and geometrical quantities and formulas for their computation. Thus +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +provides ready solutions to many standard problems. +\end_layout + +\begin_layout Standard +Another unique feature of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is that it can export results of calculations into other computer algebra system. You can save your data in to the file in the format of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +maple +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mathematica +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +macsyma +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +in order to use this system to proceed analysis of the data. The \SpecialChar LaTeX + +\begin_inset space \space{} + +\end_inset + +output format is supported as well. In addition +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is compatible with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +graphics shells providing niece book-quality output with Greek letters, integral signs etc. +\end_layout + +\begin_layout Standard +The main built-in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +capabilities are: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{list} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset Formula $\bullet$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +labelwidth +\end_layout + +\end_inset + +=8mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +leftmargin +\end_layout + +\end_inset + +=10mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Connection, torsion and nonmetricity. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Curvature. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Spinorial formalism. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Irreducible decomposition of the curvature, torsion, and nonmetricity in any dimension. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Einstein equations. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Scalar field with minimal and non-minimal interaction. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Electromagnetic field. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Yang-Mills field. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Dirac spinor field. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Geodesic equation. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Null congruences and optical scalars. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Kinematics for time-like congruences. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Ideal and spin fluid. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Newman-Penrose formalism. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Gravitational equations for the theory with arbitrary gravitational Lagrangian in Riemann and Riemann-Cartan spaces. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{list} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +I would like to stress that current +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +version is intended for calculations in a concrete coordinate map only. It cannot operate with tensors as with objects having abstract symbolic indices. +\end_layout + +\begin_layout Standard +This book consist of two main parts. First part contains detailed description of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +as a programming system. Second part describes all built-in objects and formulas for their computation. +\end_layout + +\begin_layout Chapter +Programming in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Throughout the chapter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are printed in typewriter font. The slanted serif-less font is used for command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The optional parts of the commands are enclosed in squared brackets +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +option +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + stands for one or several repetitions of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + etc. Examples are separated form the text by horizontal lines +\begin_inset Formula $\stackrel{\rule{0.1mm}{1mm}\rule[1mm]{3mm}{0.1mm}} +{\rule{0.1mm}{1mm}\rule{3mm}{0.1mm}}$ +\end_inset + + and the user input can be easily distinguished from the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +output by the prompt +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +<- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + which precedes every input line. +\end_layout + +\begin_layout Section +Session, Tasks and Commands +\end_layout + +\begin_layout Standard +To start +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +it is necessary to start +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + + On some systems you have to use +\family typewriter + +\shape up +load!_package grg; +\family default +\series default +\shape default + +\begin_inset Newline newline +\end_inset + +since +\family typewriter + +\shape up +load +\family default +\series default +\shape default + is not defined. +\begin_inset Newline newline +\end_inset + + +\begin_inset Newline newline +\end_inset + +Sometimes it +\begin_inset Newline newline +\end_inset + +is better to use two commands +\begin_inset Newline newline +\end_inset + + +\family typewriter + +\shape up +load grg32; grg; +\family default +\series default +\shape default + +\begin_inset Newline newline +\end_inset + +or +\begin_inset Newline newline +\end_inset + + +\family typewriter + +\shape up +load grg; grg; +\family default +\series default +\shape default + +\begin_inset Newline newline +\end_inset + +(See section +\begin_inset CommandInset ref +LatexCommand ref +reference "configsect" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + for details.) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + enter the command +\family typewriter +load grg; +\family default +\series default +\shape default + +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + REDUCE 3.5, 15 Oct 93, patched to 15 Jun 95 ... +\end_layout + +\begin_layout Standard +1: load grg; +\end_layout + +\begin_layout Standard +This is GRG 3.2 release 2 (Feb 9, 1997) ... +\end_layout + +\begin_layout Standard +System directory: c: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +reduce +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +grg32 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + System variables are upper-cased: E I PI SIN ... Dimension is 4 with Signature (-,+,+,+) +\end_layout + +\begin_layout Standard +<- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Symbol +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +<- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +prompt which shows that now +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +waits for your input. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\emph on +task +\emph default + (we prefer this term instead of usual +\emph on +program +\emph default +) consist of the sequence of commands terminated by semicolon +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Reading the input +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +splits it on +\emph on +atoms +\emph default +. There are several types of atoms: +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Atoms +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{list} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset Formula $\bullet$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +labelwidth +\end_layout + +\end_inset + +=4mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +leftmargin +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parindent +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +The identifier or symbol is a sequence of letters and digits starting with a letter: +\end_layout + +\begin_layout Verbatim + i I alpha1 beta ABC123D Find +\end_layout + +\begin_layout Standard +The identifiers in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +may have trailing tilde character +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + +. Any other character may be incorporated in the identifier if preceded by the exclamation sign: +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Identifiers +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Verbatim + beta~ LIMIT!+ +\end_layout + +\begin_layout Standard +The identifiers in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +play the role of the variables and functions in mathematical expressions and words in commands. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Integer numbers +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Numbers +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Verbatim + 0 123 104341 +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +String is a sequence of characters enclosed in double quotes +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Strings +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Verbatim + "file.txt" "This is a string" "dir *.doc" +\end_layout + +\begin_layout Standard +The strings in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are used for file names and operating system commands. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Nine special two-character atoms +\end_layout + +\begin_layout Verbatim + ** _| / +\backslash + |= ~~ .. <= >= -> +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Any other characters are considered as single-character atoms. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{list} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The format of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +commands is free. They can span one or several lines and any number of spaces and tabulations can be inserted between two neighbor atoms. +\end_layout + +\begin_layout Standard + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +enlargethispage{3mm} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session may consist of several independent tasks. The command +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Tasks +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Quit +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Quit; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + terminates both +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session and returns the control to the operating system level. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Stop +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Stop; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + terminates current +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task and brings the session control menu: +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Session control menu +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Stop; +\end_layout + +\begin_layout Standard +Quit GRG - 0 Start Task - 1 Exit to REDUCE - 2 +\end_layout + +\begin_layout Standard +Type 0, 1 or 2: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\begin_inset Newpage newpage +\end_inset + + +\end_layout + +\begin_layout Standard +\noindent +The option +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +0 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + terminates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session similarly to the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Quit; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The choice +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + starts new task by bringing +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +to its initial state: all variables, declarations, substitutions and results of calculations are cleared and all switches resume their initial positions. +\begin_inset Foot +status collapsed + + +\begin_layout Standard +Usually +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +does good job by resuming initial state and new task turns out to be independent of previous ones. But on some rare occasions the initial state cannot be completely recovered and it is better to restart +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +completely. +\end_layout + +\end_inset + + Finally the option +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + terminates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task and returns control to the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +command level. In this case +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +can be restarted later by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +grg; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard +The commands in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are case insensitive, i.e. command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Quit; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is equivalent to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +quit; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +QUIT; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + etc. But notice that unlike +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +variables and functions in mathematical expressions in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\emph on +are case sensitive +\emph default +. +\end_layout + +\begin_layout Subsection +Switches +\end_layout + +\begin_layout Standard + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Switches +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Switches in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are used to control various system modes of operation. They are denoted by identifiers and the commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +On +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Off +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +On +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + + +\family typewriter +Off +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + turns the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + on and off respectively. Any switch defined by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is available in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +as well. In addition +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +defines a couple of its own switches. The full list of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +switches is presented in appendix A. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show Switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or equivalently +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +? +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints current +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + position +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Show Switch TORSION; TORSION is Off. <- On torsion,gcd; <- switch torsion; TORSION is On. <- switch exp; GCD is On +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Switches in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are case insensitive. +\end_layout + +\begin_layout Subsection + +\family typewriter +Batch File Execution +\end_layout + +\begin_layout Standard + +\family typewriter +Usually +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +works in the interactive mode which is not always convenient. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Input +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Batch file execution +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Input +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + reads the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and executes commands stored in it. The file names in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are always denoted by strings and exact specification of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is operating system dependent. The word +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Input +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is optional, thus in order to run batch file it suffices to enter its name +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The execution of batch file commands can be suspended by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Pause +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Pause; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + After this command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +enters the interactive mode. One can enter one or several commands interactively and then resume batch file execution by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Next +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Next; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +In general no any special end-of-file symbol or command is required in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +batch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + but is necessary the symbol +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +end-of-file symbol +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +$ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +$ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is recognized by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +as the end-of-file mark. +\end_layout + +\begin_layout Standard + +\family typewriter +If during the batch file execution an error occurs +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +enter interactive mode and ask user to input the command which is supposed to replace the erroneous one. After the receiving of +\emph on +one +\emph default + command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +automatically resumes the batch file execution. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Pause; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be used if it is necessary to execute +\emph on +several +\emph default + commands instead of one. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Output +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Output +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +outfile +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + redirects all +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +output into the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +outfile +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +outfile +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be closed by the equivalent commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +EndO +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +End of Output +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +EndO; +\begin_inset Newline newline +\end_inset + +End of Output; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +It is convenient to run long-time +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +tasks in background. The way of doing this depend on the operating system. For example to execute +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task in background in UNIX it is necessary to use the following command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +reduce < task.grg > grg.out & + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Here we assume that the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +invoking command is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +reduce +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and the file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +task.grg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + contains the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task commands: +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +load grg; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +grg command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +grg command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; ... +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +grg command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; quit; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The output of the session will be written into the file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grg.out +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +Since no proper reaction on errors is possible during the background execution it is good idea to turn the switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +BATCH +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + on. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +BATCH +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + This makes +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +to terminate the session immediately in the case of any error. +\end_layout + +\begin_layout Subsection + +\family typewriter +Operating System Commands +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +System +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +System +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + executes the operating system +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The same command without parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +System; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + temporary suspends +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session and passes the control to the operating system command level. The details may depend on the concrete operating system. In particular in UNIX the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +system; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + may fail but UNIX has some general mechanism for suspending running programs: you can press +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Ẑ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to suspend any program and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +%+ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to resume its execution. +\end_layout + +\begin_layout Subsection + +\family typewriter +Comments +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +% +\backslash +reversemarginpar +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The comment commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Comment +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Comment +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +any text +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +% +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +any text +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are used to supply additional information to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +tasks +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "Unload" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and data saved by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command. The comment can be also attached to the end of any +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +grg command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + % +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +any text +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +% +\backslash +normalmarginpar +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Timing +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Time +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show Time +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Time; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints time elapsed since the beginning of current +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task including the percentage of so called garbage collections. The garbage collection time can be also printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +GC Time +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show GC Time +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + GC Time; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +If percentage of garbage collections grows and exceeds say 30% then memory of your system is running short and you probably need more RAM. +\end_layout + +\begin_layout Section + +\family typewriter +Declarations +\end_layout + +\begin_layout Standard + +\family typewriter +Any object, variable or function in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +must be declared. This allows to locate misprints and makes the system more reliable. Since +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +always work in some concrete coordinate system (map) the coordinate declaration is the most important one and must be present in every +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task. +\end_layout + +\begin_layout Subsection + +\family typewriter +Dimension and Signature +\end_layout + +\begin_layout Standard + +\family typewriter +During installation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +always defines default value of the dimension and signature. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Dimension!default +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Signature!default +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +tuning +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to find out how to change the default dimension and signature. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The information about this default value is printed +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Dimension +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Signature +\end_layout + +\end_inset + + upon +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +start in the form of the following (or similar) message line: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + Dimension is 4 with Signature (-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The following command overrides the default dimension and signature +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Dimension +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Dimension +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +dim +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Signature +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +pm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +dim +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the number +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or greater and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +pm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + ++ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +pm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be preceded or succeeded by a number which denotes several repetitions of this +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +pm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. For example the declarations +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Dimension 5 with Signature (+,+,-,-,-); Dimension 5 with (2+,-3); + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +are equivalent and defines 5-dimensional space with the signature +\begin_inset Formula ${\rm diag}{\scriptstyle(+1,+1,-1,}$ +\end_inset + + +\begin_inset Formula ${\scriptstyle-1,-1)}$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +The important point is that the dimension declaration must be +\emph on +very first in the task +\emph default + and goes before any other command. Current dimension and signature can be printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Status; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Coordinates +\end_layout + +\begin_layout Standard + +\family typewriter +The coordinate declaration command must be present in every +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Only few commands such as informational commands, other declarations, switch changing commands may precede the coordinate declaration. The only way to have a tusk without the coordinate declaration is to load the file where coordinates where saved by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +UnloadLoad +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to find out how to save data and declarations into a file. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + but no any computation can be done before coordinates are declared. Current coordinate list can be printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdindx{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +}{ +\end_layout + +\end_inset + +Coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write Coordinates; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Float table +wide false +sideways false +status open + + +\begin_layout Standard +\align center + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Constants!predefined +\end_layout + +\end_inset + + +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +E I PI INFINITY +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Mathematical constants +\begin_inset Formula $e,i,\pi$ +\end_inset + +, +\begin_inset Formula $\infty$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +FAILED +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +ECONST +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Charge of the electron +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +DMASS +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Dirac field mass +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +SMASS +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Scalar field mass +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +GCONST +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Gravitational constant +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +CCONST +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Cosmological constants +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +LC0 LC1 LC2 LC3 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Parameters of the quadratic +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +LC4 LC5 LC6 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +gravitational Lagrangian +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +MC1 MC2 MC3 +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +AC0 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Nonminimal interaction constant +\end_layout + +\end_inset + + + + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Predefined constants +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "predefconstants" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Constants +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Constants +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Any constant must be declared by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Constants +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Constants +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The list of currently declared constants can be printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdindx{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +}{ +\end_layout + +\end_inset + +Constants +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write Constants; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + There are also a number of built-in constants which are listed in table +\begin_inset CommandInset ref +LatexCommand ref +reference "predefconstants" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Subsection + +\family typewriter +Functions +\end_layout + +\begin_layout Standard + +\family typewriter +Functions in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are the analogues of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\emph on +operators +\emph default + but we prefer to use this traditional mathematical term. The function must be declared by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the function identifier. The optional list of parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines function with +\emph on +implicit +\emph default + dependence. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + must be either coordinate or constant. The construction +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +(*) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is a shortcut which declares the function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + depending on +\emph on +all coordinates +\emph default +. +\end_layout + +\begin_layout Standard + +\family typewriter +The following example declares three functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun3 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, which was declared without implicit coordinate list, must be always used in mathematical expressions together with the explicit arguments like +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun1(x+y) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + etc. The functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun3 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can appear in expressions in similar fashion but also as a single symbol +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +fun3 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- Constant a; <- Functions fun1, fun2(x,y), fun3(*); <- Write functions; Functions: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +fun1 fun2(x,y) fun3(t,x,y,z) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- d fun1(x+a); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +DF(fun1(a + x),x) d x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- d fun2; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +DF(fun2,x) d x + DF(fun2,y) d y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- d fun3; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +DF(fun3,t) d t + DF(fun3,x) d x + DF(fun3,y) d y + DF(fun3,z) d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The functions may have particular properties with respect to their arguments permutation and sign. The corresponding declarations are +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Symmetric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Antisymmetric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Odd +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Even +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Symmetric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +Antisymmetric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +Odd +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +Even +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Notice that these commands are valid only after function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + was declared by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +In addition to user-defined there is also large number of functions predefined in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + +. All these functions can be used in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +without declaration. The complete list of these functions depends on +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +versions. Any function defined in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +package (module) is available too if the package is loaded before +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +was started or during +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +packages +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to find out how to load the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +packages. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + For example the package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +specfn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + contains definitions for various special functions. +\end_layout + +\begin_layout Standard + +\family typewriter +Finally there is also special declaration +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Generic Functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Generic Functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + This command is valid iff the package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +dfpart.red +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is installed on your +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +system. Here unlike the usual function declaration the list of parameters must be always present and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be any identifier preferably distinct from any other variable. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +genfun +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to find out about the generic functions. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The role of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is also completely different and is explained later. +\end_layout + +\begin_layout Standard + +\family typewriter +The list of declared functions can be printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdindx{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +}{ +\end_layout + +\end_inset + +Functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write Functions; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Generic functions in this output are marked by the label +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +* +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Subsection + +\family typewriter +Affine Parameter +\end_layout + +\begin_layout Standard + +\family typewriter +The variable which plays the role of affine parameter in the geodesic equation must be declared by the command +\begin_inset CommandInset label +LatexCommand label +name "affpar" + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Affine Parameter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and can be printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdindx{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +}{ +\end_layout + +\end_inset + +Affine Parameter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write Affine Parameter; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset VSpace vfill +\end_inset + + +\begin_inset Newpage newpage +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Case Sensitivity +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset CommandInset label +LatexCommand label +name "case" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Usually +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is case insensitive which means for example that expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +x-X +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + will be evaluated by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +as zero. On the contrary all coordinates, constants and functions in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are case sensitive, e.g. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +alpha +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Alpha +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +ALPHA +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are all different. Notice that commands and switches in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + +3.2 remain case insensitive. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Internal +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +case +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Therefore all predefined by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +constants and all built-in objects must be used exactly as they presented in this manual +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +GCONST +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SMASS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + etc. The situation with the constants and functions which predefined by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is different. The point is that in spite of its default case insensitivity internally +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +converts everything into some default case which may be upper or lower. Therefore depending on the particular +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +system they must be typed either as +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +E I PI INFINITY SIN COS ATAN + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +or in lower case +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +e i pi infinity sin cos atan + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +For the sake of definiteness throughout this book we chose the first upper case convention. +\end_layout + +\begin_layout Standard + +\family typewriter +When +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +starts it informs you about internal case of your particular +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +system by printing the message +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + System variables are upper-cased: E I PI SIN ... +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + System variables are lower-cased: e i pi sin ... +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + You can find out about the internal case using the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Status; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset VSpace vfill +\end_inset + + +\begin_inset Newpage newpage +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Complex Conjugation +\end_layout + +\begin_layout Standard + +\family typewriter +By default all variables and functions in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are considered to be real excluding the imaginary unit constant +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +I +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + (or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +i +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + as explained above). But if two identifiers differ only by the trailing character +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + they are considered as a pair of complex variables which are conjugated to each other. In the following example coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + comprise such a pair: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates u, v, z, z +\begin_inset space ~ + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +z & z +\begin_inset space ~ + +\end_inset + + - conjugated pair. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Re(z); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +z + z +\begin_inset space ~ + +\end_inset + + ——– 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Im(z +\begin_inset space ~ + +\end_inset + +); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +I*(z - z +\begin_inset space ~ + +\end_inset + +) ———— 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section + +\family typewriter +Objects +\end_layout + +\begin_layout Standard + +\family typewriter +Objects play a fundamental role in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +. They represent mathematical quantities such as metric, connection, curvature and any other spinor or tensor geometrical and physical fields and equations. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +has quite large number of built-in objects and knows many formulas for their calculation. But you are not obliged to use the built-in quantities and can declare your own. The purpose of the declaration is to tell +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +basic properties of a new quantity. +\end_layout + +\begin_layout Subsection + +\family typewriter +Built-in Objects +\end_layout + +\begin_layout Standard +\noindent + +\family typewriter +An object is characterized by the following properties and attributes: +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Built-in objects +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{list} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset Formula $\bullet$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +labelwidth +\end_layout + +\end_inset + +=4mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +leftmargin +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parindent +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parsep +\end_layout + +\end_inset + +=0mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Name +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Identifier or symbol +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Type of the component +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +List of indices +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Symmetries with respect to index permutation +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Density and pseudo-tensor property +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Built-in ways of calculation +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Value +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{list} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The object +\emph on +name +\emph default + is a sequence of words which are usually the common English name of corresponding quantity. The name is case insensitive and is used to denote a particular object in commands. So called +\emph on +group names +\emph default + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Group names +\end_layout + +\end_inset + + refer to a collection of closely related objects. In particular the name Curvature Spinors +\family typewriter +\series default +\shape default + (see page +\begin_inset CommandInset ref +LatexCommand pageref +reference "curspincoll" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +) refers to the irreducible components of the curvature tensor in spinorial representation. Actual content of the group may depend on the environment. In particular the group Curvature Spinors +\family typewriter +\series default +\shape default + includes three objects in the Riemann space (Weyl spinor, traceless Ricci spinor and scalar curvature) while in the space with torsion we have six irreducible curvature spinors. +\end_layout + +\begin_layout Standard + +\family typewriter +The object +\emph on +identifier +\emph default + or +\emph on +symbol +\emph default + is an identifier which denotes the object in mathematical expressions. Object symbols are case sensitive. +\end_layout + +\begin_layout Standard + +\family typewriter +The object +\emph on +type +\emph default + is the type of its component: objects can be scalar, vector or +\begin_inset Formula $p$ +\end_inset + +-form valued. The +\emph on +density +\emph default + and +\emph on +pseudo-tensor +\emph default + properties of the object characterizes its behaviour under coordinate and frame transformations. +\end_layout + +\begin_layout Standard + +\family typewriter +Objects can have the following types of indices: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{list} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset Formula $\bullet$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +labelwidth +\end_layout + +\end_inset + +=4mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +leftmargin +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parindent +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Upper and lower holonomic coordinate indices. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Upper and lower frame indices. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Upper and lower spinorial indices. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Upper and lower conjugated spinorial indices. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Enumerating indices. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{list} +\end_layout + +\end_inset + + The major part of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +built-in objects has frame indices. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "metric" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about the frame in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The frame in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +can be arbitrary but you can easily specify the frame to be holonomic or say orthogonal. Then built-in object indices become holonomic or orthogonal respectively. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +deals only with the SL(2,C) spinors which are restricted to the 4-dimensional spaces of Lorentzian signature. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +spinors +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + about the spinorial formalism in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The corresponding SL(2,C) indices take values 0 and 1. The conjugated indices are transformed with the help of the complex conjugated SL(2,C) matrix. If some spinor is totally symmetric in the group of +\begin_inset Formula $n$ +\end_inset + + spinorial indices (irreducible spinor) then these indices can be replaced by a single so called +\emph on +summed spinorial index +\emph default + of rank +\begin_inset Formula $n$ +\end_inset + + which take values from 0 to +\begin_inset Formula $n$ +\end_inset + +. The summed spinorial indices provide the most economic way to store the irreducible spinor components. +\end_layout + +\begin_layout Standard + +\family typewriter +Enumerating indices just label a collection of values and have no any covariant meaning. Accordingly there is no difference between upper and lower enumerating indices. +\end_layout + +\begin_layout Standard + +\family typewriter +Notice that an index of any type in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +always runs from 0 up to some maximal value which depend on the index type and dimensionality: +\begin_inset Formula $d-1$ +\end_inset + + for frame and coordinate indices, +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Dimension +\end_layout + +\end_inset + + and +\begin_inset Formula $n$ +\end_inset + + the spinor indices of the rank +\begin_inset Formula $n$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +understands various types of index symmetries: symmetry, antisymmetry, cyclic symmetry and Hermitian symmetry. These symmetries can apply not only to single indices but to any group of indices as well. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Index symmetries +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Canonical order of indices +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +uses object symmetries to decrease the amount of memory required to store the object components. It stores only components with the indices in certain +\emph on +canonical +\emph default + order and any other component are automatically restored if necessary by appropriate index permutation. The canonical order of indices is defined as follows: for symmetry, antisymmetry or Hermitian symmetry indices are sorted in such a way that index values grows from left to the right. For cyclic symmetry indices are shifted to minimize the numerical value of the whole list of indices. +\end_layout + +\begin_layout Standard + +\family typewriter +Finally there are two special types of objects: equations and connection 1-forms. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Equations +\end_layout + +\end_inset + + Equations have all the same properties as any other object but in addition they have left and right hand side and are printed in the form of equalities. The connections are used by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +to construct covariant derivatives. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Connections +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +conn2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + about the connections. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + There are only four types of connections: holonomic connection 1-form, frame connection 1-form, spinor connection 1-form and conjugated spinor connection 1-form. +\end_layout + +\begin_layout Standard + +\family typewriter +Almost all built-in objects have associated built-in +\emph on +ways of calculation +\emph default + (one or several). +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Ways of calculation +\end_layout + +\end_inset + + Each way is nothing but a formula which can be used to obtain the object value. +\end_layout + +\begin_layout Standard + +\family typewriter +Every object can be in two states. Initially when +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +starts all objects are in +\emph on +indefinite +\emph default + state, i.e. nothing is known about their value. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Object value +\end_layout + +\end_inset + + Since +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +always works in some concrete frame and coordinate system the object value is a table of the components. As soon as the value of certain object is obtained either by direct assignment or using some built-in formula (way of calculation) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +remember this value and store it in some internal table. Later this value can be printed, re-evaluated used in expression etc. The object can be returned to its initial indefinite state using the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +uses object symmetries to reduce total number of components to store. +\end_layout + +\begin_layout Standard + +\family typewriter +The complete list of built-in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +objects is given in appendix C. The chapter 3 also describes built-in objects but in the usual mathematical style. The equivalent commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +? +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints detailed information about the object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + including object name, identifier, list of indices, type of the component, current state (is the value of an object known or not), symmetries and ways of calculation. Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is either object name or its identifier. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show * +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Show *; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints complete list of built-in object names. This list is quite long and the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +c +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +*; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + gives list of objects whose names begin with the character +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +c +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +– +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +). +\end_layout + +\begin_layout Standard + +\family typewriter +Finally the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show All +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Show All; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints list of objects whose values are currently known. +\end_layout + +\begin_layout Standard + +\family typewriter +Notice that some built-in objects has limited scope. In particular some objects exists only in certain dimensionality, the quantities which are specific to spaces with torsion are defined iff switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +TORSION +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is turned on etc. +\end_layout + +\begin_layout Standard + +\family typewriter +Let us consider some examples. We begin with the curvature tensor +\begin_inset Formula $R^a{}_{bcd}$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Show Riemann Tensor; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Riemann tensor RIM'a.b.c.d is Scalar Value: unknown Symmetries: a(3,4) Ways of calculation: Standard way (D,OMEGA) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + This object has name Riemann Tensor +\family typewriter +\series default +\shape default + and identifier RIM +\family typewriter +\series default +\shape default +. The object is Scalar +\family typewriter +\series default +\shape default + (0-form) valued and has four frame indices. Frame indices are denoted by the lower-case characters and their upper or lower position are denoted by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +' +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + respectively. The Riemann tensor is antisymmetric in two last indices which is denoted by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a(3,4) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +The curvature 2-form +\begin_inset Formula $\Omega^a{}_b$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- ? OMEGA; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Curvature OMEGA'e.f is 2-form Value: unknown Ways of calculation: Standard way (omega) From spinorial curvature (OMEGAU*,OMEGAD) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + has name Curvature +\family typewriter +\series default +\shape default + and the identifier OMEGA +\family typewriter +\series default +\shape default + and is 2-form valued. +\end_layout + +\begin_layout Standard + +\family typewriter +The traceless Ricci spinor (the quantity which is usually denoted in the Newman-Penrose formalism as +\begin_inset Formula $\Phi_{AB\dot{C}\dot{D}}$ +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- ? Traceless Ricci Spinor; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Traceless ricci spinor RC.AB.CD +\begin_inset space ~ + +\end_inset + + is Scalar Value: unknown Symmetries: h(1,2) Ways of calculation: From spinor curvature (OMEGAU,SD,VOL) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Spinorial indices are denoted by upper case characters with the trailing +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + for conjugated indices. Usual spinorial indices are denoted by a +\emph on +single +\emph default + upper case letter while summed indices are denoted by several characters. Thus, the traceless Ricci spinor has two summed spinorial indices of rank 2 each taking the values from 0 to 2. The spinor is hermitian +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +h(1,2) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +The Einstein equation is an example of equation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- ? Einstein Equation; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Einstein equation EEq.g.h is Scalar Equation Value: unknown Symmetries: s(1,2) Ways of calculation: Standard way (G,RIC,RR,TENMOM) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + and 1-form +\begin_inset Formula $\Gamma^\alpha{}_\beta$ +\end_inset + + is an example of the connection +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +enlargethispage{2mm} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Show Holonomic Connection; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reversemarginpar +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Holonomic connection GAMMAxy is 1-form Holonomic Connection Value: unknown Ways of calculation: From frame connection (T,D,omega) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + The coordinate indices are denoted by the lower-case letters with labels +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +˖̂ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +_ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + denoting upper and lower index position respectively. Notice that above the first +\begin_inset Quotes eld +\end_inset + +Holonomic connection +\family typewriter +\series default +\shape default + +\begin_inset Quotes erd +\end_inset + + is the name of the object while second +\begin_inset Quotes eld +\end_inset + +Holonomic Connection +\family typewriter +\series default +\shape default + +\begin_inset Quotes erd +\end_inset + + means that +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +recognizes it as the connection and will use +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +GAMMA +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to construct covariant derivatives for quantities having the coordinate indices. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +cder +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + about the covariant derivatives. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + You can define any number of other holonomic connections and use them in the covariant derivatives on the equal footing with the built-in object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +GAMMA +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +normalmarginpar +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The notation in which command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints information about a particular object is the same as in the new object declaration and is explained in details below. +\end_layout + +\begin_layout Subsection + +\family typewriter +Macro Objects +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Macro Objects +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "macro" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +There is also another class of built-in objects which are called +\emph on +macro objects +\emph default +. The main difference between the usual and macro objects is that macro quantities has no permanent storage to their components instead they are calculated dynamically only when its component is required in some expression. In addition they do not have names and are denoted only by the identifier only. Usually macro objects play auxiliary role. The complete list of macro objects can be found in appendix B. +\end_layout + +\begin_layout Standard + +\family typewriter +The example of macro objects are the Christoffel symbols of second and first kind +\begin_inset Formula $\{{}^\alpha_{\beta\gamma}\}$ +\end_inset + + and +\begin_inset Formula $[{}_{\alpha,\beta\gamma}]$ +\end_inset + + having identifiers +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +CHR +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +CHRF +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + respectively +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Show CHR; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +CHRxyz is Scalar Macro Object Symmetries: s(2,3) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- ? CHRF; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +CHRFuvw is Scalar Macro Object Symmetries: s(2,3) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +New Object Declaration +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +has very large number of built-in quantities but you are not obliged to use them in your calculations instead you can define new quantities. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +New Object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +New Object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ilst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ctype +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Symmetries +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + declares a new object. The words +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +New +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are optional (but not both) so the above command are equivalent to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ilst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ctype +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Symmetries +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +New +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ilst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ctype +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Symmetries +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is an identifier of a new object. The identifier can contain letters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +– +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +– +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + but neither digits nor any other symbols. The identifier must be unique and cannot coincide with the identifier of any other built-in or user-defined object. +\end_layout + +\begin_layout Standard + +\family typewriter +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ilist +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the list of indices having the form +\begin_inset CommandInset label +LatexCommand label +name "indices" + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ipos +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +itype +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ipos +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines the index position and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +itype +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + specifies its type. The coordinate holonomic and frame indices are denoted by single lower-case letters with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ipos +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +' +\family typewriter +\series default +\shape default + +\family roman + +\begin_inset space \space{} + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +upper frame index +\begin_inset Newline newline +\end_inset + + +\family typewriter +. +\family roman +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +lower frame index +\begin_inset Newline newline +\end_inset + + +\family typewriter +˖̂ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\family roman + +\begin_inset space \space{} + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +upper holonomic index +\begin_inset Newline newline +\end_inset + + +\family typewriter +_ +\family roman +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +lower holonomic index +\family roman +\series default +\shape default + The frame and holonomic indices in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +take values from 0 to +\begin_inset Formula $d-1$ +\end_inset + + where +\begin_inset Formula $d$ +\end_inset + + is the current space dimensionality. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Dimension +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family roman +Spinorial indices are denoted by upper case letters with trailing +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + for conjugated spinorial indices: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + etc. Summed spinorial index of rank +\begin_inset Formula $n$ +\end_inset + + is denoted by +\begin_inset Formula $n$ +\end_inset + + upper-case letters. For example +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +ABC +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + denotes summed spinorial index of the rank 3 (runs from 0 to 3) and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + denotes conjugated summed index of the rank 2 (values 0, 1, 2). The upper position for spinorial indices are denoted either by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +' +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +˖̂ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and lower one by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +_ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family roman +Finally the enumerating indices are denoted by a single lower-case letter followed either by digits or by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +dim +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. For example the index declared as +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +i2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + runs from 0 to 2 while specification +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a13 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + denotes index whose values runs from 0 to 13. The specification +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +idim +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + denotes enumerating index which takes the values from 0 to +\begin_inset Formula $d-1$ +\end_inset + +. Upper of lower position for enumerating indices are identical, thus in this case symbols +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +' . +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +^{ +\backslash +_} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are equivalent. +\end_layout + +\begin_layout Standard + +\family roman +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ctype +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines the type of new object component: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Scalar +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Density +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +dens +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Newline newline +\end_inset + + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +p +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +-form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Density +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +dens +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Newline newline +\end_inset + +Vector +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Density +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +dens +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + This part of the declaration can be omitted and then the object is assumed to be scalar-valued. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +dens +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines pseudo-scalar and density properties of the object with respect to coordinate and frame transformations: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +sgnL +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +*sgnD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +*L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +^{ +\backslash +parm} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +*D +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +^{ +\backslash +parm} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +m +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +D +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the coordinate transformation determinant +\begin_inset Formula ${\rm det}(\partial x^{\alpha'}/\partial x^\beta)$ +\end_inset + + and frame transformation determinant +\begin_inset Formula ${\rm det}(L^a{}_b)$ +\end_inset + + respectively. If +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +sgnL +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +sgnD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is specified then under appropriate transformation the object must be multiplied on the sign of the corresponding determinant (pseudo tensor). The specification +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +^{ +\backslash +parm} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +D +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +^{ +\backslash +parm} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +m +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + means that the quantity must be multiplied on the appropriate degree of the corresponding determinant (tensor density). The parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +p +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +m +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + may be given by expressions (must be enclosed in brackets) but value of these expressions must be always integer and positive in the case of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +p +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +The symmetry specification +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is a list +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where each element +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + describes symmetries for one group of indices and has the form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sym +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sym +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + determines type of the symmetry +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +s +\begin_inset space \space{} + +\end_inset + + +\family roman +symmetry +\begin_inset Newline newline +\end_inset + + +\family typewriter +a +\begin_inset space \space{} + +\end_inset + + +\family roman +antisymmetry +\begin_inset Newline newline +\end_inset + + +\family typewriter +c +\begin_inset space \space{} + +\end_inset + + +\family roman +cyclic symmetry +\begin_inset Newline newline +\end_inset + + +\family typewriter +h +\begin_inset space \space{} + +\end_inset + + +\family roman +Hermitian symmetry +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is either index number +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +i +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or list of index numbers +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +i +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or another symmetry specification of the form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Notice that +\begin_inset Formula $n$ +\end_inset + +th object index can be present only in one of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family roman +Let us consider an object having four indices. Then the following symmetry specifications are possible +\end_layout + +\begin_layout Standard + +\family roman + +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Standard + +\family roman + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +s(1,2,3,4) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family roman +total symmetry +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family roman +[1mm] +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a(1,2),s(3,4) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family roman +antisymmetry in first pair of indices and +\end_layout + +\end_inset + + + + +\begin_inset Text + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family roman +symmetry in second pair +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family roman +[1mm] +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +s((1,2),(3,4)) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family roman +symmetry in pair permutation +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family roman +[1mm] +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +s(a(1,2),a(3,4)) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family roman +antisymmetry in first and second pair of indices +\end_layout + +\end_inset + + + + +\begin_inset Text + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family roman +and symmetry in pair permutation +\end_layout + +\end_inset + + + + +\end_inset + + +\begin_inset Newline newline +\end_inset + +The last example is the well known symmetry of Riemann curvature tensor. The specification +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a(1,2),s(2,3) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is erroneous since second index present in both parts of the specification which is not allowed. +\end_layout + +\begin_layout Standard + +\family roman +Declaration for new equations is completely similar +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +New Equation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +New +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Equation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ilst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ctype +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Symmetries +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +slst +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family roman + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +knows four types of connections: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +New Connection +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "conn2" + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{list} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset Formula $\bullet$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +labelwidth +\end_layout + +\end_inset + +=4mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +leftmargin +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parindent +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family roman + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Frame Connection 1-form +\begin_inset Formula $\omega^a{}_b$ +\end_inset + + having first upper and second lower frame indices +\end_layout + +\begin_layout Standard + +\family roman + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Holonomic Connection 1-form +\begin_inset Formula $\Gamma^\alpha{}_\beta$ +\end_inset + + having first upper and second lower coordinate indices +\end_layout + +\begin_layout Standard + +\family roman + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Spinor Connection 1-form +\begin_inset Formula $\omega_{AB}$ +\end_inset + + with lower spinor index of rank 2 +\end_layout + +\begin_layout Standard + +\family roman + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Conjugated Spinor Connection +\begin_inset Formula $\omega_{\dot{A}\dot{B}}$ +\end_inset + + 1-form with lower conjugated spinor index of rank 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{list} +\end_layout + +\end_inset + + Each of these connections are used to construct covariant derivatives with respect to corresponding indices. In addition they are properly transformed under the coordinate change and frame rotation. There are complete set of built-in connections but you can declare a new one by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +New +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Connection +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +'a.b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is 1-form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +New +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Connection +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +m̂_n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is 1-form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +New +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Connection +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +.AB +\begin_inset space \space{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is 1-form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +New +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Connection +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +.AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +is 1-form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Notice that any new connection must belong to one of the listed above types and have indicated type and position of indices. This representation of connection is chosen in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +for the sake of definiteness. +\end_layout + +\begin_layout Standard + +\family typewriter +There is one special case when new object can be declared without explicit +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +New Object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + declaration. Let us consider the following example: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- www=d x; <- Show www; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +www is 1-form Value: known +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + If we assign the value to some identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +www +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + in our example) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "assig" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about assignment command. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and this identifier is not reserved yet by any other object then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +automatically declares a new object without indices labeled by the identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and having the type of the expression in the right-hand side of the assignment (1-form in our example). Notice that the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + must not include digits since digits represent indices and any new object with indices must be declared explicitly. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Forget +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + completely removes the user-defined object with the identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +Finally let us consider some examples: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- New RNEW'a.bcd is scalar density sgnD with a(3,4); <- Show RNEW; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +RNEW'a.bxy is Scalar Density sgnD Value: unknown Symmetries: a(3,4) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Null Metric; <- Connection omnew.AA; <- Show omnew; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +omnew.AB is 1-form Spinor Connection Value: unknown +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Here the first declaration defines a new scalar valued pseudo tensor +\begin_inset Formula $\mbox{\comm{RNEW}}^a{}_{b\gamma\delta}$ +\end_inset + + which is antisymmetric in the last pair of indices. Second declaration introduce new spinor connection +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +omnew +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Notice that new connection is automatically declared 1-form and the type of connection is derived by the type of new object indices (lower spinorial index of rank 2 in our example). +\end_layout + +\begin_layout Section + +\family typewriter +Assignment Command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Assignment (command) +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "assig" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The assignment command sets the value to the particular components of the object. In general it has the form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +Name +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +comp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + = +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or for equations +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +Name +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +comp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + = +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +lhs +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +rhs +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +Name +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the optional object name. If the object has no indices then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +comp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the object identifier. If the object has indices then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +comm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + consist of identifier with additional digits denoting indices. For example the following command assigns standard spherical flat value to the frame +\begin_inset Formula $\theta^a$ +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Frame T0 = d t, T1 = d r, T2 = r*d theta, T3 = r*SIN(theta)*d phi; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +and the command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +RIM0123 = 100; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +assigns the value to the +\begin_inset Formula $R^0{}_{123}$ +\end_inset + + component of the Riemann tensor. Notice that in this notation each digit is considered as one index, thus it does not work if the value of some index is greater than 9 (e.g. if dimensionality is 10 or greater). In this case another notation can be used in which indices are added to the object identifier as a list of digits enclosed in brackets +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +Name +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset space ~ + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + In particular the command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +RIM(0,1,2,3) = 100; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +is equivalent to the example above. +\end_layout + +\begin_layout Standard + +\family typewriter +The assignment set value only to the certain components of an object leaving other components unchanged. But if before assignment the object was in indefinite state (no value is known) then assignment turns it to the definite state and all other components of the object are assumed to be zero. +\end_layout + +\begin_layout Standard + +\family typewriter +The digits standing for object indices in the left-hand side of an assignment can be replaced by identifiers +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Assignment (command)!tensorial +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +Name +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +ID +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset space ~ + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Such assignment is called +\emph on +tensorial +\emph default + one. For example the following tensorial assignment set the value to the curvature 2-form +\begin_inset Formula $\Omega^a{}_b$ +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +OMEGA(a,b) = d omega(a,b) + omega(a,m) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +omega(m,b); + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +This command is equivalent to +\begin_inset Formula $d\times d$ +\end_inset + + of assignments where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + take values from 0 to +\begin_inset Formula $d-1$ +\end_inset + + ( +\begin_inset Formula $d$ +\end_inset + + is the space dimensionality). +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Dimension +\end_layout + +\end_inset + + Notice that identifiers in the left-hand side of tensorial assignment must not coincide with any predefined or declared by the user constant or coordinate. It is possible to mix digits and identifiers: +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +FT(0,a) = 0; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +FT +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is identifier of the built-in object EM Tensor +\family typewriter +\series default +\shape default + which is the electromagnetic strength tensor +\begin_inset Formula $F_{ab}$ +\end_inset + + and this command sets the electric part of the tensor to zero. +\end_layout + +\begin_layout Standard + +\family typewriter +The assignment command takes into account symmetries of the objects. For example EM Tensor +\family typewriter +\series default +\shape default + is antisymmetric and in order to assign value say to the components +\begin_inset Formula $F_{01}=-F_{10}$ +\end_inset + + it suffices to do this just for one of them +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- EM Tensor FT01=111, FT(3,2)=222; <- Write FT; EM tensor: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = 111 t x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = -222 y z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + We can see that +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +automatically transforms indices to the +\emph on +canonical +\emph default + order. This rule works in the case or tensorial assignment as well +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- Function ff; <- EM Tensor FT(a,b)=ff(a,b); <- Write FT; EM tensor: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ff(0,1) t x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ff(0,2) t y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ff(0,3) t z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ff(1,2) x y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ff(1,3) x z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ff(2,3) y z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- FT(2,1); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +- ff(1,2) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + In this case both parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + runs from 0 to 3 but +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +assigns the value only to the components having indices in the canonical order +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $<$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +follows this rule also if in the left-hand side of tensorial assignment digits are mixed with parameters which may sometimes produce unexpected result: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- Function ee; <- FT(0,a)=ee(a); <- Write FT; EM tensor: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ee(1) t x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ee(2) t y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FT = ee(3) t z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Erase FT; <- FT(3,a)=ee(a); <- Write FT; EM tensor: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +0 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Observe the difference between these two assignments (the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Erase FT; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + destroys the previously assigned value). In fact second assignment assigns no values since +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +3 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are not in the canonical order +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +3 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $\geq$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + running from 0 to 3. Notice the difference from the case when all indices in the left-hand side are given by the explicit numerical values. In this case +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +automatically transforms the indices to their canonical order and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +FT(3,2)=222; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is equivalent to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +FT(2,3)=-222; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +Finally there is one more form of the tensorial assignment which can be applied to the summed spinorial indices. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Assignment (command)!summed spinor indices +\end_layout + +\end_inset + + Let us consider the spinorial analogue of electromagnetic strength tensor +\begin_inset Formula $\Phi_{AB}$ +\end_inset + +. This spinor is irreducible (i.e. symmetric in +\begin_inset Formula $\scriptstyle AB$ +\end_inset + +). The corresponding +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +built-in object Undotted EM Spinor +\family typewriter +\series default +\shape default + (identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +FIU +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) has one summed spinorial index of rank 2. Let us consider two different assignment commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates u, v, z, z +\begin_inset space ~ + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +z & z +\begin_inset space ~ + +\end_inset + + - conjugated pair. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Null Metric; <- Function ee; <- FIU(a)=ee(a); <- Write FIU; Undotted EM spinor: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FIU = ee(0) 0 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FIU = ee(1) 1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FIU = ee(2) 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Erase FIU; <- FIU(a+b)=ee(a,b); <- Write FIU; Undotted EM spinor: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FIU = ee(0,0) 0 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FIU = ee(0,1) 1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +FIU = ee(1,1) 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + In the first case +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is treated as a summed index and runs from 0 to 2 but in the second case +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are considered as usual single SL(2,C) spinorial indices each having values 0 and 1. +\end_layout + +\begin_layout Standard + +\family typewriter +The notation for the object components in the left-hand side of assignment do not distinguishes upper and lower indices. Actually the indices are always assumed to be in the default position. You can always check the default index types and positions using the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + For example the Riemann Tensor +\family typewriter +\series default +\shape default + has first upper and three lower frame indices and the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +RIM0123=100; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +RIM(0,1,2,3)=100; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + both assign value to the +\begin_inset Formula $R^0{}_{123}$ +\end_inset + + component of the tensor where indices are represented with respect to the current frame. +\end_layout + +\begin_layout Section + +\family typewriter +Geometry +\end_layout + +\begin_layout Standard + +\family typewriter +The number of built-in objects in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is rather large. They all described in chapter 3 and appendices B and C. In this section we consider only the most important ones. +\end_layout + +\begin_layout Subsection + +\family typewriter +Metric, Frame and Line-Element +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Metric +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Frame +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "metric" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The line-element in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is defined by the following equation +\begin_inset Formula \begin{equation} +ds^2 = g_{ab}\,\theta^a\!\otimes\theta^b +\end{equation} +\end_inset + +where +\begin_inset Formula $\theta^a=h^a_\mu dx^\mu$ +\end_inset + + is the frame 1-form and +\begin_inset Formula $g_{ab}$ +\end_inset + + is the frame metric. The corresponding built-in objects are +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Frame +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + (identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +T +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Metric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + (identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +G +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +). There are also the +\begin_inset Quotes eld +\end_inset + +inverse +\begin_inset Quotes erd +\end_inset + + counterparts +\begin_inset Formula $\partial_a=h_a^\mu\partial_\mu$ +\end_inset + + (Vector Frame +\family typewriter +\series default +\shape default +, identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +D +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) and +\begin_inset Formula $g^{ab}$ +\end_inset + + (Inverse Metric +\family typewriter +\series default +\shape default +, identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +GI +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +). To determine the metric properties of the space you can assign some values to both the metric and the frame. There are two well known special cases. First is the usual coordinate formalism in which frame is holonomic +\begin_inset Formula $\theta^a=dx^\alpha$ +\end_inset + +. In this case there is no difference between frame and coordinate indices. Another representation is known as the tetrad (in dimension 4) formalism. In this case frame metric equals to some constant matrix +\begin_inset Formula $g_{ab}=\eta_{ab}$ +\end_inset + + and significant information about line-element +\begin_inset Quotes eld +\end_inset + +is encoded +\begin_inset Quotes erd +\end_inset + + in the frame. +\end_layout + +\begin_layout Standard + +\family typewriter +In general both metric and frame can be nontrivial but not necessarily. If no any value is given by user to the frame when +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +automatically assumes that frame is +\emph on +holonomic +\emph default + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Frame!default value +\end_layout + +\end_inset + + +\begin_inset Formula \begin{equation} +\theta^a=dx^\alpha +\end{equation} +\end_inset + +Thus if we assign the value to metric only we automatically get standard coordinate formalism. On the contrary if no value is assigned to the metric then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +automatically assumes +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Signature +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "defaultmetric" + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Metric!default value +\end_layout + +\end_inset + + +\begin_inset Formula \begin{equation} +g_{ab} = {\rm diag}(+1,-1,\dots) +\end{equation} +\end_inset + +where +\begin_inset Formula $+1$ +\end_inset + + and +\begin_inset Formula $-1$ +\end_inset + + on the diagonal of the matrix correspond to the current signature specification. +\end_layout + +\begin_layout Standard + +\family typewriter +Notice that current signature is printed among other information by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Status; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and current line-element is printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +ds2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +ds2; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or equivalently +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Line-Element +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Line-Element; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Finally if neither frame nor metric are specified by user then both these quantities acquire default value and we automatically obtain flat space of the default signature: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Dimension 4 with Signature(-,+,+,+); <- Coordinates t, x, y, z; <- ds2; Assuming Default Metric. Metric calculated By default. 0.05 sec Assuming Default Holonomic Frame. Frame calculated By default. 0.05 sec +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 2 2 2 2 ds = - d t + d x + d y + d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Spinors +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset CommandInset label +LatexCommand label +name "spinors" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Spinorial representations exist in spaces of various dimensions and signatures but in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +spinors are restricted to the 4-dimensional spaces of Lorentzian signature +\begin_inset Formula ${\scriptstyle(-,+,+,+)}$ +\end_inset + + or +\begin_inset Formula ${\scriptstyle(+,-,-,-)}$ +\end_inset + + only. Another restriction is that in the spinorial formalism the metric must be the +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Metric!Standard Null +\end_layout + +\end_inset + + +\emph on +standard null metric +\emph default +: +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Standard null metric +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Spinors +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Spinors!Standard null metric +\end_layout + +\end_inset + + +\begin_inset Formula \begin{equation} +g_{ab}=g^{ab}=\pm\left(\begin{array}{rrrr} +0 & -1 & 0 & 0 \\ +-1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 \\ +0 & 0 & 1 & 0 +\end{array}\right) +\end{equation} +\end_inset + +where upper sign correspond to the signature +\begin_inset Formula ${\scriptstyle(-,+,+,+)}$ +\end_inset + + and lower sign to the signature +\begin_inset Formula ${\scriptstyle(+,-,-,-)}$ +\end_inset + +. There is special command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Null Metric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Null Metric; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + which assigns this standard value to the metric. +\end_layout + +\begin_layout Standard + +\family typewriter +Thus spinorial frame (tetrad) in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +must be null +\begin_inset Formula \begin{equation} +ds^2 = \pm(-\theta^0\!\otimes\theta^1 +-\theta^1\!\otimes\theta^0 ++\theta^2\!\otimes\theta^3 ++\theta^3\!\otimes\theta^2) +\end{equation} +\end_inset + +and conjugation rules for this tetrad must be +\begin_inset Formula \begin{equation} +\overline{\theta^0}=\theta^0,\quad +\overline{\theta^1}=\theta^1,\quad +\overline{\theta^2}=\theta^3,\quad +\overline{\theta^3}=\theta^2 +\end{equation} +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +For the sake of efficiency the sigma-matrices +\begin_inset Formula $\sigma^a\!{}_{A\dot{B}}$ +\end_inset + + for such a tetrad are chosen in the simplest form. The only nonzero components of the matrices are +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Sigma matrices +\end_layout + +\end_inset + + +\begin_inset Formula \begin{eqnarray} +&&\sigma_0{}^{1\dot{1}}= +\sigma_1{}^{0\dot{0}}= +\sigma_2{}^{1\dot{0}}= +\sigma_3{}^{0\dot{1}}=1 \\[1mm] && +\sigma^0{}_{1\dot{1}}= +\sigma^1{}_{0\dot{0}}= +\sigma^2{}_{1\dot{0}}= +\sigma^3{}_{0\dot{1}}=\mp1 +\end{eqnarray} +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Connection, Torsion and Nonmetricity +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset CommandInset label +LatexCommand label +name "conn" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +As was explained above +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +recognizes four types of connections: holonomic +\begin_inset Formula $\Gamma^\alpha{}_\beta$ +\end_inset + +, frame +\begin_inset Formula $\omega^a{}_b$ +\end_inset + +, spinorial +\begin_inset Formula $\omega_{AB}$ +\end_inset + + and conjugated spinorial +\begin_inset Formula $\omega_{\dot{A}\dot{B}}$ +\end_inset + +. Accordingly there are four built-in objects: Holonomic Connection +\family typewriter +\series default +\shape default + (id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +GAMMA +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +), Frame Connection +\family typewriter +\series default +\shape default + (id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +omega +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +), Undotted Connection +\family typewriter +\series default +\shape default + (id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +omegau +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +), Dotted Connection +\family typewriter +\series default +\shape default + (id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +omegad +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +). Connections are used in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +in covariant derivatives. In addition they are properly transformed under frame and coordinate transformations. +\end_layout + +\begin_layout Standard + +\family typewriter +By default the connection in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are assumed to be Riemannian. In particular in this case holonomic connection is nothing but Christoffel symbols +\begin_inset Formula $\Gamma^\alpha{}_\beta= +\{{}^\alpha_{\beta\pi}\}dx^\pi$ +\end_inset + +. If it is necessary to work with torsion and/or nonmetricity +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +TORSION +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +NONMETR +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + then the switches +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +TORSION +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and/or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +NONMETR +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + must be turned on. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +conn2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + about the built-in connections. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + In this case the Riemannian analogues or the aforementioned four connections are available as well. +\end_layout + +\begin_layout Section + +\family typewriter +Expressions +\end_layout + +\begin_layout Standard + +\family typewriter +Expressions in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +can be algebraic (scalar), vector or p-form valued. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +knows all the usual mathematical operations on algebraic expressions, exterior forms and vectors. +\end_layout + +\begin_layout Subsection + +\family typewriter +Operations and Operators +\end_layout + +\begin_layout Standard + +\family typewriter +The operations known to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are presented in the form of the table. Operations are subdivided into six groups separated by horizontal lines. Operations in each group have equal level of precedence and the precedence level decreases from the top to the bottom of the table. As in usual mathematical notation we can use brackets +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +verb"( )" +\end_layout + +\end_inset + + to change operation precedence. +\end_layout + +\begin_layout Standard + +\family typewriter +Other constructions which can be used in expression are described below. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Float table +wide false +sideways false +status open + + +\begin_layout Standard +\align center + +\family typewriter + +\begin_inset Tabular + + + + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\family default + +\series bold +Operation +\family typewriter +\series default +\shape default + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\family default + +\series bold +Description +\family typewriter +\series default +\shape default + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\family default + +\series bold +Grouping +\family typewriter +\series default +\shape default + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +[ +\begin_inset Formula $v_1$ +\end_inset + +, +\begin_inset Formula $v_2$ +\end_inset + +] +\family typewriter +\series default +\shape default + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Vector bracket +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +@ +\family typewriter +\series default +\shape default + +\begin_inset Formula $x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Holonomic vector +\begin_inset Formula $\partial_x$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +d +\family typewriter +\series default +\shape default + +\begin_inset Formula $a$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Exterior differential +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +d +\family typewriter +\series default +\shape default + +\begin_inset Formula $\omega$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +d +\family typewriter +\series default +\shape default + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset Formula $a$ +\end_inset + + +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + (d( +\family typewriter +\series default +\shape default + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset Formula $a$ +\end_inset + +)) +\family typewriter +\series default +\shape default + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dd +\end_layout + +\end_inset + + +\family typewriter +\series default +\shape default + +\begin_inset Formula $a$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Dualization +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dd +\end_layout + +\end_inset + + +\family typewriter +\series default +\shape default + +\begin_inset Formula $\omega$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\family typewriter +\series default +\shape default + +\begin_inset Formula $e$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Complex conjugation +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $a_1$ +\end_inset + +** +\family typewriter +\series default +\shape default + +\begin_inset Formula $a_2$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Exponention +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $a_1$ +\end_inset + +˖̂ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $a_2$ +\end_inset + + +\family typewriter +\series default +\shape default + +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $e$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +/ +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $a$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Division +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $e$ +\end_inset + +/ +\family typewriter +\series default +\shape default + +\begin_inset Formula $a_1$ +\end_inset + +/ +\family typewriter +\series default +\shape default + +\begin_inset Formula $a_2$ +\end_inset + + +\begin_inset Formula $\Leftrightarrow$ +\end_inset + + ( +\family typewriter +\series default +\shape default + +\begin_inset Formula $e$ +\end_inset + +/ +\family typewriter +\series default +\shape default + +\begin_inset Formula $a_1$ +\end_inset + +)/ +\family typewriter +\series default +\shape default + +\begin_inset Formula $a_2$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $a$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +* +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Multiplication +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $v$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +| +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $a$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Vector acting on scalar +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $v$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +ii +\end_layout + +\end_inset + + +\begin_inset Formula $\omega_1$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset Formula $\omega_2$ +\end_inset + +* +\family typewriter +\series default +\shape default + +\begin_inset Formula $a$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $v$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +ip +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $\omega$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Interior product +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $\Updownarrow$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $v_1$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +. +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $v_2$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Scalar product +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $v$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +ii +\end_layout + +\end_inset + +( +\family typewriter +\series default +\shape default + +\begin_inset Formula $\omega_1$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + +( +\family typewriter +\series default +\shape default + +\begin_inset Formula $\omega_2$ +\end_inset + +* +\family typewriter +\series default +\shape default + +\begin_inset Formula $a$ +\end_inset + +)) +\family typewriter +\series default +\shape default + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $v$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +. +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $o$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $o_1$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +. +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $o_2$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $\omega_1$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $\omega_2$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Exterior product +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter ++ +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Prefix plus +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +- +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Prefix minus +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $e_1$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + ++ +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $e_2$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Addition +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $e_1$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +- +\family typewriter +\series default +\shape default + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $e_2$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Subtraction +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset CommandInset label +LatexCommand label +name "operators" + +\end_inset + + +\begin_inset Caption Standard + +\begin_layout Plain Layout +Operation and operators. Here: +\begin_inset Formula $e$ +\end_inset + + is any expression, +\begin_inset Formula $a$ +\end_inset + + is any scalar valued (algebraic) expressions, +\begin_inset Formula $v$ +\end_inset + + is any vector valued expression, +\begin_inset Formula $x$ +\end_inset + + is a coordinate, +\begin_inset Formula $o$ +\end_inset + + is any 1-form valued expression, +\begin_inset Formula $\omega$ +\end_inset + + is any form valued expression. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Variables and Functions +\end_layout + +\begin_layout Standard + +\family typewriter +Operator listed in the table 2.2 act on the following types of the operands: +\end_layout + +\begin_layout Itemize + +\family typewriter + +\begin_inset Argument item:1 +status collapsed + + +\begin_layout Standard + +\family typewriter +(i) +\end_layout + +\end_inset + +integer numbers (e.g. 0 +\family typewriter +\series default +\shape default +, 123 +\family typewriter +\series default +\shape default +), +\end_layout + +\begin_layout Itemize + +\family typewriter + +\begin_inset Argument item:1 +status collapsed + + +\begin_layout Standard + +\family typewriter +(ii) +\end_layout + +\end_inset + +symbols or identifiers (e.g. I +\family typewriter +\series default +\shape default +, phi +\family typewriter +\series default +\shape default +, RIM0103 +\family typewriter +\series default +\shape default +), +\end_layout + +\begin_layout Itemize + +\family typewriter + +\begin_inset Argument item:1 +status collapsed + + +\begin_layout Standard + +\family typewriter +(iii) +\end_layout + +\end_inset + +functional expressions (e.g. SIN(x) +\family typewriter +\series default +\shape default +, G(0,1) +\family typewriter +\series default +\shape default + etc). +\end_layout + +\begin_layout Standard + +\family typewriter +Valid identifier must belong to one of the following types: +\end_layout + +\begin_layout Itemize + +\family typewriter +Coordinate. +\end_layout + +\begin_layout Itemize + +\family typewriter +User-defined or built-in constant. +\end_layout + +\begin_layout Itemize + +\family typewriter +Function declared with the implicit dependence list. +\end_layout + +\begin_layout Itemize + +\family typewriter +Component of an object. +\end_layout + +\begin_layout Standard + +\family typewriter +Any valid functional expression must belong to one of the following types: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +itemsep +\end_layout + +\end_inset + +=0.5mm +\end_layout + +\begin_layout Itemize + +\family typewriter +User-defined function. +\end_layout + +\begin_layout Itemize + +\family typewriter +Function defined in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +(operator). +\end_layout + +\begin_layout Itemize + +\family typewriter +Component of built-in or user-defined object in functional notation. +\end_layout + +\begin_layout Itemize + +\family typewriter +Some special functional expressions listed below. +\end_layout + +\begin_layout Subsection + +\family typewriter +Derivatives +\end_layout + +\begin_layout Standard + +\family typewriter +The derivatives in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are written as +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +DF( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the differentiated expression, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the differentiation variable and integer number +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the repetition of the differentiation. For example +\begin_inset Formula \[ +\mbox{\tt DF(f(x,y),x,2,y)}=\frac{\partial^3f(x,y)}{\partial^2x\partial y} +\] +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +There are also another type of derivatives +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +DFP( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See section +\begin_inset CommandInset ref +LatexCommand ref +reference "genfun" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about the generic functions. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + They are valid only after Generic Function +\family typewriter +\series default +\shape default + declaration if the package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +dfpart +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is installed on your system. +\end_layout + +\begin_layout Subsection + +\family typewriter +Complex Conjugation +\end_layout + +\begin_layout Standard + +\family typewriter +Symbol +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + in the sum of terms is an abbreviation: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $=$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + +\begin_inset Newline newline +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + - +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $=$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + - +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Re +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Im +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + gives real and imaginary parts of an expression: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Re( +\begin_inset Formula $e$ +\end_inset + +) +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $=$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +( +\begin_inset Formula $e$ +\end_inset + ++ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + +)/2 +\begin_inset Newline newline +\end_inset + + Im( +\begin_inset Formula $e$ +\end_inset + +) +\begin_inset space \space{} + +\end_inset + + +\begin_inset Formula $=$ +\end_inset + + +\begin_inset space \space{} + +\end_inset + +I*(- +\begin_inset Formula $e$ +\end_inset + ++ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cc +\end_layout + +\end_inset + + +\begin_inset Formula $e$ +\end_inset + +)/2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Sums and Products +\end_layout + +\begin_layout Standard + +\family typewriter +The following expressions represent sum and product +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Sum( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset Newline newline +\end_inset + +Prod( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the summed expression and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines summation variables. The range of summation can be +\begin_inset CommandInset label +LatexCommand label +name "iter" + +\end_inset + + specified by two methods. First +\begin_inset Quotes eld +\end_inset + +long +\begin_inset Quotes erd +\end_inset + + notation is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + = +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +low +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +.. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +up +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and the identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + runs from +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +low +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + up to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +up +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Both +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +low +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +up +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be given by arbitrary expressions but value of these expressions must be integer. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +low +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be omitted +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + = +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +up +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and in this case +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + runs from 0 to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +up +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + should not coincide with any built-in or user-defined variable. +\end_layout + +\begin_layout Standard + +\family typewriter +In +\begin_inset Quotes eld +\end_inset + +short +\begin_inset Quotes erd +\end_inset + + notation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is just identifier +\begin_inset CommandInset label +LatexCommand label +name "siter" + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and its range is determined using the following rules +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{list} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset Formula $\bullet$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +labelwidth +\end_layout + +\end_inset + +=4mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +leftmargin +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parindent +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Mixed letter-digit +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + runs from 0 to +\begin_inset Formula $d-1$ +\end_inset + + where +\begin_inset Formula $d$ +\end_inset + + is the space dimensionality. +\end_layout + +\begin_layout Verbatim + Aid j2s +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + consisting of lower-case letters runs from +\begin_inset Formula $0$ +\end_inset + + to +\begin_inset Formula $d-1$ +\end_inset + + +\end_layout + +\begin_layout Verbatim + j a abc kkk +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + consisting of upper-case letters runs from +\begin_inset Formula $0$ +\end_inset + + to the number of letters in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, e.g. the following identifiers run from 0 to 1 and from 0 to 3 respectively +\end_layout + +\begin_layout Verbatim + B ABC +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Letters with one trailing digit run from 0 to the value of this digit. Both +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + below runs from 0 to 3: +\end_layout + +\begin_layout Verbatim + j3 A3 +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Letters with two digits run from the value of the first digit to the value of the second digit. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + below run from 2 to 3: +\end_layout + +\begin_layout Verbatim + j23 A23 +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Letters with 3 or more digits are incorrect +\end_layout + +\begin_layout Verbatim + j123 +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{list} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Two or more summation parameters are separated either by commas or by one of the relational operators +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +< > <= => + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +This means that only the terms satisfying these relations will be included in the sum. For example +\begin_inset Formula \[ +\mbox{\tt Sum(i24<=ABC,k=1..d-1,f(i24,ABC,k))} = +\sum_{i=2}^{4} \sum_{\scriptstyle a=0\atop\scriptstyle i\leq a}^{3} \sum^{d-1}_{k=1} f(i,a,k) +\] +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +enlargethispage{5mm} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +'s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sum +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Prod +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +Use +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SUM +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +PROD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +sum +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +prod +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + depending on +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +internal case as explained on page +\begin_inset CommandInset ref +LatexCommand pageref +reference "case" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + should not be confused with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + +'s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SUM +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +PROD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + which are also available in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +'s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sum +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + apply to any scalar, vector or form-valued expressions and always expanded by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +into the appropriate explicit sum of terms. On the contrary +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SUM +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defined in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +can be applied to the algebraic expressions only. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +leaves such expression unchanged and passes it to the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +algebraic evaluator. Unlike +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sum +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + the summation limits in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SUM +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be given by algebraic expressions. If value of these expressions is integer then result of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SUM +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + will be the same as for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sum +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + but if summation limits are symbolic sometimes +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is capable to find a closed expression for such a sum but not always. See the following example +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- Function f; <- Constants n, m; <- Sum(k=1..3,f(k)); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +f(3) + f(2) + f(1) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- SUM(f(n),n,1,3); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +f(3) + f(2) + f(1) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- SUM(n,n,1,m); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +m*(m + 1) ———– 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- SUM(f(n),n,1,m); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +SUM(f(n),n,1,m) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Newpage newpage +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Einstein Summation Rule +\end_layout + +\begin_layout Standard + +\family typewriter +According to the Einstein summation rule if +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +encounters some unknown repeated identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + then summation over this +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +id +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is performed. The range of the summation variable is determined according to the +\begin_inset Quotes eld +\end_inset + +short +\begin_inset Quotes erd +\end_inset + + notation explained in the previous section. +\end_layout + +\begin_layout Subsection + +\family typewriter +Object Components and Index Manipulation +\end_layout + +\begin_layout Standard + +\family typewriter +The components of built-in or user-defined object can be denoted in expressions by two methods which are similar to the notation used in the left-hand side of the assignment command. The first method uses the object identifier with additional digits denoting the indices T0 +\family typewriter +\series default +\shape default +, RIM0213 +\family typewriter +\series default +\shape default +. The second method uses the functional notation T(0) +\family typewriter +\series default +\shape default +, RIM(0,2,1,3) +\family typewriter +\series default +\shape default +, OMEGA(j,k) +\family typewriter +\series default +\shape default +. +\end_layout + +\begin_layout Standard + +\family typewriter +In functional notation the default index type and position +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Index manipulations +\end_layout + +\end_inset + + can be changed using the markers: ' +\family typewriter +\series default +\shape default + upper frame, . +\family typewriter +\series default +\shape default + lower frame, ˖̂ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + upper holonomic, _ +\family typewriter +\series default +\shape default + lower holonomic. For example expression RIM(a,b,m,n) +\family typewriter +\series default +\shape default + gives components of Riemann tensor with the default indices +\begin_inset Formula $R^a{}_{bmn}$ +\end_inset + + (first upper frame and three lower frame indices) while expression RIM('a,'b,_m,_n) +\family typewriter +\series default +\shape default + gives +\begin_inset Formula $R^{ab}{}_{\mu\nu}$ +\end_inset + + with two upper frame and two lower coordinate indices. For enumerating indices position markers are ignored and only ' +\family typewriter +\series default +\shape default + and . +\family typewriter +\series default +\shape default + works for spinorial indices. +\end_layout + +\begin_layout Standard + +\family typewriter +In the spinorial formalism +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +spinors +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + about spinorial formalism. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + each frame index can be replaced by a pair if spinorial indices according to the formulas: +\begin_inset Formula \[ +A^a\sigma_a{}^{B\dot{D}}=A^{B\dot{D}},\qquad +B_a\sigma^a\!{}_{B\dot{D}}=B_{B\dot{D}} +\] +\end_inset + +Accordingly any frame index can be replaced by a pair of spinorial indices. +\begin_inset CommandInset label +LatexCommand label +name "sumspin" + +\end_inset + + Similarly one summed spinorial index or rank +\begin_inset Formula $n$ +\end_inset + + can be replaced by +\begin_inset Formula $n$ +\end_inset + + single spinor indices. There is only one restriction. If an object has several frame and/or summed spinorial indices then +\emph on +all +\emph default + must be represented in such expanded form. In the following example the null frame +\begin_inset Formula $\theta^a$ +\end_inset + + is printed in the usual and spinorial +\begin_inset Formula $\theta^{B\dot C}$ +\end_inset + + representations. The relationship +\begin_inset Formula $\theta^a\sigma_a{}^{B\dot C}-\theta^{B\dot C}=0$ +\end_inset + + is verifies as well +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates u, v, z, z +\begin_inset space ~ + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +z & z +\begin_inset space ~ + +\end_inset + + - conjugated pair. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Null Metric; <- Frame T(a)=d x(a); <- ds2; +\begin_inset Newpage newpage +\end_inset + +2 ds = (-2) d u d v + 2 d z d z +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- T(a); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +a=0 : d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +a=1 : d v +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +a=2 : d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +a=3 : d z +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- T(B,C); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +B=0 C=0 : d v +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +B=0 C=1 : d z +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +B=1 C=0 : d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +B=1 C=1 : d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- T(a)*sigmai(a,B,C)-T(B,C); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +0 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Parts of Equations and Solutions +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Equations!in expressions +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The functional expressions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +LHS( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +eqcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset Newline newline +\end_inset + +RHS( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +eqcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + give access to the left-hand and right-hand side of an equation respectively. Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +eqcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the component of the equation as explained in the previous section. +\end_layout + +\begin_layout Standard + +\family typewriter +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +LHS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +RHS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + also provide access to the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +'th +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "solutions" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about solutions. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + solution if +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +eqcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sol( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Subsection + +\family typewriter +Lie Derivatives +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Lie derivatives +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The Lie derivative is given by the expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Lie( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +v +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +objcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +objcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the component of an object in functional notation. For example the following expression is the Lie derivative of the metric +\begin_inset Formula $\pounds_vg_{ab}$ +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Lie(vec,G(a,b)); + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The index manipulations in the Lie derivatives are permitted. In particular the expression +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Lie(vec,G(m,b)); + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +is the Lie derivative of the frame +\begin_inset Formula $\pounds_vg^\mu{}_{b} +\equiv \pounds_vh^\mu_a$ +\end_inset + + and must vanish. +\end_layout + +\begin_layout Subsection + +\family typewriter +Covariant Derivatives and Differentials +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Covariant derivatives +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Covariant differentials +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "cder" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The covariant differential +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Dc( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +objcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\shape up + +\shape default +, +\shape default + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +conn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and covariant derivative +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Dfc( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +v +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +objcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\shape up + +\shape default +, +\shape default + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +conn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +objcomp +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is an object component in functional notation and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +v +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is a vector-valued expression. The optional parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +conn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are the identifiers of connections. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "conn" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about the built-in connections. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + If +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +conn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is omitted then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +uses default connection for each type of indices: frame, coordinate, spinor and conjugated spinor. If +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +conn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is indicated then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +uses this connection instead of default one for appropriate type of indices. For example expression +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Dc(OMEGA(a,b)) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +is the covariant differential of the curvature 2-form +\begin_inset Formula $D\Omega^a{}_b$ +\end_inset + +. This expression should vanish in Riemann space and should be proportional to the torsion in Riemann-Cartan space. Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +will use default object Frame connection +\family typewriter +\series default +\shape default + (id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +omega +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +). The expression +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Dc(OMEGA(a,b),romega) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +is similar but it uses another built-in connection Riemann frame connection +\family typewriter +\series default +\shape default + (id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +romega +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) which are different if torsion or nonmetricity are nonzero. The index manipulations are allowed in the covariant derivatives. For example the expression +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Dfc(v,RIC(m̂,_n)) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +gives the covariant derivative of the curvature of the Ricci tensor with first coordinate upper and second coordinate lower indices +\begin_inset Formula $\nabla_vR^\mu{}_\nu$ +\end_inset + +. +\end_layout + +\begin_layout Subsection + +\family typewriter +Symmetrization +\end_layout + +\begin_layout Standard + +\family typewriter +The functional expressions works iff the switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +EXPANDSYM +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +EXPANDSYM +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is on +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Asy( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +i +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset Newline newline +\end_inset + +Sy( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +i +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset Newline newline +\end_inset + +Cy( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +i +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + They produce antisymmetrization, symmetrization and cyclic symmetrization of the expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with respect to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +i +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + without corresponding +\begin_inset Formula $1/n$ +\end_inset + + or +\begin_inset Formula $1/n!$ +\end_inset + +. +\end_layout + +\begin_layout Subsection + +\family typewriter +Substitutions +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Substitutions +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "subs" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +SUB( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sub +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is similar to the analogous expression in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +with two generalizations: (i) it applies not only to algebraic but to form and vector valued expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + as well, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "solutions" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about solutions. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + (ii) as in Let +\family typewriter +\series default +\shape default + command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sub +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be either the relation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +l +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + += +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +r +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\family typewriter +\series default +\shape default + or solution Sub( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\family typewriter +\series default +\shape default +. +\end_layout + +\begin_layout Subsection + +\family typewriter +Conditional Expressions +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Conditional expressions +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Boolean expressions +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The conditional expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +If( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +cond +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + chooses +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +e2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + depending on the value of the boolean expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +cond +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +Boolean expression appears in (i) the conditional expression +\begin_inset CommandInset label +LatexCommand label +name "bool" + +\end_inset + + If +\family typewriter +\series default +\shape default +, (ii) in For all Such That +\family typewriter +\series default +\shape default + substitutions. Any nonzero expression is considered as +\family default + +\series bold +true +\family typewriter +\series default +\shape default + and vanishing expression as +\family default + +\series bold +false +\family typewriter +\series default +\shape default +. Boolean expressions may contain the following usual relations and logical operations: < > <= >= = |= not and or +\family typewriter +\series default +\shape default +. They also may contain the following predicates +\begin_inset VSpace 2mm* +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +OBJECT( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +obj +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Is +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +obj +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + an object identifier or not +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +ON( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Test position of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +OFF( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\end_layout + +\end_inset + + +\begin_inset Text + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +ZERO( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Is the value of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + zero or not +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +HASVALUE( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Whether the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + has any value or not +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +NULLM( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Is the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + the standard null metric +\end_layout + +\end_inset + + + + +\end_inset + + +\begin_inset VSpace 2mm* +\end_inset + + +\begin_inset Newline newline +\end_inset + +Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is an object identifier. +\end_layout + +\begin_layout Standard + +\family typewriter +The expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +ERROR( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +message +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + causes an error with the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +message +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. It can be used to test any required conditions during the batch file execution. +\end_layout + +\begin_layout Subsection + +\family typewriter +Functions in Expressions +\end_layout + +\begin_layout Standard + +\family typewriter +Any function which appear in expression must be either declared by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + declaration or be defined in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +(in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +functions are called operators). In general arguments of functions in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +must be algebraic expression with one exception. If one (and only one) argument of some function +\begin_inset Formula $f$ +\end_inset + + is form-valued +\begin_inset Formula $\omega=a d x + b d y$ +\end_inset + + then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +applies +\begin_inset Formula $f$ +\end_inset + + to the algebraic multipliers of the form +\begin_inset Formula $f(\omega) = f(a) d x+ f(b) d y$ +\end_inset + +. The same rule works for vector-valued arguments. Let us consider the example in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + +operator +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +LIMIT +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is applied to the form-valued expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- www=(x+y)2̂/(x2̂-1)*d x+(x+y)/(x-z)*d y; <- www; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 2 x + 2*x*y + y x + y (—————–) d x + (——-) d y 2 x - z x - 1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- LIMIT(www,x,INFINITY); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +d x + d y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +I would like to remind also that depending on the particular +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +system +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +operators must be used in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +in upper +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +LIMIT +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or lower case +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +limit +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "case" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + for more details. +\end_layout + +\begin_layout Standard + +\family typewriter +Any function or operator defined in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +package can be used in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +as well. Some examples are considered in section +\begin_inset CommandInset ref +LatexCommand ref +reference "packages" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Subsection + +\family typewriter +Expression Evaluation +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Expression evaluation +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +evaluates expressions in several steps: +\end_layout + +\begin_layout Standard + +\family typewriter +(1) All +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +-specific constructions such as +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sum +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Prod +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Re +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Im +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + etc are explicitly expanded. +\end_layout + +\begin_layout Standard + +\family typewriter +(2) If expression contains components of some built-in or user defined object they are replaced by the appropriate value. If the object is in indefinite state +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "find" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + (no value of the object is known) then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +tries to calculate its value by the method used by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command. The automatic object calculation can be prevented by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +AUTO +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + turning the switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +AUTO +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + off. If due to some reason the object cannot be calculated then expression evaluation is terminated with the error message. +\end_layout + +\begin_layout Standard + +\family typewriter +(3) After all object components are replaced by their values +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +performs all +\begin_inset Quotes eld +\end_inset + +geometrical +\begin_inset Quotes erd +\end_inset + + operations: exterior and interior products, scalar products etc. If expression is form-valued when it is reduced to the form +\begin_inset Formula $a\,dx^0\wedge dx^1\dots+b\,d x^1\wedge+\dots$ +\end_inset + + where +\begin_inset Formula $a$ +\end_inset + + and +\begin_inset Formula $b$ +\end_inset + + are algebraic expressions (similarly for the vector-valued expressions). +\end_layout + +\begin_layout Standard + +\family typewriter +(4) The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +algebraic simplification routine is applied to the algebraic expressions +\begin_inset Formula $a$ +\end_inset + +, +\begin_inset Formula $b$ +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +In the anholonomic mode the basis +\begin_inset Formula $b^i\wedge b^j\dots$ +\end_inset + + is used instead. See section +\begin_inset CommandInset ref +LatexCommand ref +reference "amode" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Final expression consist of exterior products of basis coordinate differentials +\begin_inset Formula $dx^i\wedge dx^j\dots$ +\end_inset + + (or basis vectors +\begin_inset Formula $\partial_{x^i}$ +\end_inset + +) multiplied by the algebraic expressions. The algebraic expressions contain only the coordinates, constants and functions. +\end_layout + +\begin_layout Subsection + +\family typewriter +Controlling Expression Evaluation +\end_layout + +\begin_layout Standard + +\family typewriter +There are many +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +switches which control algebraic expression evaluation. The number of these switches and details of their work depend on the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +version. Here we consider some of these switches. All examples below are made with the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +3.5. On other +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +versions result may be a bit different. +\end_layout + +\begin_layout Standard + +\family typewriter +Switches EXP +\family typewriter +\series default +\shape default + and MCD +\family typewriter +\series default +\shape default + control expansion and reduction of rational expressions to a common denominator respectively. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- (x+y)2̂; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 2 x + 2*x*y + y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Off EXP; <- (x+y)2̂; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 (x + y) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- On EXP; <- 1/x+1/y; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +x + y ——- x*y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Off MCD; <- 1/x+1/y; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +-1 -1 x + y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + These switches are normally on. +\end_layout + +\begin_layout Standard + +\family typewriter +Switches PRECISE +\family typewriter +\series default +\shape default + and REDUCED +\family typewriter +\series default +\shape default + control evaluation of square roots: +\begin_inset CommandInset label +LatexCommand label +name "PRECISE" + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "REDUCED" + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- SQRT(-8*x2̂*y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2*SQRT( - 2*y)*x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- On REDUCED; <- SQRT(-8*x2̂*y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2*SQRT(y)*SQRT(2)*I*x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Off REDUCED; <- On PRECISE; <- SQRT(-8*x2̂*y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2*SQRT(y)*SQRT(2)*I*x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- On REDUCED, PRECISE; <- SQRT(-8*x2̂*y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2*SQRT(y)*SQRT(2)*ABS(x) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Combining rational expressions the system by default calculates the least common multiple of denominators but turning the switch LCM +\family typewriter +\series default +\shape default + off prevents this calculation. +\end_layout + +\begin_layout Standard + +\family typewriter +Switch GCD +\family typewriter +\series default +\shape default + (normally off) makes the system search and cancel the greatest common divisor of the numerator and denominator of rational expressions. Turning GCD +\family typewriter +\series default +\shape default + on may significantly slow down the calculations. There is also another switch EZGCD +\family typewriter +\series default +\shape default + which uses other algorithm for g.c.d. calculation. +\end_layout + +\begin_layout Standard + +\family typewriter +Switches COMBINELOGS +\family typewriter +\series default +\shape default + and EXPANDLOGS +\family typewriter +\series default +\shape default + control the evaluation of logarithms +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- On EXPANDLOGS; <- LOG(x*y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +LOG(x) + LOG(y) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- LOG(x/y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +LOG(x) - LOG(y) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Off EXPANDLOGS; <- On COMBINELOGS; <- LOG(x)+LOG(y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +LOG(x*y) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +By default all polynomials are considered by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +as the polynomials with integer coefficients. The switches RATIONAL +\family typewriter +\series default +\shape default + and COMPLEX +\family typewriter +\series default +\shape default + allow rational and complex coefficients in polynomials respectively: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- (x2̂+y2̂+x*y/3)/(x-1/2); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 2 2*(3*x + x*y + 3*y ) ———————– 3*(2*x - 1) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- On RATIONAL; <- (x2̂+y2̂+x*y/3)/(x-1/2); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 1 2 x + —*x*y + y 3 ——————- 1 x - — 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Off RATIONAL; <- 1/I; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +1 — I +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- (x2̂+y2̂)/(x+I*y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 2 x + y ——— I*y + x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- On COMPLEX; <- 1/I; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +- I +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- (x2̂+y2̂)/(x+I*y); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +x - I*y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Switch RATIONALIZE +\family typewriter +\series default +\shape default + removes complex numbers from the denominators of the expressions but it works even if COMPLEX +\family typewriter +\series default +\shape default + is off. +\end_layout + +\begin_layout Standard + +\family typewriter +Turning off switch EXP +\family typewriter +\series default +\shape default + and on GCD +\family typewriter +\series default +\shape default + one can make the system to factor expressions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Off EXP; <- On GCD; <- x2̂+y2̂+2*x*y; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 (x + y) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Similar effect can be achieved by turning on switch FACTOR +\family typewriter +\series default +\shape default +. Unfortunately this works only when +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +prints expressions and internally expressions remain in the expanded form. To make +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +to work with factored expressions internally one must turn on FACTOR +\family typewriter +\series default +\shape default + and AEVAL +\family typewriter +\series default +\shape default +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +AEVAL +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +switch AEVAL +\family typewriter +\series default +\shape default + make +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +to use an alternative +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +routine for algebraic expression evaluation and simplification. This routine works well with FACTOR +\family typewriter +\series default +\shape default + on. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See section +\begin_inset CommandInset ref +LatexCommand ref +reference "tuning" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about configuration files. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Possibly it is good idea to turn switch AEVAL +\family typewriter +\series default +\shape default + on by default. This can be done using +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +configuration files. +\end_layout + +\begin_layout Subsection + +\family typewriter +Substitutions +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Substitutions +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The substitution commands in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +are the same as the corresponding +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +instructions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Let +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Match +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +For All Let +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +For All +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Such That +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +cond +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Let +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sub +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +For All +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Such That +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +cond +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Match +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sub +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "solutions" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about solutions. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sub +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is either relation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +l +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + += +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +r +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\family typewriter +\series default +\shape default + or the solution in the form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sol( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. After the substitution is activated every appearance of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +l +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + will be replaced by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +r +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The For All +\family typewriter +\series default +\shape default + substitutions have additional list of parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and will work for any value of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The optional condition +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +cond +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + imposes restrictions on the value of the parameters +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +cond +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the boolean expression (see page +\begin_inset CommandInset ref +LatexCommand pageref +reference "bool" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +). +\end_layout + +\begin_layout Standard + +\family typewriter +The substitution can be deactivated by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Clear +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +For All +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space \thinspace{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Such That +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +cond +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Clear +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +sub +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Notice that the variables +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + must be exactly the same as in the corresponding For All Let +\family typewriter +\series default +\shape default + command. +\end_layout + +\begin_layout Standard + +\family typewriter +The difference between +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Match +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Let +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is that the former matches the degrees of the expressions exactly while +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Let +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + matches all powers which are greater than one indicated in the substitution: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Const a; <- (a+1)8̂; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +8 7 6 5 4 3 2 a + 8*a + 28*a + 56*a + 70*a + 56*a + 28*a + 8*a + 1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Let a3̂=1; <- (a+1)8̂; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 85*a + 86*a + 85 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Clear a3̂; <- Match a3̂=1; <- (a+1)8̂; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +8 7 6 5 4 2 a + 8*a + 28*a + 56*a + 70*a + 28*a + 8*a + 57 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Substitutions can be used for various purposes, for example: (i) to define additional mathematical relations such as trigonometric ones; (ii) to +\begin_inset Quotes eld +\end_inset + +assign +\begin_inset Quotes erd +\end_inset + + value to the user-defined and built-in constants; (iii) to define differentiation rules for functions. +\end_layout + +\begin_layout Standard + +\family typewriter +After some substitution is activated it applies to every evaluated expression but value of the objects calculated +\emph on +before +\emph default + remain unchanged. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + re-simplifies the value of the object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the object name, or identifier, or the group object name. Let us consider a simple +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task which calculates the volume 4-form of some metric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- Constant a; <- Tetrad T0=d t, T1=d x, T2=SIN(a)*d y+COS(a)*d z, T3=-COS(a)*d y+SIN(a)* d z; <- Find and Write Volume; Volume : +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 2 VOL = (SIN(a) + COS(a) ) d t +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + We see that +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +do not know the appropriate trigonometric rule. Thus we are going to apply substitution +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- For all x let SIN(x)2̂ = 1-COS(x)2̂; <- Write Volume; Volume : +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +VOL = d t +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + The situation has been improved. But actually, the +\emph on +internal +\emph default + representation of VOL +\family typewriter +\series default +\shape default + remains unchanged. Write +\family typewriter +\series default +\shape default + by default re-simplifies expressions before printing. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swinda{ +\end_layout + +\end_inset + +WRS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + By turning switch WRS +\family typewriter +\series default +\shape default + off we can prevent this re-simplification: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Off WRS; <- Write Volume; Volume : 2 2 VOL = (SIN(a) + COS(a) ) d t +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Now we can apply +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Evaluate Volume; <- Write Volume; Volume : +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +VOL = d t +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +d z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + We see that the internal value of VOL +\family typewriter +\series default +\shape default + now has been replaced by re-simplified expression. +\end_layout + +\begin_layout Standard + +\family typewriter +Notice that the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Evaluate All; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + applies +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to all objects whose value is currently known. +\end_layout + +\begin_layout Subsection + +\family typewriter +Generic Functions +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Generic Functions +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "genfun" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Unfortunately +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +lacks the notion of partial derivative of a function. The expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +DF(f(x,y),x) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is treated by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +as the +\begin_inset Quotes eld +\end_inset + +derivative of the expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +f(x,y) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with respect to the variable +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +x +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Quotes erd +\end_inset + + rather than the +\begin_inset Quotes eld +\end_inset + +derivative of the function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with respect to its first argument +\begin_inset Quotes erd +\end_inset + +. Due to this +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +cannot handle chain differentiation rule etc. This problem is fixed by the package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +dfpart +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + written by H. +\begin_inset space ~ + +\end_inset + +Melenk. This package introduces notion of generic function and partial derivative +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +DFP +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. If +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +dfpart +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is installed on your +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +system +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +provides the interface to these facilities. +\end_layout + +\begin_layout Standard + +\family typewriter +Let us consider an example. First we declare one usual and two generic functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- Function f; <- Generic Function g(a,b), h(b); <- Write Functions; Functions: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +g*(a,b) h*(b) f +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Generic functions must be always declared with the list of parameters ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + in our example). These parameters play the role of labels which denotes arguments of the generic function and the partial derivatives with respect to these arguments are defined. Due to this generic functions allow the chain differentiation rule +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- DF(f(SIN(x),y),x); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +DF(f(SIN(x),y),x) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- DF(g(SIN(x),y),x); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +COS(x)*g (SIN(x),y) a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + Here subscript +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + denotes the derivative of the function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with respect to the first argument. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +enlargethispage{5mm} +\end_layout + +\end_inset + + The operator +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +DFP +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is introduced to denotes such derivatives in expressions: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- DF(g(x,y)*h(y),b); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +0 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- DFP(g(x,y)*h(y),b); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +g (x,y)*h(y) + h (y)*g(x,y) b b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Newpage newpage +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +If switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +DFPCOMMUTE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +DFPCOMMUTE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is turned on then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +DFP +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + derivatives commute. +\end_layout + +\begin_layout Section + +\family typewriter +Using Built-in Formulas In Calculations +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +has large number of built-in objects and almost each object has built-in formulas or so called +\emph on +ways of calculation +\emph default + which can be used to find the value of the object. This section explains how these formulas (ways) can be used. +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Ways of calculation +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "find" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Almost each +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +built-in object has associated +\emph on +ways of calculation +\emph default +. Each way is nothing but a formula or equation which allows to compute the value of the object. All these formulas are described in the usual mathematical style in chapter 3. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or equivalently +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +? +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints information about object's ways of calculation. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + applies built-in formulas to calculate the object value +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +way +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the object name, or identifier, or group object name. The optional specification +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +way +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + indicates the particular way if the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + has several built-in ways of calculation. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +enlargethispage{3mm} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Consider the curvature 2-form +\begin_inset Formula $\Omega^a{}_b$ +\end_inset + + (object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Curvature +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +OMEGA +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +): +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Show Curvature; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Curvature OMEGA'a.b is 2-form Value: unknown Ways of calculation: Standard way (omega) From spinorial curvature (OMEGAU*,OMEGAD) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\noindent + +\family typewriter +We can see that this object has two built in ways of calculation. First way named Standard way +\family typewriter +\series default +\shape default + is the usual equation +\begin_inset Formula $\Omega^a{}_b=d\omega^a{}_b+\omega^a{}_m\wedge\omega^m{}_b$ +\end_inset + +. Second way under the name From spinorial curvature +\family typewriter +\series default +\shape default + uses spinor +\begin_inset Formula $\tsst$ +\end_inset + + tensor relationship to compute the curvature 2-form using its spinor analogues +\begin_inset Formula $\Omega_{AB}$ +\end_inset + + and +\begin_inset Formula $\Omega_{\dot{A}\dot{B}}$ +\end_inset + + as the source data. The ways of calculation are printed by the command Show +\family typewriter +\series default +\shape default + in the form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +wayname +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +SI +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +wayname +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the way name and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See Eq. ( +\begin_inset CommandInset ref +LatexCommand ref +reference "omes" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +) on +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +omes +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +SI +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are the identifiers of the +\emph on +source +\emph default + objects which are present in the right-hand side of the equation. The value of these objects must be known before the formula can be applied. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +% +\backslash +enlargethispage{5mm} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +way +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command allows one to choose the particular way which can be done by two methods. In the first form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +way +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is just the name exactly as it printed by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +wayname +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or Using standard way +\family typewriter +\series default +\shape default + or By standard way +\family typewriter +\series default +\shape default + if the way name is Standard way +\family typewriter +\series default +\shape default +. Another method to specify the way is to indicate the appropriate source object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +From +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Newline newline +\end_inset + +Using +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the name or the identifier of the source object. For example second (spinorial) way of calculation for the curvature 2-form can be chosen by the following equivalent commands +\begin_inset VSpace -1mm +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find curvature from spinorial curvature; Find curvature using OMEGAU; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +while first way is activated by the commands +\begin_inset VSpace -1mm* +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find curvature by standard way; Find curvature using omega; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Recall that object identifiers are case sensitive and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +omega +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the identifier of the frame connection 1-form +\begin_inset Formula $\omega^a{}_b$ +\end_inset + + and should not be confused with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +OMEGA +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +way +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + specification in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be omitted and in this case +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +uses the following algorithm to choose a particular way of calculation. Observe that the identifier of the undotted curvature 2-form +\begin_inset Formula $\Omega_{AB}$ +\end_inset + + is marked by the symbol +\begin_inset Formula $*$ +\end_inset + +. This label marks so called +\emph on +main +\emph default + objects. If no way of calculation is specified when +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +tries to choose the way, browsing the way list form top to the bottom, for which the value of the +\emph on +main +\emph default + object is already known. If no switch way exists then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +just picks up the first way in the list. Therefore in our example the command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find curvature; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +will use the second way if the value of the object +\begin_inset Formula $\Omega_{AB}$ +\end_inset + + (id. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +OMEGAU +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) is known and second way otherwise. +\end_layout + +\begin_layout Standard + +\family typewriter +As soon as some way of calculation is chosen +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +tries to calculate the values of the source objects which are present in the right-hand side of corresponding equations. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +tries to do this by applying the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command without way specification to these objects. Thus a single +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can cause quite long chain of calculations. This recursive work is reflected by the appropriate tracing messages. The tracing can be eliminated by turning off switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +TRACE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +TRACE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Here we present the sample +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session which computes curvature 2-form for the flat gravitational waves +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Cord u, v, z, z +\begin_inset space ~ + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +z & z +\begin_inset space ~ + +\end_inset + + - conjugated pair. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Null Metric; <- Function H(u,z,z +\begin_inset space ~ + +\end_inset + +); <- Frame T0=d u, T1=d v+H*d u, T2=d z, T3=d z +\begin_inset space ~ + +\end_inset + +; <- ds2; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 2 ds = ( - 2*H) d u + (-2) d u d v + 2 d z d z +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Find Curvature; Sqrt det of metric calculated. 0.16 sec Volume calculated. 0.16 sec Vector frame calculated From frame. 0.16 sec Inverse metric calculated From metric. 0.16 sec Frame connection calculated. 0.22 sec Curvature calculated. 0.22 sec <- Write Curvature; Curvature: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +1 OMEGA = ( - DF(H,z,2)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z + ( - DF(H,z,z +\begin_inset space ~ + +\end_inset + +)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z +\begin_inset space ~ + +\end_inset + + 2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +1 OMEGA = ( - DF(H,z,z +\begin_inset space ~ + +\end_inset + +)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z + ( - DF(H,z +\begin_inset space ~ + +\end_inset + +,2)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z +\begin_inset space ~ + +\end_inset + + 3 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2 OMEGA = ( - DF(H,z,z +\begin_inset space ~ + +\end_inset + +)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z + ( - DF(H,z +\begin_inset space ~ + +\end_inset + +,2)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z +\begin_inset space ~ + +\end_inset + + 0 +\begin_inset Newpage newpage +\end_inset + +3 OMEGA = ( - DF(H,z,2)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z + ( - DF(H,z,z +\begin_inset space ~ + +\end_inset + +)) d u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d z +\begin_inset space ~ + +\end_inset + + 0 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Finally we want to emphasize that ways associated with some object may depend on the concrete environment. In particular the Standard way +\family typewriter +\series default +\shape default + for the curvature 2-form is always available but second way which is essentially related to spinors works +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +pref{ +\end_layout + +\end_inset + +spinors +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + about the spinorial formalism. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + only in the 4-dimensional spaces of Lorentzian signature and iff the metric is null. If some way is not valid in the current environment it simply disappears from the way list printed by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +It should be noted also that the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command works only if the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is in the indefinite state and is rejected if the value of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is already known. If you want to re-calculate the object then previous value must be cleared by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command. +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + destroys the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + value and returns it to initial indefinite state. It can be used also to free the memory. +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Zero +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Zero +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Zero +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + assigns zero values to all +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + components. +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Normalize +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Normalize +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Normalize +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + applies to equations. It replaces equalities of the form +\begin_inset Formula $l=r$ +\end_inset + + by the equalities +\begin_inset Formula $l-r=0$ +\end_inset + + and re-simplifies the result. +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Evaluate +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + re-simplifies existing value of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. This command is useful if we want to apply new substitutions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "subs" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about substitutions. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + to the object whose value is already known. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Evaluate All; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + re-simplifies all objects whose value is currently known. +\end_layout + +\begin_layout Section + +\family typewriter +Printing Result of Calculations +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints value of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + id the object name or identifier. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Group name +\end_layout + +\end_inset + + Group names denoting a collection of several objects +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "macro" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about macro objects. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and macro object identifiers can be used in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command as well. In addition word +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +All +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can be used to print all currently known objects. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + can print declarations as well if +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is functions +\family typewriter +\series default +\shape default +, constants +\family typewriter +\series default +\shape default +, or affine parameter +\family typewriter +\series default +\shape default +. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space ~ + +\end_inset + +to +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or equivalently +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space ~ + +\end_inset + +> +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + writes result into the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Notice that +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + always destroys previous contents of the file. Therefore we have another command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +Write > +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + which redirects all output into the file. The standard output can be restored by the commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +End of Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +EndW +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +EndW; +\begin_inset Newline newline +\end_inset + +End of Write; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +enlargethispage{3mm} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +By default +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + re-simplifies the expressions before printing them. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +WRS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "subs" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about substitutions. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + This is convenient when substitutions are activated but slows down the printing especially for very large expressions. The re-simplification can be abolished by turning off switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +WRS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. If switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +WMATR +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is turned on then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +WMATR +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +prints all 2-index scalar-valued objects in the matrix form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- On wmatr; <- Find and Write metric; Assuming Default Metric. Metric calculated By default. 0.06 sec Metric: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +[-1 0 0 0] [ ] [0 1 0 0] [ ] [0 0 1 0] [ ] [0 0 0 1] +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + prints frame, spinor and enumerating indices as numerical subscripts while holonomic indices are printed as the coordinate identifiers. If frame is holonomic and there is no difference between frame and coordinate indices then by default all frame indices are also labelled by the appropriate identifiers. But is switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +HOLONOMIC +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swinda{ +\end_layout + +\end_inset + +HOLONOMIC +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is turned off they are still printed as numbers. +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Print +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Command +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Print +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command described in the previous section prints value of an object. This value must be calculated beforehand by the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Find +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command or established by the assignment. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Print +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + evaluates expression and immediately prints its value. It has several forms +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Print +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +For +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +For +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Print +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is expression to be evaluated and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + indicates that expression must be evaluated for several value of some variable. The specification +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +iter +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is completely the same as is the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sum +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + expression and is described in details in section +\begin_inset CommandInset ref +LatexCommand ref +reference "iter" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + on page +\begin_inset CommandInset ref +LatexCommand pageref +reference "iter" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. It consists of the list of parameters separated by commas +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or relational operators < > => =< +\family typewriter +\series default +\shape default +. For example the command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +G(a,b) for a + + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +MACSYMA +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +macsyma +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +MAPLE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +maple +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +MATH +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mathematica +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +REDUCE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +GRG +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + + + +\end_inset + + +\begin_inset Newline newline +\end_inset + +Notice the last switch allows one to print the data in the form which can be later inserted into +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task. +\end_layout + +\begin_layout Section + +\family typewriter +Advanced Facilities +\end_layout + +\begin_layout Subsection + +\family typewriter +Solving Equations +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Solve +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "solutions" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +provides simple interface to the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +algebraic equation solver. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Solve +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +l +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +r +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space ~ + +\end_inset + +for +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + resolves equations +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +l +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +r +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with respect to expressions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. This command has also other form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Solve +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +equation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + for +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +equation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the name or identifier of some built-in or user-defined equation. Both form of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Solve +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command works with form and scalar valued equations as well but +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + must be algebraic. The resulting solutions are stored in the special object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Solutions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + (identifier +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sol +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +). They can be printed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdindx{ +\end_layout + +\end_inset + +Write +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +}{ +\end_layout + +\end_inset + +Solutions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Write Solutions; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Left and right hand sides of +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +'th solution can be used in expression as +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +LHS(Sol( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +)) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +RHS(Sol( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +)) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The expression +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Sol( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + referring to the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +'th solution can be used in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SUB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Let +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + substitutions as well: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- Solve x2-2*x=5, y=9 for x, y; <- Write Solutions; Solutions: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Sol(0) : y = 9 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Sol(1) : x = - SQRT(6) + 1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Sol(2) : y = 9 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +Sol(3) : x = SQRT(6) + 1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- SUB(Sol(1),(x-1)2); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +6 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Let Sol(3); <- (x-1)2; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +6 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Solutions can be cleared by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdindx{ +\end_layout + +\end_inset + +Erase +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +}{ +\end_layout + +\end_inset + +Solutions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Erase Solutions; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + One need to stress that +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Solve +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is capable to solve algebraic relations only. Solving algebraic relations +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +knows already that the function +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +ASIN +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is inverse to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SIN +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Inverse +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Inverse +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + tells the system that functions +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +f2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are inverse to each other. +\end_layout + +\begin_layout Subsection + +\family typewriter +Saving Data for Later Use +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset CommandInset label +LatexCommand label +name "UnloadLoad" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +It is very convenient to have facilities to save results of calculations in a form fitted for restoring and further manipulation. For this purpose +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +has two special commands: Unload +\family typewriter +\series default +\shape default + and Load +\family typewriter +\series default +\shape default +. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "Unload" + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + > +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + To +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + writes +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + value into +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + in some special format. Here +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is name or identifier of an object. +\end_layout + +\begin_layout Standard + +\family typewriter +The data can be later restored with help of the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The command Unload +\family typewriter +\series default +\shape default + always overwrites previous +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + contents. To save several objects in one file one must use the following sequence of commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +EndU +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +End of Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Unload > +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; ... Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; End Of Unload; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Here command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Unload > +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + opens +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and End Of Unload; +\family typewriter +\series default +\shape default + closes it. The last command has the short form +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +EndU; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + In fact presented above sequence of commands can be abbreviated as +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space ~ + +\end_inset + +> +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +One needs to stress that only the commands Unload …; +\family typewriter +\series default +\shape default + can be used between Unload > … +\family typewriter +\series default +\shape default + and End Of Unload; +\family typewriter +\series default +\shape default +. If this rule does not hold then Load +\family typewriter +\series default +\shape default + may fail to restore the file. The only additional command which can be used among these Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\family typewriter +\series default +\shape default + commands is the comment % +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +text +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\family typewriter +\series default +\shape default +. This command insertes the comment +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +text +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + into the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Later when +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + will be restored by the Load +\family typewriter +\series default +\shape default + the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +text +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + message will be printed. This allows one to attach comments to unreadable files produced by Unload +\family typewriter +\series default +\shape default + command. +\end_layout + +\begin_layout Standard + +\family typewriter +As in other commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command is either the name or identifier of an object. Names Coordinates +\family typewriter +\series default +\shape default +, Constants +\family typewriter +\series default +\shape default + and Functions +\family typewriter +\series default +\shape default + can also be used to save declarations. And finally, the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Unload All > +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + saves all objects whose value is currently known +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See section +\begin_inset CommandInset ref +LatexCommand ref +reference "amode" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about anholonomic basis. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and all declarations. Moreover, in the anholonomic basis mode this command saves full information about an anholonomic basis. +\end_layout + +\begin_layout Standard + +\family typewriter +When data or coordinates declarations are restored from a file they replace current values. Function and constant declarations are added to current declarations. +\end_layout + +\begin_layout Standard + +\family typewriter +One should realize that serious troubles may appear when different coordinates are used in the current session and in the restored file. Even the order of coordinates is extremely important. We strongly recommend saving all declarations (especially coordinates) in addition to other objects. It ensures at least that will +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +print a warning message if some contradictions are detected between current declarations and declarations stored into a file. The best way to avoid these troubles is to use the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Unload All > +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Loading the file saved by this command at the very beginning of a new +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +task completely restores the previous +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +state with all data and declarations. +\end_layout + +\begin_layout Standard + +\family typewriter +Sometimes one needs to prevent the Load +\family typewriter +\series default +\shape default +/Unload +\family typewriter +\series default +\shape default + operations with coordinates. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +UNLCORD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + If switch UNLCORD +\family typewriter +\series default +\shape default + is turned off (normally on) then all Load +\family typewriter +\series default +\shape default + and Unload +\family typewriter +\series default +\shape default + operations with coordinates are blocked. +\end_layout + +\begin_layout Standard + +\family typewriter +Since Unload +\family typewriter +\series default +\shape default + writes data in human-unreadable form there is the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show File +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +File +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +File +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or equivalently +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +? +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +File +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset space ~ + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +File +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + which prints short information about objects and declarations contained in the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. It also prints comments contained in the file. +\end_layout + +\begin_layout Subsection + +\family typewriter +Coordinate Transformations +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Coordinate transformations +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +New Coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +New Coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +new +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rpt{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +old +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + introduces new coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +new +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and defines how old coordinates +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +old +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are expressed in terms of new ones. If the specified transformation is nonsingular +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +converts all existing objects to the new coordinate system. +\end_layout + +\begin_layout Standard + +\family typewriter +The New Coordinates +\family typewriter +\series default +\shape default + command properly transforms all objects having coordinate indices. The transformation of frame indices depend on the switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +HOLONOMIC +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +HOLONOMIC +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + In general case when frame is not holonomic then objects having frame indices remain unchanged and only their components are transformed into the new coordinate system. But if frame is holonomic then by default all frame indices are transformed similarly to the coordinate ones. Notice that in such situation the frame after transformation once again will be holonomic in the new coordinate system. But if switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +HOLONOMIC +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is turned off the system distinguishes frame and coordinate indices in spite of the current frame type. In such situation the holonomic frame ceases to be holonomic after coordinate transformation. +\end_layout + +\begin_layout Subsection + +\family typewriter +Frame Transformations +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Frame transformations +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Spinorial rotations are performed by the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Make Spinorial Rotation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Spinorial Rotation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Make +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Spinorial Rotation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + (( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{00}$ +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{01}$ +\end_inset + +), ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{10}$ +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{11}$ +\end_inset + +)) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where expressions +\begin_inset Formula $\mbox{\parm{expr}}_{AB}$ +\end_inset + + comprise the SL(2,C) transformation matrix +\begin_inset Formula \[ +\phi'_A=L_A{}^B\phi_B,\ \ +\mbox{\parm{expr}}_{AB}=L_A{}^B +\] +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +If the specified matrix is really a SL(2,C) one then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + +performs appropriate transformation on all objects whose value is currently known. +\end_layout + +\begin_layout Standard + +\family typewriter +Matrix specification in the command can be omitted +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Make +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Spinorial Rotation; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + In this case the SL(2,C) matrix +\begin_inset Formula $L_A{}^B$ +\end_inset + + must be specified as the value of a special object Spinorial Transformation LS.A'B +\family typewriter +\series default +\shape default + (identifier LS +\family typewriter +\series default +\shape default +). +\end_layout + +\begin_layout Standard + +\family typewriter +Command for frame rotation is analogously +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Make Rotation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Rotation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Make +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Rotation +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + (( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{00}$ +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{01}$ +\end_inset + +,...), ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{10}$ +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{11}$ +\end_inset + +,...),...) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + with the nonsingular +\begin_inset Formula $d\times d$ +\end_inset + + rotation matrix +\begin_inset Formula \[ +A'^a=L^a{}_bA^b,\ \ \mbox{\parm{expr}}_{ab}=L^a{}_b +\] +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +verifies that this matrix is a valid +\emph on +rotation +\emph default + by checking that frame metric +\begin_inset Formula $g_{ab}$ +\end_inset + + +\emph on +remains unchanged +\emph default + under this transformation +\begin_inset Formula \[ +g'_{ab} = L^m{}_a L^n{}_b g_{mn} = g_{ab} +\] +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Once again the matrix specification can be omitted and transformation +\begin_inset Formula $L^a{}_b$ +\end_inset + + can be specified as the value of the object Frame Transformation L'a.b +\family typewriter +\series default +\shape default + (identifier L +\family typewriter +\series default +\shape default +) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Make +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Rotation; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Frame rotation commands correctly transform frame and spinor connection 1-forms. +\end_layout + +\begin_layout Standard + +\family typewriter +Finally, there is a special form of the frame transformation command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Change Metric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Change Metric +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + + (( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{00}$ +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{01}$ +\end_inset + +,...), ( +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{10}$ +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +expr +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula ${}_{11}$ +\end_inset + +,...),...) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + The only difference between this command and Make Rotation +\family typewriter +\series default +\shape default + is that Change Metric +\family typewriter +\series default +\shape default + does not impose any restriction on the transformation matrix and transformed metric does not necessary coincides with the original one. +\end_layout + +\begin_layout Standard + +\family typewriter +Sometimes it is convenient to keep some object unchanged under the frame transformation. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Hold +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Hold +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + makes the system to keep the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + unchanged during frame and spinor transformations. The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Release +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Release +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + discards the action of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Hold +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command. +\end_layout + +\begin_layout Subsection + +\family typewriter +Algebraic Classification +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Algebraic classification +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Classify +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Classify +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + performs algebraic classification of the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +object +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + specified by its name or identifier. Currently +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +knows algorithms for classifying the following irreducible spinors +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Tabular + + + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $X_{ABCD}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Weyl spinor type +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $X_{AB\dot{C}\dot{D}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Traceless Ricci spinor type +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $X_{AB}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Electromagnetic stress spinor type +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter + +\begin_inset Formula $X_{A\dot{B}}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Standard + +\family typewriter +Vector in the spinorial representation +\end_layout + +\end_inset + + + + +\end_inset + + +\begin_inset Newline newline +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reversemarginpar +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The Classify +\family typewriter +\series default +\shape default + command can be applied to any built-in or user-defined object having one of the listed above +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +seethis{ +\end_layout + +\end_inset + +See page +\begin_inset CommandInset ref +LatexCommand pageref +reference "sumspin" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + about summed spinor indices. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + types of indices. Notice that all spinors must be irreducible (totally symmetric in dotted and undotted indices) and +\begin_inset Formula $X_{AB\dot{C}\dot{D}}$ +\end_inset + +, +\begin_inset Formula $X_{A\dot{B}}$ +\end_inset + + must be Hermitian. Groups of the irreducible indices must be represented as a single summed index. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +normalmarginpar +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +uses the algorithm by F. +\begin_inset space ~ + +\end_inset + +W. +\begin_inset space ~ + +\end_inset + +Letniowski and R. +\begin_inset space ~ + +\end_inset + +G. +\begin_inset space ~ + +\end_inset + +McLenaghan [Gen. Rel. Grav. 20 (1988) 463-483] for Petrov-Penrose classification of Weyl spinor +\begin_inset Formula $X_{ABCD}$ +\end_inset + +. The obvious simplification of this algorithm is applied to the spinor analog of electromagnetic strength tensor +\begin_inset Formula $X_{AB}$ +\end_inset + +. The spinor +\begin_inset Formula $X_{AB\dot{C}\dot{D}}$ +\end_inset + + is classified by the algorithm by G. +\begin_inset space ~ + +\end_inset + +C. +\begin_inset space ~ + +\end_inset + +Joly, M. +\begin_inset space ~ + +\end_inset + +A. +\begin_inset space ~ + +\end_inset + +H. +\begin_inset space ~ + +\end_inset + +McCallum and W. +\begin_inset space ~ + +\end_inset + +Seixas [Class. Quantum Grav. 7 (1990) 541-556, Class. Quantum Grav. 8 (1991) 1577-1585]. +\end_layout + +\begin_layout Standard + +\family typewriter +The classification process is accompanied by the tracing messages which can be eliminated by turning +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swinda{ +\end_layout + +\end_inset + +TRACE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + off the switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +TRACE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. On the contrary if one turns on +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +SHOWEXPR +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + the switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SHOWEXPR +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +prints all expressions which appear during the classification to let you check whether the decision about nonvanishing of these expressions is really correct or not. This facility is important also in classifying +\begin_inset Formula $X_{AB\dot{C}\dot{D}}$ +\end_inset + + and +\begin_inset Formula $X_{A\dot{B}}$ +\end_inset + + since algebraic type for this objects may depend on the +\emph on +sign +\emph default + of some expressions which cannot be determined by +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +correctly. +\end_layout + +\begin_layout Subsection + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +Packages and Functions in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Using +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +packages +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "packages" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Any procedure or function defined in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +package can be used in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +. The package must be loaded either before +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is started or during +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session by one of the equivalent commands +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Use Package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Use +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset Newline newline +\end_inset + +Load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + where +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + is the package name. Notice that an identifier must be used for the package name unlike the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +Load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command described in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +enlargethispage{5mm} +\end_layout + +\end_inset + + section +\begin_inset CommandInset ref +LatexCommand ref +reference "UnloadLoad" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. Let us consider some examples. The +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +specfn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + contains definitions of various special functions and below we demonstrate 11th Legendre polynomial +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- package specfn; <- LEGENDREP(11,x); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +10 8 6 4 2 x*(88179*x - 230945*x + 218790*x - 90090*x + 15015*x - 693) ——————————————————————- 256 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Newpage newpage +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Another example demonstrates the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +taylor +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + <- Coordinates t, x, y, z; <- www=d(E(x+y)*SIN(x)); <- www; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +x + y x + y (E *(COS(x) + SIN(x))) d x + (E *SIN(x)) d y +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- load taylor; <- TAYLOR(www,x,0,5); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +y y y y y 2 E 4 E 5 6 y y 2 (E + 2*E *x + E *x - —-*x - —-*x + O(x )) d x + (E *x + E *x 6 15 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +y y E 3 E 5 6 + —-*x - —-*x + O(x )) d y 3 30 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +You can also define your own operators and procedures in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +and later use them in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +. In the following example file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +lasym.red +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + contains a definition of little +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +procedure which computes a leading term of asymptotic expansion of the rational function at large values of some variable. This file is inputted in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +before +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is started +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{slisting} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +1: in +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +lasym.red +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +procedure leadingterm(w,x); lterm(num(w),x)/lterm(den(w),x); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +leadingterm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +end; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +2: load grg; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +This is GRG 3.2 release 2 (Feb 9, 1997) ... +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +System directory: c: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +red35 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +grg32 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + System variables are upper-cased: E I PI SIN ... Dimension is 4 with Signature (-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- Coordinates t, r, theta, phi; <- OMEGA01=(123*r3+2*r+t)/(r+t)5*d theta +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +d phi; <- OMEGA01; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +3 123*r + 2*r + t (————————————————-) d theta +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d phi 5 4 3 2 2 3 4 5 r + 5*r *t + 10*r *t + 10*r *t + 5*r*t + t +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +<- LEADINGTERM(OMEGA01,r); +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +par +\end_layout + +\end_inset + +123 (—–) d theta +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +w +\end_layout + +\end_inset + + d phi 2 r +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{slisting} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Anholonomic Basis Mode +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Anholonomic basis mode +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Basis +\end_layout + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "amode" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +may work in both holonomic and anholonomic basis modes. In the first default case, values of all expressions are represented in a natural holonomic (coordinate) basis: +\begin_inset Formula $d x^\mu,~d x^\mu\wedge x^\nu\dots$ +\end_inset + + for exterior forms and +\begin_inset Formula $\partial_\mu=\partial/\partial x^\mu$ +\end_inset + + for vectors. In the second case an arbitrary basis +\begin_inset Formula $b^i=b^i_\mu d x^\mu$ +\end_inset + + is used for forms and inverse vector basis +\begin_inset Formula $e_i=e_i^\mu\partial_\mu$ +\end_inset + + for vectors ( +\begin_inset Formula $b^i_\mu e^\mu_j=\delta^i_j$ +\end_inset + +). You can specify this basis assigning a value to built-in object Basis +\family typewriter +\series default +\shape default + (identifier b +\family typewriter +\series default +\shape default +). If Basis +\family typewriter +\series default +\shape default + is not specified by user then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +assumes that it coincides with the frame +\begin_inset Formula $b^i=\theta^i$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +Frame should not be confused with basis. Frame +\begin_inset Formula $\theta^a$ +\end_inset + + is used only for +\begin_inset Quotes eld +\end_inset + +external +\begin_inset Quotes erd +\end_inset + + purposes to represent tensor indices while basis +\begin_inset Formula $b^i$ +\end_inset + + and vector basis +\begin_inset Formula $e_i$ +\end_inset + + is used for +\begin_inset Quotes eld +\end_inset + +internal +\begin_inset Quotes erd +\end_inset + + purposes to represent form and vector valued object components. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Anholonomic +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Anholonomic; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + switches the system to the anholonomic basis mode and the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Holonomic +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + +Holonomic; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + switches it back to the standard holonomic mode. +\end_layout + +\begin_layout Standard + +\family typewriter +Working in anholonomic mode +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +creates some internal tables for efficient calculation of exterior differentiation and complex conjugation. In anholonomic mode the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Unload +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Unload All > +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +automatically saves these tables into the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. Subsequent +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +" +\end_layout + +\end_inset + +; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +restores the tables and automatically switches the current mode to anholonomic one. Note that automatic anholonomic mode saving/restoring works only if All +\family typewriter +\series default +\shape default + is used in Unload +\family typewriter +\series default +\shape default + command. +\end_layout + +\begin_layout Standard + +\family typewriter +One can find out the current mode with the help of the command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Show Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cmdind{ +\end_layout + +\end_inset + +Status +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +command{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +opt{ +\end_layout + +\end_inset + +Show +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + Status; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Subsection + +\family typewriter +Synonymy +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Synonymy +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Sometimes +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +commands may be rather long. For instance, in order to find the curvature 2-form +\begin_inset Formula $\Omega_{ab}$ +\end_inset + + from the spinorial curvature +\begin_inset Formula $\Omega_{AB}$ +\end_inset + + and +\begin_inset Formula $\Omega_{\dot{A}\dot{B}}$ +\end_inset + + the following command should be used +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find Curvature From Spinorial Curvature; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Certainly, this command is clear but typing of such long phrases may be very dull. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +has synonymy mechanism which allows one to make input much shorter. +\end_layout + +\begin_layout Standard + +\family typewriter +The synonymous words in commands and object names are considered to be equivalent. The complete list of predefined +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +synonymy is given in appendix D. Here we present just the most important ones +\end_layout + +\begin_layout Verbatim + Connection Con +\end_layout + +\begin_layout Verbatim + Constants Const Constant +\end_layout + +\begin_layout Verbatim + Coordinates Cord +\end_layout + +\begin_layout Verbatim + Curvature Cur +\end_layout + +\begin_layout Verbatim + Dotted Do +\end_layout + +\begin_layout Verbatim + Equation Equations Eq +\end_layout + +\begin_layout Verbatim + Find F Calculate Calc +\end_layout + +\begin_layout Verbatim + Functions Fun Function +\end_layout + +\begin_layout Verbatim + Next N +\end_layout + +\begin_layout Verbatim + Show ? +\end_layout + +\begin_layout Verbatim + Spinor Spin Spinorial Sp +\end_layout + +\begin_layout Verbatim + Switch Sw +\end_layout + +\begin_layout Verbatim + Symmetries Sym Symmetric +\end_layout + +\begin_layout Verbatim + Undotted Un +\end_layout + +\begin_layout Verbatim + Write W +\end_layout + +\begin_layout Standard + +\family typewriter +Words in each line are considered as equivalent in all commands. Thus the above command can be abbreviated as +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +F cur from sp cur; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Section +\begin_inset CommandInset ref +LatexCommand ref +reference "tuning" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + explains how to change built-in synonymy and how to define a new one. +\end_layout + +\begin_layout Subsection + +\family typewriter +Compound Commands +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Compound commands +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Sometime one may need to perform several consecutive actions with one object. In this case we can use so called +\emph on +compound commands +\emph default + to shorten the input. Internally +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +replaces each compound command by several usual ones. For example the compound command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find and Write Einstein Equation; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +to a pair of usual ones +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find Einstein Equation; Write Einstein Equation; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Actions (commands) can be attached to the end of the compound command as well: +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find, Write Curvature and Erase It; +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +udr +\end_layout + +\end_inset + + Find & Write & Erase Curvature; +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +udr +\end_layout + +\end_inset + + Find Curvature; Write Curvature; Erase Curvature; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Note that we have used , +\family typewriter +\series default +\shape default + and & +\family typewriter +\series default +\shape default + instead of and +\family typewriter +\series default +\shape default + in this example. All these separators are equivalent in compound commands. +\end_layout + +\begin_layout Standard + +\family typewriter +Now let us consider the case when one needs to perform a single action with several objects. The command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Write Frame, Vector Frame and Metric; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +is equivalent to +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Write Frame; Write Vector Frame; Write Metric; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Way specification can be attached to the Find +\family typewriter +\series default +\shape default + command: +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find QT, QP From Torsion using spinors; +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +udr +\end_layout + +\end_inset + + Find QT From Torsion using spinors; Find QP From Torsion using spinors; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +One can combine several actions and several objects. For example, the command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find omega, Curvature by Standard Way and Write and Erase Them; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +is equivalent to the sequence of +\begin_inset Formula $(2{\rm\ objects})\times(3{\rm\ commands}) =6$ +\end_inset + + commands +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Find omega by Standard Way; Find Curvature by Standard Way; Write omega; Write Curvature; Erase omega; Erase Curvature; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +Note that the way specification is attached only to +\begin_inset Quotes eld +\end_inset + +left +\begin_inset Quotes erd +\end_inset + + commands (Find +\family typewriter +\series default +\shape default + in our case). +\end_layout + +\begin_layout Standard + +\family typewriter +The compound commands mechanism works only with Find +\family typewriter +\series default +\shape default +, Erase +\family typewriter +\series default +\shape default +, Write +\family typewriter +\series default +\shape default + and Evaluate +\family typewriter +\series default +\shape default + commands. +\end_layout + +\begin_layout Standard + +\family typewriter +And finally, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +always replaces Re- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\family typewriter +\series default +\shape default + by Erase and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +parm{ +\end_layout + +\end_inset + +command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +; +\family typewriter +\series default +\shape default +. For example +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +Re-Calculate Maxwell Equations; +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset space \qquad{} + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +udr +\end_layout + +\end_inset + + Erase and Calculate Maxwell Equations; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +You can see how +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +expand compound commands into the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +swind{ +\end_layout + +\end_inset + +SHOWCOMMANDS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + usual ones by turning switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +SHOWCOMMANDS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + on. +\end_layout + +\begin_layout Section + +\family typewriter +Tuning +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset CommandInset label +LatexCommand label +name "tuning" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +can be tuned according to your needs and preferences. The configuration files allow one to change some default settings and the environment variable +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +grg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines the system directory which can be used as the depository for frequently used files. +\end_layout + +\begin_layout Subsection + +\family typewriter +Configuration Files +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset CommandInset label +LatexCommand label +name "configsect" + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The configuration files allows one to establish +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +begin{list} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset Formula $\bullet$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +labelwidth +\end_layout + +\end_inset + +=8mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +leftmargin +\end_layout + +\end_inset + +=10mm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Default dimension and signature. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Initial position of switches. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +packages which must be preloaded. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Synonymy. +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +item +\end_layout + +\end_inset + +Default +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +start up method. +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +end{list} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +There are two configuration files. First +\emph on +global +\emph default + configuration file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grgcfg.sl +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines the settings +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Global configuration file +\end_layout + +\end_inset + + during system installation when +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is compiled. These global settings become permanent and can be changed only if +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is recompiled. The +\emph on +local +\emph default + configuration file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grg.cfg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + allows one to override global settings locally. +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Local configuration file +\end_layout + +\end_inset + + When +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +starts it search the file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grg.cfg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + in current directory (folder) and if it is present uses the corresponding settings. +\end_layout + +\begin_layout Standard + +\family typewriter +Below we are going to explain how to change settings in both global and local configuration files but before doing this we must emphasize that this need some care. First, the configuration files use LISP command format which differs from usual +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +commands. Second, is something is wrong with configuration file then no clear diagnostic is provided. Finally, if global configuration is damaged you will not be able to compile +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + +. The best strategy is to make a back-up copy of the configuration files before start editing them. Notice that lines preceded by the percent sign +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +% +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + are ignored by the system (comments). +\end_layout + +\begin_layout Standard + +\family typewriter +Both local +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grg.cfg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and global +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grgcfg.sl +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + configuration files have similar structure and can include the following commands. +\end_layout + +\begin_layout Standard + +\family typewriter +Command +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Signature!default +\end_layout + +\end_inset + + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Dimension!default +\end_layout + +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +(signature!> - + + + +) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +establishes default dimension 5 with the signature +\begin_inset Formula $\scriptstyle(-,+,+,+,+)$ +\end_inset + +. Do not forget +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +! +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and spaces between +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + ++ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + and +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. This command +\emph on +must be present +\emph default + in the global configuration file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grgcfg.sl +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + otherwise +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +cannot be compiled. +\end_layout + +\begin_layout Standard + +\family typewriter +The commands +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +(on!> page) (off!> allfac) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +change default switch position. In this example we turn on the switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +PAGE +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + (this switch is defined in DOS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +only and allows one to scroll back and forth through input and output) and turn off switch +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +ALLFAC +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. +\end_layout + +\begin_layout Standard + +\family typewriter +The command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +(package!> taylor) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +makes the system to load +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +package +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +taylor +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + during +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +start. +\end_layout + +\begin_layout Standard + +\family typewriter +The command of the form +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +Synonymy +\end_layout + +\end_inset + + +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +(synonymous!> ( affine aff ) ( antisymmetric asy ) ( components comp ) ( unload save ) ) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +defines synonymous words. The words in each line will be equivalent in all +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +commands. +\end_layout + +\begin_layout Standard + +\family typewriter +Finally the command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +(setq ![autostart!] nil) + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +alters default +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +start up method. It makes sense only in the global configuration file +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +file{ +\end_layout + +\end_inset + +grgcfg.sl +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +. By default +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +is launched by single command +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +load grg; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +which firstly load the program into memory and then automatically starts it. Unfortunately on some systems this short method does not work properly: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +shows wrong timing during computations, the +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +quit; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + command returns the control to +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +reduce +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +session instead of terminating the whole program. If the aforementioned option is activated then +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +must be launched by two commands +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +load grg; grg; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +which fixes the problems. Here first command just loads the program into memory and second one starts it manually. Notice that one can always use commands +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +load grg32; grg; + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +to start +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +manually. Command +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +load grg32; +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + always loads +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +into memory without starting it independently on the option under consideration. +\end_layout + +\begin_layout Subsection + +\family typewriter +System Directory +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset Index idx +status collapsed + +\begin_layout Plain Layout +System directory +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter +The environment variable +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +grg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + or +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +GRG +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + defines so called +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +grg +\end_layout + +\end_inset + + +\begin_inset space \space{} + +\end_inset + +system directory (folder). The way of setting this variable is operating system dependent. For example the following commands can be used to set +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +comm{ +\end_layout + +\end_inset + +grg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + variable in DOS, UNIX and VAX/VMS respectively: +\begin_inset listings +lstparams "float" +inline false +status collapsed + +\begin_layout Plain Layout + +\begin_inset Caption Standard + +\begin_layout Standard + +\family typewriter +set grg=d: +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +xxx +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +yyy +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +bs +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + DOS +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + setenv grg /xxx/yyy/ +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + UNIX +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + define grg SYS +\begin_inset Formula $USER:[xxx.yyy] {\rm VAX/VMS} +{listing} +The value of the variable \comm{grg} must point +out to some directory. +In DOS and UNIX the directory +name must include trailing \comm{\bs} or \comm{/} +respectively. The command\cmdind{Show Status}\cmdind{Status} +\command{\opt{Show} Status;} +prints current system directory. + +When \grg\ tries to input some batch file containing +\grg\ commands it first searches it in the current working +directory and if the file is absent then it tries +to find it in the system directory. Therefore if you have +some frequently used files you can define the system directory +and move these files there. In this case it is not necessary +to keep them in each working directory. Notice \grg\ uses +the same strategy when opening local configuration file +\file{grg.cfg}. Thus if system directory is defined and it +contains the file \file{grg.cfg} the settings contained in +this file effectively overrides global settings without +recompiling \grg. + + +\section{Examples} + +In this section we want to demonstrate how \grg\ can be applied +to solve simple but realistic problem. +We want to calculate the Ricci tensor for the Robertson-Walker +metric by three different methods. + +First \grg\ task (program) +\begin{listing} + Coordinates t,r,theta,phi; + Function a(t); + Frame T0=d t, T1=a*d r, T2=a*r*d theta, T3=a*r*SIN(theta)*d phi; + ds2; + Find and Write Ricci Tensor; + RIC(\_j,\_k); +\end{listing} +defines the Robertson-Walker metric using the tetrad +formalism with the orthonormal Lorentzian tetrad $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +theta +\end_layout + +\end_inset + +a +\begin_inset Formula $. +Using built-in formulas for the Ricci tensor the only one command +is required to accomplish out goal +{\tt Find and Write Ricci Tensor;}. The command {\tt ds2;} +just shows the metric we are dealing with. Notice that +command {\tt Find ...} gives the \emph{tetrad} components of the Ricci +tensor $ +\end_inset + +R +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. Thus, in addition we print coordinate +components of the tensor $ +\end_inset + +R +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mu +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +nu +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ by the command +{\tt RIC(\_j,\_k);}. The hard-copy of the corresponding +\grg\ session is presented below \enlargethispage{4mm} +\begin{slisting} +<- Coordinates t, r, theta, phi; +<- Function a(t); +<- Frame T0=d t, T1=a*d r, T2=a*r*d theta, T3=a*r*SIN(theta)*d phi; +<- ds2; +Assuming Default Metric. +Metric calculated By default. 0.16 sec + + 2 2 2 2 2 2 2 2 2 2 2 + ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi + +<- Find and Write Ricci Tensor; +Sqrt det of metric calculated. 0.21 sec +Volume calculated. 0.21 sec +Vector frame calculated From frame. 0.21 sec +Inverse metric calculated From metric. 0.21 sec +Frame connection calculated. 0.38 sec +Curvature calculated. 0.49 sec +Ricci tensor calculated From curvature. 0.54 sec +Ricci tensor: + + - 3*DF(a,t,2) +RIC = ---------------- + 00 a +\newpage + 2 + DF(a,t,2)*a + 2*DF(a,t) +RIC = -------------------------- + 11 2 + a + + 2 + DF(a,t,2)*a + 2*DF(a,t) +RIC = -------------------------- + 22 2 + a + + 2 + DF(a,t,2)*a + 2*DF(a,t) +RIC = -------------------------- + 33 2 + a + +<- RIC(_j,_k); + + - 3*DF(a,t,2) +j=0 k=0 : ---------------- + a + + 2 +j=1 k=1 : DF(a,t,2)*a + 2*DF(a,t) + + 2 2 +j=2 k=2 : r *(DF(a,t,2)*a + 2*DF(a,t) ) + + 2 2 2 +j=3 k=3 : SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) +\end{slisting} +Tracing messages demonstrate that \grg\ automatically +applied several built-in equations to obtain required value of +$ +\end_inset + +R +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. The metric is automatically assumed to be +Lorentzian $ +\end_inset + +g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + diag +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +(-1,1,1,1) +\begin_inset Formula $. +First \grg\ computed the frame connection 1-form $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $. +Next the curvature 2-form $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ was computed using +standard equation (\ref{omes}) on page \pageref{omes}. +Finally the Ricci tensor was obtained using +relation (\ref{rics}) on page \pageref{rics}. + +Second \grg\ task is similar to the first one: +\begin{listing} + Coordinates t,r,theta,phi; + Function a(t); + Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; + ds2; + Find and Write Ricci Tensor; +\end{listing} +The only difference is that now we work in the coordinate +formalism by assigning value to the metric rather than +frame. The frame is assumed to be holonomic automatically. +\begin{slisting} +<- Coordinates t, r, theta, phi; +<- Function a(t); +<- Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; +<- ds2; +Assuming Default Holonomic Frame. +Frame calculated By default. 0.11 sec + + 2 2 2 2 2 2 2 2 2 2 2 + ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi + +<- Find and Write Ricci Tensor; +Sqrt det of metric calculated. 0.22 sec +Volume calculated. 0.22 sec +Vector frame calculated From frame. 0.22 sec +Inverse metric calculated From metric. 0.27 sec +Frame connection calculated. 0.33 sec +Curvature calculated. 0.60 sec +Ricci tensor calculated From curvature. 0.60 sec +Ricci tensor: + + - 3*DF(a,t,2) +RIC = ---------------- + t t a + + 2 +RIC = DF(a,t,2)*a + 2*DF(a,t) + r r + + 2 2 +RIC = r *(DF(a,t,2)*a + 2*DF(a,t) ) + theta theta + + 2 2 2 +RIC = SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) + phi phi +\end{slisting} +Once again \grg\ uses the same built-in formulas to compute +the Ricci tensor but now all quantities have holonomic +indices instead of tetrad ones. + +Finally the third task demonstrate how \grg\ can be used +without built-in equations. Once again we use coordinate +formalism and declare two new objects the Christoffel symbols +\comm{Chr} and Ricci tensor \comm{Ric} +(since \grg\ is case sensitive they are different from the built-in +objects \comm{CHR} and \comm{RIC}). Next we use +well-known equations to compute these quantities +\begin{listing} + Coordinates t,r,theta,phi; + Function a(t); + Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; + ds2; + New Chr^a_b_c with s(2,3); + Chr(j,k,l)= 1/2*GI(j,m)*(@x(k)|G(l,m)+@x(l)|G(k,m)-@x(m)|G(k,l)); + New Ric_a_b with s(1,2); + Ric(j,k) = @x(n)|Chr(n,j,k) - @x(k)|Chr(n,j,n) + + Chr(n,m,n)*Chr(m,j,k) - Chr(n,m,k)*Chr(m,n,j); + Write Ric; +\end{listing} +The hard-copy of the corresponding session is +\begin{slisting} +<- Coordinates t, r, theta, phi; +<- Function a(t); +<- Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; +<- ds2; +Assuming Default Holonomic Frame. +Frame calculated By default. 0.16 sec + + 2 2 2 2 2 2 2 2 2 2 2 + ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi + +<- New Chr^a_b_c with s(2,3); +<- Chr(j,k,l)=1/2*GI(j,m)*(@x(k)|G(l,m)+@x(l)|G(k,m)-@x(m)|G(k,l)); +Inverse metric calculated From metric. 0.27 sec +<- New Ric_a_b with s(1,2); +<- Ric(j,k)=@x(n)|Chr(n,j,k)-@x(k)|Chr(n,j,n)+Chr(n,m,n)*Chr(m,j,k) + -Chr(n,m,k)*Chr(m,n,j); +<- Write Ric; +The Ric: + + - 3*DF(a,t,2) +Ric = ---------------- + t t a + + 2 +Ric = DF(a,t,2)*a + 2*DF(a,t) + r r +\newpage + 2 2 +Ric = r *(DF(a,t,2)*a + 2*DF(a,t) ) + theta theta + + 2 2 2 +Ric = SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) + phi phi +\end{slisting} + + + +\chapter{Formulas} +\parindent=0pt +\arraycolsep=1pt +\parskip=1.6mm plus 1mm minus 1mm + +This chapter describes in usual mathematical manner all \grg\ +built-in objects and formulas. The description is extremely short +since it is intended for reference only. +If not stated explicitly we use lower case greek letters +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +alpha +\end_layout + +\end_inset + +, +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +beta +\end_layout + +\end_inset + +,… +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ for +holonomic (coordinate) indices; $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + + a,b,c,d,m,n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ for +anholonomic frame indices and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + + i,j,k,l +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +for enumerating indices. + +To establish the relationship between \grg\ built-in object6s +and mathematical quantities we use the following notation +\[\mbox{\tt Frame Connection omega'a.b} = \omega^a{}_b +\] +This equality means that there is built-in object named +{\tt Frame Connection} having identifier {\tt omega} +which represent the frame connection 1-form $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $. +If the name is omitted then we deal with \emph{macro} object +(see page \pageref{macro}). The notation for indices +in the left-hand side of such equalities is the same +as in the {\tt New object} declaration and +is explained on page \pageref{indices}. + +This chapter contains not only definitions of all built-in +objects but all formulas which \grg\ knows and can apply +to find their value. If an object has +several formulas for its computation when each formula +is given together with the corresponding name which is printed +in the typewriter font. +In the case then an object has only one associated +formula the way name is usually omitted. + + +\section{Dimension and Signature} + +Let us denote the space-time dimensionality by $ +\end_inset + +d +\begin_inset Formula $ +and $ +\end_inset + +n +\begin_inset Formula $'th element of the signature specification +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + diag +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+1,-1,…) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ by $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + diag +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +n +\begin_inset Formula $ +($ +\end_inset + +n +\begin_inset Formula $ runs from 0 to $ +\end_inset + +d-1 +\begin_inset Formula $). + +There are several macro objects which gives access to +the dimension and signature +\object{dim}{d} +\object{sdiag.idim}{{\rm diag}_i} +\object{sgnt \mbox{=} sign}{s=\prod^{d-1}_{i=0}{\rm diag}_i} +\object{mpsgn}{{\rm diag}_0} +\object{pmsgn}{-{\rm diag}_0} + +The macros (two equivalent ones) which give access to +coordinates +\object{X\^m \mbox{=} x\^m}{x^\mu} + + +\section{Metric, Frame and Basis} + +Frame $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +theta +\end_layout + +\end_inset + +a +\begin_inset Formula $ and metric $ +\end_inset + +g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ plays the +fundamental role in \grg. Together they determine the +space-time line element +\begin{equation} +ds^2 = g_{ab}\,\theta^a\!\otimes\theta^b = + g_{\mu\nu}\,dx^\mu\!\otimes dx^\nu +\end{equation} + +The corresponding objects are +\object{Frame T'a}{\theta^a=h^a_\mu dx^\mu} +\object{Metric G.a.b}{g_{ab}} +and ``inverse'' objects are +\object{Vector Frame D.a}{\partial_a=h^\mu_a\partial_\mu} +\object{Inverse Metric GI'a'b}{g^{ab}} + +The frame can be computed by two ways. First, {\tt By default} +frame is assumed to be holonomic +\begin{equation} +\theta^a = dx^\alpha +\end{equation} +and {\tt From vector frame} +\begin{equation} +\theta^a= |h_a^\mu|^{-1} d x^\mu +\end{equation} + +The vector frame can be obtained {\tt From frame} +\begin{equation} +\partial_a= |h^a_\mu|^{-1} \partial_\mu +\end{equation} + +The metric can be computed {\tt By default} \index{Metric!default value} +\begin{equation} +g_{ab} = {\rm if}\ a=b\ {\rm then}\ {\rm diag}_a\ {\rm else}\ 0 +\end{equation} +or {\tt From inverse metric} +\begin{equation} +g_{ab} = |g^{ab}|^{-1} +\end{equation} + +The inverse metric can be computed {\tt From metric} +\begin{equation} +g^{ab} = |g_{ab}|^{-1} +\end{equation} + +The holonomic metric $ +\end_inset + +g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mu +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +nu +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and frame $ +\end_inset + +ha +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mu +\end_layout + +\end_inset + + +\begin_inset Formula $ +are given by the macro objects: +\object{g\_m\_n}{g_{\mu\nu}} +\object{gi\^m\^n}{g^{\mu\nu}} +\object{h'a\_m}{h^a_\mu} +\object{hi.a\^m}{h_a^\mu} + +The metric determinants and related densities +\object{Det of Metric detG}{g={\rm det}|g_{ab}|} +\object{Det of Holonomic Metric detg}{{\rm det}|g_{\mu\nu}|} +\object{Sqrt Det of Metric sdetG}{\sqrt{sg}} + +The volume $ +\end_inset + +d +\begin_inset Formula $-form +\object{Volume VOL}{\upsilon = \sqrt{sg}\,\theta^0\wedge\dots\wedge\,\theta^{d-1} +=\frac{1}{d!}{\cal E}_{a_0\dots a_{d-1}}\,\theta^{a_0}\wedge\dots\wedge\,\theta^{a_{d-1}}} + +The so called s-forms play the role of basis in the space of the +2-forms +\object{S-forms S'a'b}{S^{ab}=\theta^a\wedge\theta^b} + +The basis and corresponding inverse vector basis are used +when \grg\ works in the anholonomic mode +\seethis{See page \pageref{amode}.} +\object{Basis b'idim }{b^i=b^i_\mu dx^\mu} +\object{Vector Basis e.idim }{e_i=b_i^\mu\partial_\mu} +The basis can be computed {\tt From frame} +\begin{equation} +b^i=\theta^i +\end{equation} +or {\tt From vector basis} +\begin{equation} +b^i = |b_i^\mu|^{-1}dx^\mu +\end{equation} +The vector basis can be computed {\tt From basis} +\begin{equation} +e_i = |b^i_\mu|^{-1}\partial_\mu +\end{equation} + + +\section{Delta and Epsilon Symbols} + +Macro objects for Kronecker delta symbols +\object{del\^m\_n}{\delta^\mu_\nu} +\object{delh'a.b}{\delta^a_b} +and totally antisymmetric tensors +\object{eps.a.b.c.d}{{\cal E}_{abcd},\quad{\cal E}_{0123}=\sqrt{sg}} +\object{epsi'a'b'c'd}{{\cal E}^{abcd},\quad{\cal E}_{0123}=\frac{s}{\sqrt{sg}}} +\object{epsh\_m\_n\_k\_l}{{\cal E}_{\mu\nu\kappa\lambda},\quad{\cal E}_{0123}=\sqrt{s\,{\rm det}|g_{\mu\nu}|}} +\object{epsih\^m\^n\^k\^l}{{\cal E}^{\mu\nu\kappa\lambda},\quad{\cal E}_{0123}=\frac{s}{\sqrt{s\,{\rm det}|g_{\mu\nu}|}}} +The definition for epsilon-tensors is given for dimension 4. +The generalization to other dimensions is obvious. + + +\section{Dualization} + +We use the following definition for the dualization +operation. For any $ +\end_inset + +p +\begin_inset Formula $-form +\begin{equation} +\omega_p=\frac{1}{p!}\omega_{\alpha_1\dots\alpha_p}dx^{\alpha_1}\wedge +\dots\wedge dx^{\alpha_p} +\end{equation} +the dual $ +\end_inset + +(d-p) +\begin_inset Formula $-form is +\begin{equation} +*\omega_p=\frac{1}{p!(d-p)!}{\cal E}_{\alpha_1\dots\alpha_{d-p}} +{}^{\beta_1\dots\beta_p}\,\omega_{\beta_1\dots\beta_p}\, +dx^{\alpha_1}\wedge\dots\wedge dx^{\alpha_{d-p}} +\end{equation} + +The equivalent relation which also uniquely defines the $ +\end_inset + +* +\begin_inset Formula $ +operation is +\begin{equation} +*(\theta^{a_1}\wedge\dots\wedge \theta^{a_p}) = +(-1)^{p(d-p)} \partial_{a_p}\ipr\dots\partial_{a_1}\ipr\,\upsilon +\end{equation} + +With such convention we have the following identities +\begin{eqnarray} +**\omega_p &=& s(-1)^{p(d-p)}\,\omega_p \\[0.5mm] +*\upsilon &=& s \\[0.5mm] +*1 &=& \upsilon +\end{eqnarray} + + +\section{Spinors} +\label{spinors1} + +The notion of spinors in \grg\ is restricted to + 4-dimensional spaces of Lorentzian signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +or $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ only. In this section the upper sign relates to the +signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and lower one to +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +In addition to work with spinors the metric must have the following +form which we call the \emph{standard null metric} \index{Metric!Standard Null} +\index{Standard null metric}\index{Spinors}\index{Spinors!Standard null metric} +\begin{equation} +g_{ab}=g^{ab}=\pm\left(\begin{array}{rrrr} +0 & -1 & 0 & 0 \\ +-1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 \\ +0 & 0 & 1 & 0 +\end{array}\right) +\end{equation} +Such value of the metric can be established by the command +\cmdind{Null Metric} +{\tt Null metric;}. + +Therefore the line-element for spinorial formalism has the form +\begin{equation} +ds^2 = \pm(-\theta^0\!\otimes\theta^1 +-\theta^1\!\otimes\theta^0 ++\theta^2\!\otimes\theta^3 ++\theta^3\!\otimes\theta^2) +\end{equation} + +We require also the conjugation rules for this null tetrad (frame) be +\begin{equation} +\overline{\theta^0}=\theta^0,\quad +\overline{\theta^1}=\theta^1,\quad +\overline{\theta^2}=\theta^3,\quad +\overline{\theta^3}=\theta^2 +\end{equation} + +For such a metric and frame we fix sigma-matrices in the +following form \index{Sigma matrices} +\begin{eqnarray} \label{sigma} +&&\sigma_0{}^{1\dot{1}}= +\sigma_1{}^{0\dot{0}}= +\sigma_2{}^{1\dot{0}}= +\sigma_3{}^{0\dot{1}}=1 \\[1mm] && +\sigma^0{}_{1\dot{1}}= +\sigma^1{}_{0\dot{0}}= +\sigma^2{}_{1\dot{0}}= +\sigma^3{}_{0\dot{1}}=\mp1 +\end{eqnarray} + +The sigma-matrices obey the rules +\begin{eqnarray} +g_{mn}\sigma^m\!{}_{A\dot B}\sigma^n\!{}_{C\dot D} &=& +\mp \epsilon_{AC}\epsilon_{\dot B\dot D} \\[1mm] +\sigma^{aM\dot N}\sigma^b\!{}_{M\dot N} &=& \mp g^{ab} +\end{eqnarray} + +The antisymmetric SL(2,C) spinor metric +\begin{equation} +\epsilon_{AB}=\epsilon^{AB} +=\epsilon_{\dot A\dot B} +=\epsilon^{\dot A\dot B}= +\left(\begin{array}{rr} +0 & 1 \\ +-1 & 0 +\end{array}\right) +\end{equation} +can be used to raise and lower spinor indices +\begin{equation} +\varphi^A=\varphi_B\,\epsilon^{BA},\qquad +\varphi_A=\epsilon_{AB}\,\varphi^B +\end{equation} + +The following macro objects represent standard +spinorial quantities +\object{DEL'A.B}{\delta^A_B} +\object{EPS.A.B}{\epsilon_{AB}} +\object{EPSI'A'B}{\epsilon^{AB}} +\object{sigma'a.A.B\cc}{\sigma^a\!{}_{A\dot B}} +\object{sigmai.a'A'B\cc}{\sigma_a{}^{A\dot B}} + +The relationship between tensors and spinors +is established by the sigma-matrices +\begin{eqnarray} +X^a &\tsst& X^{A\dot A}=A^a\sigma_a{}^{A\dot A} \\ +X_a &\tsst& X_{A\dot A}=A_a\sigma^a\!{}_{A\dot A} +\end{eqnarray} +where sigma-matrices are given by Eq. (\ref{sigma}) +We shall denote similar equations by the sign $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +tsst +\end_layout + +\end_inset + + +\begin_inset Formula $ +conserving alphabetical relationship between tensor indices in the +left-hand side and spinorial one in the right-hand side: +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + + a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +tsst +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + + b +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +tsst +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset Formula $. + +There is one quite important special case. Any real +antisymmetric tensor $ +\end_inset + +X +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ are equivalent to the +pair of conjugated irreducible (symmetric) spinors +\begin{eqnarray} +&& X_{ab}=X_{[ab]} \tsst X_{A\dot AB\dot B}= +\epsilon_{AB} X_{\dot A\dot B} + \epsilon_{\dot A\dot B}X_{AB} +\nonumber\\[1mm] +&& X_{AB}=\frac{1}{2}X_{A\dot AB\dot B}\epsilon^{\dot A\dot B},\ + X_{\dot A\dot B}=\frac{1}{2}X_{A\dot AB\dot B}\epsilon^{AB} +\end{eqnarray} +The explicit form of these relations for the sigma-matrices +(\ref{sigma}) is +\begin{equation} +\begin{array}{rclrcl} +X_0 &=& X_{13} & X_{\dot0} &=& X_{12} \\[1mm] +X_1 &=&-\frac{1}{2}(X_{01}-X_{23})\qquad & X_{\dot1} &=& +-\frac{1}{2}(X_{01}+X_{23}) \\[1mm] +X_2 &=& -X_{02} & X_{\dot2} &=& -X_{03} +\end{array}\label{asys} +\end{equation} +and the ``inverse'' relation +\begin{equation} +\begin{array}{rclrcl} +X_{01} &=& -X_1-X_{\dot1},\qquad & X_{23} &=& X_1-X_{\dot1}, \\[1mm] +X_{02} &=& -X_2, & X_{12} &=& X_{\dot0}, \\[1mm] +X_{03} &=& -X_{\dot 2}, & X_{13} &=& X_0 +\end{array}\label{asyt} +\end{equation} + +We shall apply the relations (\ref{asys}) and (\ref{asyt}) to various +antisymmetric quantities. In particular the {\tt Spinorial S-forms} +\object{Undotted S-forms SU.AB}{S_{AB}} +\object{Dotted S-forms SD.AB\cc}{S_{\dot A\dot B}} +The {\tt Standard way} to compute these quantities uses +relations (\ref{asys}) +\begin{equation} + S_{ab}=\theta_a\wedge\theta_b \tsst +\epsilon_{AB} S_{\dot A\dot B} + \epsilon_{\dot A\dot B}S_{AB} +\end{equation} +Spinorial S-forms are self dual +\begin{equation} +*S_{AB}=iS_{AB},\qquad +*S_{\dot A\dot B}=-iS_{\dot A\dot B} +\end{equation} +and exteriorly orthogonal +\begin{equation} +S_{AB}\wedge S_{CD}=-\frac{i}2\upsilon(\epsilon_{AC}\epsilon_{BD}+ +\epsilon_{AD}\epsilon_{BC}),\quad S_{AB}\wedge S_{\dot C\dot D}=0 +\end{equation} + +There is one subtle pint concerning tensor quantities in the +spinorial formalism. Since spinorial null tetrad is complex +with the conjugation rule $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +overline +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +theta +\end_layout + +\end_inset + +2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +theta +\end_layout + +\end_inset + +3 +\begin_inset Formula $ +all tensor quantities represented in this frame also becomes +complex with similar conjugation rules for any tensor index. +There is special macro object {\tt cci} which performs such +``index conjugation'': {\tt cci{0}=0}, {\tt cci(1)=1}, +{\tt cci{2}=3}, {\tt cci(3)=2}. Therefore the correct expression +for the $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +overline +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +theta +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is {\tt \cc T(cci(a))} but not +{\tt \cc T(a)}. + + + +\section{Connection, Torsion and Nonmetricity} +\label{conn1} + +Covariant derivatives and differentials for +quantities having frame and coordinate indices are +\begin{eqnarray} +DX^a{}_b &=& dX^a{}_b ++ \omega^a{}_m\wedge X^m{}_b - \omega^m{}_b\wedge X^a{}_m \\[1mm] +DX^\mu{}_\nu &=& dX^\mu{}_\nu ++ \Gamma^\mu{}_\pi\wedge X^\pi{}_\nu - \Gamma^\pi{}_\nu\wedge X^\mu{}_\pi +\end{eqnarray} + +The corresponding built-in connection 1-forms are +\object{Frame Connection omega'a.b}{\omega^a{}_b=\omega^a{}_{b\mu}dx^\mu} +\object{Holonomic Connection GAMMA\^m\_n} +{\Gamma^\mu{}_\nu=\Gamma^\mu{}_{\nu\pi}dx^\pi} + +Frame connection can be computed {\tt From holonomic connection} +\begin{equation} +\omega^a{}_b = \Gamma^a{}_b + dh^\mu_b\,h^a_\mu +\end{equation} +and inversely holonomic connection can be obtained +{\tt From frame connection} +\begin{equation} +\Gamma^\mu{}_\nu=\omega^\mu{}_\nu + dh^b_\nu\,h^\mu_b +\end{equation} + +By default these connections are Riemannian (i.e. symmetric and +metric compatible). To work with nonsymmetric +connection with torsion the switch \comm{TORSION}\swinda{TORSION} +must be turned on. Then the torsion 2-form is +\object{Torsion THETA'a}{\Theta^a=\frac12Q^a{}_{pq}S^{pq},\quad +Q^a{}_{bc}=\Gamma^a{}_{bc}-\Gamma^a_{cb}} +Finally to work with non metric-compatible +spaces with nonmetricity the switch \comm{NONMETR}\swinda{NONMETR} +must be turned on. The nonmetricity 1-form is +\object{Nonmetricity N.a.b}{N_{ab}=N_{ab\mu}dx^\mu, +\quad N_{ab\mu}=-\nabla_\mu g_{ab}} +In general any torsion or nonmetricity related object is +defined iff the corresponding switch is on. + +If either \comm{TORSION} or \comm{NONMETR} is on then Riemannian +versions of the connection 1-forms are available as well +\object{Riemann Frame Connection romega'a.b} +{\rim{\omega}{}^a{}_b} +\object{Riemann Holonomic Connection RGAMMA\^m\_n} +{\rim{\Gamma}{}^\mu{}_\nu} + +The Riemann holonomic connection can be obtained +{\tt From Riemann frame connection} +\begin{equation} +\rim{\Gamma}{}^\mu{}_\nu=\rim{\omega}{}^\mu{}_\nu + dh^b_\nu\,h^\mu_b +\end{equation} + + + +If torsion is nonzero but nonmetricity vanishes +(\comm{TORSION} is on, \comm{NONMETR} is off) then +the difference between the connection and Riemann connection +is called the contorsion 1-form +\object{Contorsion KQ'a.b}{\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b= +\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_{b\mu}dx^\mu= +\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} + +If nonmetricity is nonzero but torsion vanishes +(\comm{TORSION} is off, \comm{NONMETR} is on) then +the difference between the connection and Riemann connection +is called the nonmetricity defect +\object{Nonmetricity Defect KN'a.b} +{\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b= +\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_{b\mu}dx^\mu= +\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} + +Finally if both torsion and nonmetricity are nonzero +(\comm{TORSION} and \comm{NONMETR} are on) then we +\object{Connection Defect K'a.b} +{K^a{}_b=K^a{}_{b\mu}dx^\mu= +\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} +\begin{equation} +K^a{}_b = \stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b ++ \stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b +\end{equation} + + +For the sake of convenience we introduce also macro objects +which compute the usual Christoffel symbols +\object{CHR\^m\_n\_p }{ \{{}^\mu_{\nu\pi}\} = +\frac{1}{2}g^{\mu\tau}(\partial_\pi g_{\nu\tau} ++\partial_\nu g_{\pi\tau} +-\partial_\tau g_{\nu\pi})} +\object{CHRF\_m\_n\_p }{ [{}_{\mu},_{\nu\pi}] = +\frac{1}{2}(\partial_\pi g_{\nu\mu} ++\partial_\nu g_{\pi\mu} +-\partial_\mu g_{\nu\pi})} +\object{CHRT\_m }{ \{{}^\pi_{\pi\mu}\} = +\frac{1}{2{\rm det}|g_{\alpha\beta}|}\partial_\mu\left( +{\rm det}|g_{\alpha\beta}|\right)} + +The connection, frame, metric, torsion and nonmetricity are +related to each other by the so called structural equations +which in the most general case read +\begin{eqnarray} +&& D\theta^a + \Theta^a = 0 \nonumber\\[2mm] +&& Dg_{ab} + N_{ab} = 0 \label{str0} +\end{eqnarray} +or in the equivalent ``explicit'' form +\begin{equation} +\begin{array}{ll} +\omega^a{}_b\wedge\theta^b = -t^a,\qquad & t^a=d\theta^a+\Theta^a,\\[2mm] +\omega_{ab}+\omega_{ba} = n_{ab},\qquad & n_{ab}=dg_{ab}+N_{ab} \label{str} +\end{array} +\end{equation} + +The solution to equations (\ref{str}) are given by the relation +\begin{equation} +\omega^a{}_b = +\frac{1}{2}\left[ -\partial^a\ipr t_b + \partial_b\ipr t^a + n^a{}_b ++\big(\partial^a\ipr(\partial_b\ipr t_c-n_{bc}) ++\partial_b\ipr n^a{}_c\big)\theta^c\right] \label{solstr} +\end{equation} + +For various specific values of $ +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + +ta +\begin_inset Formula $ equations +(\ref{str}) and (\ref{solstr}) can be used for different purposes. + +In the most general case (\ref{solstr}) is the {\tt Standard way} to +compute connection 1-form $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $. +The torsion and nonmetricity are included in +these equations depending on the switches \comm{TORSION} and +\comm{NONMETR}. + +The same equation (\ref{solstr}) with $ +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +=dg +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + +ta=d +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +theta +\end_layout + +\end_inset + +a +\begin_inset Formula $ is the {\tt Standard way} to find Riemann +frame connection $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rim{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $. + +If torsion is nonzero then $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ can be computed +{\tt From contorsion} +\begin{equation} +\omega^a{}_b = \rim{\omega}{}^a{}_b ++ \stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b \label{a1} +\end{equation} +where $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rim{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ is given by Eq. (\ref{solstr}). + +Similarly if nonmetricity is nonzero then $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ can be computed +{\tt From nonmetricity defect} +\begin{equation} +\omega^a{}_b = \rim{\omega}{}^a{}_b ++ \stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b \label{a2} +\end{equation} +where $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rim{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ is given by Eq. (\ref{solstr}). + +Finally if both torsion and nonmetricity are +nonzero then $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ can be computed +{\tt From connection defect} +\begin{equation} +\omega^a{}_b = \rim{\omega}{}^a{}_b + K^a{}_b \label{a3} +\end{equation} +where $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rim{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ is given by Eq. (\ref{solstr}). + +The Riemannian part of connection in Eqs. (\ref{a1}), +(\ref{a2}), (\ref{a3}) are directly computed by Eq. (\ref{solstr}) +(not via the object \comm{romega}). + +The contorsion $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptscriptstyle +\end_layout + +\end_inset + + Q +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +K +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +! +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ +is obtained {\tt From torsion} by (\ref{solstr}) +with $ +\end_inset + +ta= +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Theta +\end_layout + +\end_inset + +a +\begin_inset Formula $, $ +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +=0 +\begin_inset Formula $. + +The nonmetricity defect $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptscriptstyle +\end_layout + +\end_inset + + N +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +K +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +! +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $ +is obtained {\tt From nonmetricity} by (\ref{solstr}) +with $ +\end_inset + +ta=0 +\begin_inset Formula $, $ +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +=N +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +Analogously (\ref{solstr}) with $ +\end_inset + +ta= +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Theta +\end_layout + +\end_inset + +a +\begin_inset Formula $, $ +\end_inset + +n +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +=N +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +is the {\tt Standard way} to compute the connection defect $ +\end_inset + +Ka +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +b +\begin_inset Formula $. + +The torsion $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Theta +\end_layout + +\end_inset + +a +\begin_inset Formula $ can be calculated {\tt From contorsion} +\begin{equation} +\Theta^a = -\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b\wedge\theta^b +\end{equation} +or {\tt From connection defect} +\begin{equation} +\Theta^a = -K^a{}_b\wedge\theta^b +\end{equation} + +The nonmetricity $ +\end_inset + +N +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be computed {\tt From nonmetricity defect} +\begin{equation} +N_{ab} = \stackrel{\scriptscriptstyle N}{K}_{ab}+ +\stackrel{\scriptscriptstyle N}{K}_{ba} +\end{equation} +or {\tt From connection defect} +\begin{equation} +N_{ab} = K_{ab}+K_{ba} +\end{equation} + + +\section{Spinorial Connection and Torsion} + +Spinorial connection is defined in \grg\ iff nonmetricity +is zero and switch \comm{NONMETR} is turned off. +The upper sign in this section correspond to the signature +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ while lower one to the signature +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +Spinorial connection is defined by the equation +\begin{equation} +DX^A_{\dot B} = dX^A{}_{\dot B} +\mp\omega^A{}_M\,X^M{}_{\dot B} +\pm\omega^{\dot M}{}_{\dot B}\,X^A{}_{\dot M} +\end{equation} +where due to antisymmetry of the frame connection +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +[ab] +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ we have {\tt Spinorial connection} +1-forms +\begin{equation} +\omega_{ab} \tsst +\epsilon_{AB} \omega_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \omega_{AB} +\end{equation} +\object{Undotted Connection omegau.AB}{\omega_{AB}} +\object{Dotted Connection omegad.AB\cc}{\omega_{\dot A\dot B}} + +The spinorial connection 1-forms +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +can be calculated {\tt From frame connection} by the +standard spinor $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +tsst +\end_layout + +\end_inset + + +\begin_inset Formula $ tensor relation (\ref{asys}). + +Inversely the frame connection $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be +found {\tt From spinorial connection} by relation (\ref{asyt}). + +Since $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is real the spinorial equivalents +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be computed from +each other {\tt By conjugation} +\begin{equation} +\omega_{\dot A\dot B}=\overline{\omega_{AB}},\qquad +\omega_{AB}=\overline{\omega_{\dot A\dot B}} +\end{equation} + +If torsion is nonzero (\comm{TORSION} is on) when we have +in addition the {\tt Riemann spinorial connection} +\object{Riemann Undotted Connection romegau.AB}{\rim{\omega}_{AB}} +\object{Riemann Dotted Connection romegad.AB\cc}{\rim{\omega}_{\dot A\dot B}} + +The Riemann spinorial connection $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rim{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +can be calculated by {\tt Standard way} +\begin{equation} +\stackrel{{\scriptscriptstyle\{\}}}{\omega}_{AB}= \label{ssolver} +\pm i*[ d S_{AB}\wedge\theta_{C\dot C} + -\epsilon_{C(A} d S_{B)M}\wedge \theta^M_{\ \ \dot C}]\theta^{C\dot C} +\end{equation} +The conjugated relation is used for $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rim{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +The {\tt Spinorial contorsion} 1-forms +\object{Undotted Contorsion KU.AB}{\stackrel{\scriptscriptstyle Q}{K}\!{}_{AB}} +\object{Dotted Contorsion KD.AB\cc}{\stackrel{\scriptscriptstyle Q}{K}\!{}_{\dot A\dot B}} +are the spinorial analogues of the contorsion 1-form +\begin{equation} +\stackrel{\scriptscriptstyle Q}{K}_{ab} \tsst +\epsilon_{AB} \stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \stackrel{\scriptscriptstyle Q}{K}_{AB} +\end{equation} + +The spinorial contorsion 1-forms +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptscriptstyle +\end_layout + +\end_inset + + Q +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +K +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptscriptstyle +\end_layout + +\end_inset + + Q +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +K +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +can be calculated {\tt From contorsion} by the +standard spinor $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +tsst +\end_layout + +\end_inset + + +\begin_inset Formula $ tensor relation (\ref{asys}). + +Inversely the contorsion $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptscriptstyle +\end_layout + +\end_inset + + Q +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +K +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be +found {\tt From spinorial contorsion} by relation (\ref{asyt}). + +The spinorial equivalents +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptscriptstyle +\end_layout + +\end_inset + + Q +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +K +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptscriptstyle +\end_layout + +\end_inset + + Q +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +K +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +can be computed from +each other {\tt By conjugation} +\begin{equation} +\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}=\overline{\stackrel{\scriptscriptstyle Q}{K}_{AB}},\qquad +\stackrel{\scriptscriptstyle Q}{K}_{AB}=\overline{\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}} +\end{equation} + +The {\tt Standard way} to find $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is +\begin{equation} +\omega_{AB} = \rim{\omega}_{AB}+\stackrel{\scriptscriptstyle Q}{K}_{AB} +\end{equation} +where $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rim{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is given directly by Eq. (\ref{ssolver}). +The conjugated Eq. is used for $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + + +\section{Curvature} + +The curvature 2-form +\object{Curvature OMEGA'a.b}{\Omega^a{}_b= +\frac{1}{2}R^a_{bcd}\,S^{cd}} +can be computed {\tt By standard way} +\begin{equation} +\Omega^a{}_b = d\omega^a{}_b + \omega^a{}_n \wedge \omega^n{}_b \label{omes} +\end{equation} + +The Riemann curvature tensor is given by the relation +\object{Riemann Tensor RIM'a.b.c.d}{R^a{}_{bcd}= +\partial_d\ipr\partial_c\ipr\Omega^a{}_b} + +The Ricci tensor +\object{Ricci Tensor RIC.a.b}{R_{ab}} +can be computed {\tt From Curvature} +\begin{equation} +R_{ab} = \partial_b\ipr\partial_m\ipr\Omega^m{}_a \label{rics} +\end{equation} +or {\tt From Riemann tensor} +\begin{equation} +R_{ab} = R^m{}_{amb} +\end{equation} + +The +\object{Scalar Curvature RR}{R} +can be computed {\tt From Ricci Tensor} +\begin{equation} +R = R_{mn}\,g^{mn} +\end{equation} + +The Einstein tensor is given by the relation +\object{Einstein Tensor GT.a.b}{G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R} + +If nonmetricity is nonzero (\comm{NONMETR} is on) then we have +\object{Homothetic Curvature OMEGAH}{\OO{h}} +\object{A-Ricci Tensor RICA.a.b}{\RR{A}_{ab}} +\object{S-Ricci Tensor RICS.a.b}{\RR{S}_{ab}} + +They can be calculated {\tt From curvature} by the +relations +\begin{equation} +\OO{h}=\Omega^n{}_n +\end{equation} +\begin{equation} +\RR{A}_{ab}= \partial_b\ipr\partial^m\ipr\Omega_{[ma]} +\end{equation} +\begin{equation} +\RR{S}_{ab}= \partial_b\ipr\partial^m\ipr\Omega_{(ma)} +\end{equation} +and the scalar curvature can be computed {\tt From A-Ricci tensor} +\begin{equation} +R = \RR{A}_{mn}g^{mn} +\end{equation} + + +\section{Spinorial Curvature} + +Spinorial curvature is defined in \grg\ iff nonmetricity +is zero and switch \comm{NONMETR} is turned off. +The upper sign in this section correspond to the signature +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ while lower one to the signature +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +The {\tt Spinorial curvature} 2-forms +\object{Undotted Curvature OMEGAU.AB}{\Omega_{AB}} +\object{Dotted Curvature OMEGAD.AB\cc}{\Omega_{\dot A\dot B}} +is related to the curvature 2-form $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ by the standard +relation +\begin{equation} +\Omega_{ab} \tsst +\epsilon_{AB} \Omega_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \Omega_{AB} +\end{equation} + +The spinorial curvature 1-forms +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +can be calculated {\tt From curvature} by the +relation (\ref{asys}). + +The frame curvature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be +found {\tt From spinorial curvature} by relation (\ref{asyt}). + +The $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be +computed from each other {\tt By conjugation} +\begin{equation} +\Omega_{\dot A\dot B}=\overline{\Omega_{AB}},\qquad +\Omega_{AB}=\overline{\Omega_{\dot A\dot B}} +\end{equation} + +The {\tt Standard way} to calculate $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is +\begin{equation} +\Omega_{AB} = d\omega_{AB} \pm \omega_A{}^M\wedge\omega_{MB} +\end{equation} +The conjugated relation is used for $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + + +\section{Curvature Decomposition} + +In general curvature consists of 11 irreducible pieces. +If nonmetricity is nonzero then one can +perform decomposition +\begin{equation} +R_{abcd}=\RR{A}_{abcd}+\RR{S}_{abcd},\qquad +\RR{A}_{abcd}=R_{[ab]cd},\qquad +\RR{S}_{abcd}=R_{(ab)cd} +\end{equation} +Here the S-part of the curvature vanishes identically if +nonmetricity is zero and we consider further decomposition +of A and S parts independently. + +First we consider the A-part of the curvature. It can be +decomposed into 6 pieces +\begin{equation} +\RR{A}_{abcd} = +\RR{w}_{abcd}+ +\RR{c}_{abcd}+ +\RR{r}_{abcd}+ +\RR{a}_{abcd}+ +\RR{b}_{abcd}+ +\RR{d}_{abcd} +\end{equation} +Here first three terms are the well-known irreducible pieces +of the Riemannian curvature while last three terms vanish if +torsion is zero. The corresponding 2-forms are +\object{Weyl 2-form OMW.a.b } +{\OO{w}_{ab} = \frac12 \RR{w}_{abcd}\,S^{cd}} +\object{Traceless Ricci 2-form OMC.a.b } +{\OO{c}_{ab} = \frac12 \RR{c}_{abcd}\,S^{cd}} +\object{Scalar Curvature 2-form OMR.a.b } +{\OO{r}_{ab} = \frac12 \RR{r}_{abcd}\,S^{cd}} +\object{Ricanti 2-form OMA.a.b } +{\OO{a}_{ab} = \frac12 \RR{a}_{abcd}\,S^{cd}} +\object{Traceless Deviation 2-form OMB.a.b } +{\OO{b}_{ab} = \frac12 \RR{b}_{abcd}\,S^{cd}} +\object{Antisymmetric Curvature 2-form OMD.a.b } +{\OO{d}_{ab} = \frac12 \RR{d}_{abcd}\,S^{cd}} + +The {\tt Standard way} to find these quantities is given +by the following formulas. +\begin{equation} +\OO{r}_{ab} = \frac{1}{d(d-1)}R\,S_{ab} +\end{equation} +\begin{equation} +\OO{c}_{ab} = \frac{1}{(d-2)}\left[ +C_{am}\,\theta^m\!\wedge\theta_b +-C_{bm}\,\theta^m\!\wedge\theta_a\right],\quad +C_{ab}=\RR{A}_{(ab)}-\frac{1}{d}g_{ab}R +\end{equation} +\begin{equation} +\OO{a}_{ab} = \frac{1}{(d-2)}\left[ +A_{am}\,\theta^m\!\wedge\theta_b +-A_{bm}\,\theta^m\!\wedge\theta_a\right],\quad +A_{ab}=\RR{A}_{[ab]} +\end{equation} +\begin{equation} +\OO{d}_{ab} = \frac{1}{12}\partial_b\ipr\partial_a\ipr +(\OO{A}_{mn}\wedge\theta^m\!\wedge\theta^n) +\end{equation} +\begin{equation} +\OO{b}_{ab} =\frac{1}{2}\left[ +\partial_b\ipr(\theta^m\!\wedge\OO{A0}_{am}) +-\partial_a\ipr(\theta^m\!\wedge\OO{A0}_{bm}) +\right] +\end{equation} +where +\[\OO{A0}_{ab} = +\OO{A}_{ab} +-\OO{c}_{ab} +-\OO{r}_{ab} +-\OO{a}_{ab} +-\OO{d}_{ab} +\] +And finally +\begin{equation} +\OO{w}_{ab} = +\OO{A}_{ab} +-\OO{c}_{ab} +-\OO{r}_{ab} +-\OO{a}_{ab} +-\OO{b}_{ab} +-\OO{d}_{ab} +\end{equation} + +If $ +\end_inset + +d=2 +\begin_inset Formula $ then $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +OO{ +\end_layout + +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ turns out to be irreducible and +coincides with the scalar curvature irreducible piece +\begin{equation} +\OO{A}_{ab} = \OO{r}_{ab} +\end{equation} + +Now we consider the decomposition of the S curvature part which +is nonzero iff nonmetricity is nonzero. First we consider +the case $ +\end_inset + +d +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +geq +\end_layout + +\end_inset + +3 +\begin_inset Formula $. In this case we have 5 irreducible components +\begin{equation} +\RR{S}_{abcd} = +\RR{h}_{abcd}+ +\RR{sc}_{abcd}+ +\RR{sa}_{abcd}+ +\RR{v}_{abcd}+ +\RR{u}_{abcd} +\end{equation} + +The corresponding 2-forms are +\object{Homothetic Curvature 2-form OSH.a.b } +{\OO{h}_{ab} = \frac12 \RR{h}_{abcd}\,S^{cd}} +\object{Antisymmetric S-Ricci 2-form OSA.a.b } +{\OO{sa}_{ab} = \frac12 \RR{sa}_{abcd}\,S^{cd}} +\object{Traceless S-Ricci 2-form OSC.a.b } +{\OO{sc}_{ab} = \frac12 \RR{sc}_{abcd}\,S^{cd}} +\object{Antisymmetric S-Curvature 2-form OSV.a.b } +{\OO{v}_{ab} = \frac12 \RR{v}_{abcd}\,S^{cd}} +\object{Symmetric S-Curvature 2-form OSU.a.b } +{\OO{u}_{ab} = \frac12 \RR{u}_{abcd}\,S^{cd}} + + +The {\tt Standard way} to compute the decomposition is +\begin{equation} +\OO{h}_{ab}=-\frac{1}{(d^2-4)}\left[ +\theta_a\wedge\partial_b\ipr\OO{h}{} ++\theta_b\wedge\partial_a\ipr\OO{h}{} +-g_{ab}\OO{h}{}d\right] +\end{equation} +\begin{equation} +\OO{sa}_{ab} =\frac{d}{(d^2-4)}\left[ +\theta_a\wedge(\RR{S}_{[bm]}\wedge\theta^m) ++\theta_b\wedge(\RR{S}_{[am]}\wedge\theta^m) +-\frac{2}{d}g_{ab}\,\RR{S}_{cd}S^{cd}\right] +\end{equation} +\begin{equation} +\OO{sc}_{ab} =\frac{1}{d}\left[ +\theta_a\wedge(\RR{S}_{(bm)}\wedge\theta^m) ++\theta_b\wedge(\RR{S}_{(am)}\wedge\theta^m)\right] \label{ccc} +\end{equation} +\begin{equation} +\OO{v}_{ab} = \frac{1}{4}\left[ +\partial_a\ipr(\OO{S0}_{bm}\wedge\theta^m) ++\partial_b\ipr(\OO{S0}_{am}\wedge\theta^m)\right] +\end{equation} +where +\[\OO{S0}_{ab} = +\OO{S}_{ab} +-\OO{h}_{ab} +-\OO{sa}_{ab} +-\OO{sc}_{ab} +\] +And finally +\begin{equation} +\OO{u}_{ab} = +\OO{S}_{ab} +-\OO{h}_{ab} +-\OO{sa}_{ab} +-\OO{sc}_{ab} +-\OO{v}_{ab} +\end{equation} + +If $ +\end_inset + +d=2 +\begin_inset Formula $ then only the h- and sc-components are nonzero. +The $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +OO{ +\end_layout + +\end_inset + +sc +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ are given by (\ref{ccc}) and +\begin{equation} +\OO{h}_{ab} = \OO{S}_{ab}-\OO{sc}_{ab} +\end{equation} + +\begin{center} +\begin{tabular}{|c|c|c|} +\hline object & exists if & and has $n$ components \\ +\hline +\vv$R_{abcd}$ & & $\frac{d^3(d-1)}{2}$ \\[1mm] +\hline\vv$\rim{R}{}_{abcd}$ & & $\frac{d^2(d^2-1)}{12}$ \\[1mm] +\hline\vv$\RR{A}_{abcd}$ & & $\frac{d^2(d-1)^2}{4}$ \\[1mm] +\hline\vv$\RR{S}_{abcd}$ & & $\frac{d^2(d^2-1)}{4}$ \\[1mm] +\hline\vv$\RR{w}_{abcd}$ & $d\geq4$ & $\frac{d(d+1)(d+2)(d-3)}{12}$ \\ +\vv$\RR{c}_{abcd}$ & $d\geq3$ & $\frac{(d+2)(d-1)}{2}$ \\ +\vv$\RR{r}_{abcd}$ & & $1$ \\[1mm] +\hline\vv$\RR{a}_{abcd}$ & $d\geq3$ & $\frac{d(d-1)}{2}$ \\ +\vv$\RR{b}_{abcd}$ & $d\geq4$ & $\frac{d(d-1)(d+2)(d-3)}{8}$ \\ +\vv$\RR{d}_{abcd}$ & $d\geq4$ & $\frac{d(d-1)(d-2)(d-3)}{24}$ \\[1mm] +\hline\vv$\RR{h}_{abcd}$ & & $\frac{d(d-1)}{2}$ \\ +\vv$\RR{sa}_{abcd}$ & $d\geq3$ & $\frac{d(d-1)}{2}$ \\ +\vv$\RR{sc}_{abcd}$ & & $\frac{(d+2)(d-1)}{2}$ \\ +\vv$\RR{v}_{abcd}$ & $d\geq4$ & $\frac{d(d+2)(d-1)(d-3)}{8}$ \\ +\vv$\RR{u}_{abcd}$ & $d\geq3$ & $\frac{(d-2)(d+4)(d^2-1)}{8}$ \\[1mm] +\hline +\end{tabular} +\end{center} + + + +\section{Spinorial Curvature Decomposition} + +Spinorial curvature is defined in \grg\ iff nonmetricity +is zero and switch \comm{NONMETR} is turned off. +The upper sign in this section correspond to the signature +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ while lower one to the signature +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +Let us introduce the spinorial analog of the curvature tensor +\begin{eqnarray} +R_{abcd}&\tsst& +\ \ R_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}} ++R_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD} \nonumber\\[1mm] +&&+R_{AB\dot{C}\dot{D}}\epsilon_{\dot{A}\dot{B}}\epsilon_{CD} ++R_{\dot{A}\dot{B} CD}\epsilon_{AB}\epsilon_{\dot{C}\dot{D}}, \\[1.5mm] +R_{ABCD}&=&-i*(\Omega_{AB}\wedge S_{CD}),\ \ +R_{AB\dot{C}\dot{D}}\ =\ i*(\Omega_{AB}\wedge S_{\dot{C}\dot{D}})\\[1.5mm] +R_{\dot{A}\dot{B}\dot{C}\dot{D}}&=&\overline{R_{ABCD}},\ \ +R_{\dot{A}\dot{B} CD}\ =\ \overline{R_{AB\dot{C}\dot{D}}} +\end{eqnarray} + +The quantities $ +\end_inset + +R +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ABCD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + +R +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + C +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + D +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be used to compute +the {\tt Curvature spinors} ($ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +equiv +\end_layout + +\end_inset + + +\begin_inset Formula $ {\tt Curvature components}) +\object{Weyl Spinor RW.ABCD}{C_{ABCD}} +\object{Traceless Ricci Spinor RC.AB.CD\cc}{C_{AB\dot C\dot D}} +\object{Scalar Curvature RR}{R} +\object{Ricanti Spinor RA.AB}{A_{AB}} +\object{Traceless Deviation Spinor RB.AB.CD\cc}{B_{AB\dot C\dot D}} +\object{Scalar Deviation RD}{D} +All these spinors are irreducible (totally symmetric). + +Weyl spinor $ +\end_inset + +C +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ABCD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ {\tt From spinor curvature} is +\begin{eqnarray} +C_{abcd}&\tsst& C_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}} + +C_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD} \\[1mm] +C_{ABCD}&=&R_{(ABCD)} \label{RW} +\end{eqnarray} + +Traceless Ricci spinor $ +\end_inset + +C +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ {\tt From spinor curvature} is +\begin{eqnarray} +C_{ab}&\tsst&C_{AB\dot{A}\dot{B}}\\[2mm] +C_{AB\dot{C}\dot{D}}&=&\pm(R_{AB\dot{C}\dot{D}}+R_{\dot{C}\dot{D} AB}) +\end{eqnarray} + +Scalar curvature {\tt From spinor curvature} is +\begin{equation} R=2(R^{MN}_{\ \ \ \ MN}+R^{\dot{M}\dot{N}}_{\ \ \ \ \dot{M}\dot{N}}) +\end{equation} + +Antisymmetric Ricci spinor $ +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ {\tt From spinor curvature} is +\begin{eqnarray} +A_{ab}&\tsst& A_{AB}\epsilon_{\dot{A}\dot{B}}+A_{\dot{A}\dot{B}}\epsilon_{AB}\\[1mm] +A_{AB}&=&\mp R^{\ \ \ \,M}_{(A|\ \ M|B)} +\end{eqnarray} + +Traceless deviation spinor $ +\end_inset + +B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ {\tt From spinor curvature} is +\begin{eqnarray} +B_{ab}&\tsst&B_{AB\dot{A}\dot{B}}\\[1mm] +B_{AB\dot{C}\dot{D}}&=&\pm i(R_{AB\dot{C}\dot{D}}-R_{\dot{C}\dot{D} AB}) +\end{eqnarray} + +Deviation trace {\tt From spinor curvature} is +\begin{equation} +D=-2i(R^{MN}_{\ \ \ \ MN}-R^{\dot{M}\dot{N}}_{\ \ \ \ \dot{M}\dot{N}}) +\end{equation} + +Note that spinors $ +\end_inset + +C +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +,B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +B +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ are Hermitian +\begin{equation} +C_{AB\dot{C}\dot{D}}=\overline{C_{CD\dot{A}\dot{B}}},\ \ +B_{AB\dot{C}\dot{D}}=\overline{B_{CD\dot{A}\dot{B}}} +\end{equation} + +Finally we introduce the decomposition for the spinorial +curvature 2-form +\begin{equation} +\Omega_{AB}= +\OO{w}_{AB}+\OO{c}_{AB}+\OO{r}_{AB} ++\OO{a}_{AB}+\OO{b}_{AB}+\OO{c}_{AB} +\end{equation} +where the {\tt Undotted curvature 2-forms} +\object{Undotted Weyl 2-form OMWU.AB }{\OO{w}_{AB}} +\object{Undotted Traceless Ricci 2-form OMCU.AB }{\OO{c}_{AB}} +\object{Undotted Scalar Curvature 2-form OMRU.AB }{\OO{r}_{AB}} +\object{Undotted Ricanti 2-form OMAU.AB }{\OO{a}_{AB}} +\object{Undotted Traceless Deviation 2-form OMBU.AB }{\OO{b}_{AB}} +\object{Undotted Scalar Deviation 2-form OMDU.AB }{\OO{d}_{AB}} +are given by +\begin{eqnarray} +\OO{w}_{AB}&=&C_{ABCD}S^{CD} \\[1mm] +\OO{c}_{AB}&=&\pm\frac12 C_{AB\dot{C}\dot{D}}S^{\dot{C}\dot{D}} \\[1mm] +\OO{r}_{AB}&=&\frac1{12}S_{AB}R \\[1mm] +\OO{a}_{AB}&=&\pm A_{(A}^{\ \ \ M}S_{M|B)} \\[1mm] +\OO{b}_{AB}&=&\mp\frac{i}2 B_{AB\dot{C}\dot{D}}S^{\dot{C}\dot{D}} \\[1mm] +\OO{d}_{AB}&=&\frac{i}{12}S_{AB}D +\end{eqnarray} + + + + + + + +\section{Torsion Decomposition} + +The torsion tensor +\begin{equation} +Q_{abc}=Q_{a[bc]},\qquad +\Theta^a=\frac{1}{2}Q^a{}_{bc}\,S^{bc} +\end{equation} +consists of three irreducible pieces +\begin{equation} +Q_{abc} = +\stackrel{\rm c}{Q}_{abc} ++\stackrel{\rm t}{Q}_{abc} ++\stackrel{\rm a}{Q}_{abc} +\end{equation} + +\begin{center} +\begin{tabular}{|c|c|c|} +\hline object & exists if & and has $n$ components \\ +\hline +\vv$Q_{abc}$ & & $\frac{d^2(d-1)}{2}$ \\[1mm] +\hline\vv$\stackrel{\rm c}{Q}_{abc}$ & $d\geq3$ & $\frac{d(d^2-4)}{3}$ \\ +\vv$\stackrel{\rm t}{Q}_{abc}$ & & $d$ \\ +\vv$\stackrel{\rm a}{Q}_{abc}$ & $d\geq3$ & $\frac{d(d-1)(d-2)}{6}$ \\[1mm] +\hline +\end{tabular} +\end{center} + +The corresponding union of three objects {\tt Torsion 2-forms} is +\object{Traceless Torsion 2-form THQC'a} +{\stackrel{\rm c}{\Theta}\!{}^a=\frac{1}{2} + \stackrel{\rm c}{Q}\!{}^a{}_{bc}\,S^{bc}} +\object{Torsion Trace 2-form THQT'a} +{\stackrel{\rm t}{\Theta}\!{}^a=\frac{1}{2} + \stackrel{\rm t}{Q}\!{}^a{}_{bc}\,S^{bc}} +\object{Antisymmetric Torsion 2-form THQA'a} +{\stackrel{\rm a}{\Theta}\!{}^a=\frac{1}{2} + \stackrel{\rm a}{Q}\!{}^a{}_{bc}\,S^{bc}} + +And the auxiliary quantities +\object{Torsion Trace QT'a}{Q^a} +\object{Torsion Trace 1-form QQ}{Q=-\partial_a\ipr\Theta^a} +\object{Antisymmetric Torsion 3-form QQA}{\stackrel{\rm a}{Q}=\theta_a\wedge\Theta^a} + +The torsion trace $ +\end_inset + +Qa=Qm +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +am +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ can be obtained {\tt From torsion +trace 1-form} +\begin{equation} +Q^a = \partial^a\ipr Q +\end{equation} + +The {\tt Standard way} for the irreducible torsion 2-forms is +\begin{equation} +\stackrel{\rm t}{\Theta}\!{}^a = -\frac{1}{(d-1)}\theta^a\wedge Q +\end{equation} +\begin{equation} +\stackrel{\rm t}{\Theta}\!{}^a = \frac{1}{3}\partial^a\ipr\stackrel{\rm a}{Q} +\end{equation} +\begin{equation} +\stackrel{\rm c}{\Theta}\!{}^a = \Theta^a +-\stackrel{\rm t}{\Theta}\!{}^a +-\stackrel{\rm a}{\Theta}\!{}^a +\end{equation} + +The rest of this section is valid in dimension 4 only. + +In this case one can introduce the torsion pseudo trace +\object{Torsion Pseudo Trace QP'a}{ +P^a = \stackrel{*}{Q}\!{}^{ma}{}_{m}, +\ \stackrel{*}{Q}\!{}^a{}_{bc} = \frac{1}{2}{\cal E}_{bc}{}^{pq} +Q^a{}_{pq}} +which can be computed {\tt From antisymmetric torsion 3-form} +\begin{equation} +P^a = \partial^a\ipr\,*\!\stackrel{\rm a}{Q} +\end{equation} + +Finally let us consider the spinorial representation of the +torsion. +Below the upper sign corresponds to the +\seethis{See \pref{spinors}\ or \ref{spinors1}.} +signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and lower one to the +signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +First we introduce the spinorial analog of the torsion tensor +\begin{equation} +Q_{abc}\tsst Q_{A\dot{A} BC}\epsilon_{\dot{B}\dot{C}} ++Q_{A\dot{A}\dot{B}\dot{C}}\epsilon_{BC} +\end{equation} +where +\begin{equation} +Q_{A\dot{A} BC}=-i*(\Theta_{A\dot{A}}\wedge S_{BC}),\qquad +Q_{A\dot{A}\dot{B}\dot{C}}=i*(\Theta_{A\dot{A}}\wedge S_{\dot{B}\dot{C}}) +\end{equation} +These spinors are reducible but the +\object{Traceless Torsion Spinor QC.ABC.D\cc}{C_{ABC\dot D}} +\[\stackrel{\rm c}{Q}_{abc}\tsst C_{ABC\dot A}\epsilon_{\dot{B}\dot{C}} ++Q_{\dot{A}\dot{B}\dot{C}A}\epsilon_{BC},\quad +C_{\dot{A}\dot{B}\dot{C} A}=\overline{C_{ABC\dot{A}}} +\] +is irreducible (symmetric in $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + + ABC +\begin_inset Formula $). And it can be +computed {\tt From torsion} by the relation +\begin{equation} +C_{ABC\dot A} = Q_{(A|\dot{A}|BC)} +\end{equation} + +The torsion trace can be calculated {\tt From torsion using spinors} +\begin{equation} +Q^a\tsst Q^{A\dot{A}},\quad +Q_{A\dot{B}}=\mp(Q^M{}_{\dot{B}MA}+Q_A{}^{\dot M}{}_{\dot M\dot{B}}) +\end{equation} + +And similarly the torsion pseudo-trace can be found +{\tt From torsion using spinors} +\begin{equation} +P^a\tsst P^{A\dot{A}},\quad +P_{A\dot{B}}=\mp i(Q^M{}_{\dot{B}MA}-Q_A{}^{\dot M}{}_{\dot M\dot{B}}) +\end{equation} + +Finally we introduce the {\tt Undotted trace 2-forms} +which are selfdual parts of the irreducible torsion 2-forms +\object{Undotted Traceless Torsion 2-form THQCU'a} +{\stackrel{\rm c}{\vartheta}\!{}^a} +\object{Undotted Torsion Trace 2-form THQTU'a} +{\stackrel{\rm t}{\vartheta}\!{}^a} +\object{Undotted Antisymmetric Torsion 2-form THQAU'a} +{\stackrel{\rm a}{\vartheta}\!{}^a} \seethis{See \pref{thetau}.} +These quantities will be used in the gravitational equations. + +This complex 2-forms can be obtained by the equations +({\tt Standard way}): +\begin{eqnarray} +\stackrel{\rm c}{\vartheta}\!{}^a &\tsst& \stackrel{\rm c}{\vartheta}\!{}^{A\dot A} +=C^A_{\ \ BC}{}^{\dot{A}}S^{BC}\\[1mm] +\stackrel{\rm t}{\vartheta}\!{}^a &\tsst& \stackrel{\rm t}{\vartheta}\!{}^{A\dot A} +=\mp\frac13 Q_{M}^{\ \ \ \dot{A}}S^{AM}\\[1mm] +\stackrel{\rm a}{\vartheta}\!{}^a &\tsst& \stackrel{\rm a}{\vartheta}\!{}^{A\dot A} +=\pm\frac{i}3 P_{M}^{\ \ \ \dot{A}}S^{AM} +\end{eqnarray} + + + +\section{Nonmetricity Decomposition} + +In general the nonmetricity tensor +\begin{equation} +N_{abc}=N_{(ab)c},\qquad N_{ab}=N_{abc}\theta^c +\end{equation} +consist of 4 irreducible pieces +\begin{equation} +N_{abcd} = +\stackrel{\rm c}{N}_{abc} ++\stackrel{\rm a}{N}_{abc} ++\stackrel{\rm t}{N}_{abc} ++\stackrel{\rm w}{N}_{abc} +\end{equation} + +\begin{center} +\begin{tabular}{|c|c|c|} +\hline object & exists if & and has $n$ components \\ +\hline +\vv$N_{abc}$ & & $\frac{d^2(d+1)}{2}$ \\[1mm] +\hline\vv$\stackrel{\rm c}{N}_{abc}$ & & $\frac{d(d-1)(d+4)}{6}$ \\ +\vv$\stackrel{\rm a}{N}_{abc}$ & $d\geq3$ & $\frac{d(d^2-4)}{3}$ \\ +\vv$\stackrel{\rm t}{N}_{abc}$ & & $d$ \\ +\vv$\stackrel{\rm w}{N}_{abc}$ & & $d$ \\[1mm] +\hline +\end{tabular} +\end{center} + +The corresponding union of objects {\tt Nonmetricity 1-forms} +consist of +\object{Symmetric Nonmetricity 1-form NC.a.b} +{\stackrel{\rm c}{N}_{ab}=\stackrel{\rm c}{N}_{abc}\theta^c} +\object{Antisymmetric Nonmetricity 1-form NA.a.b} +{\stackrel{\rm a}{N}_{ab}=\stackrel{\rm a}{N}_{abc}\theta^c} +\object{Nonmetricity Trace 1-form NT.a.b} +{\stackrel{\rm t}{N}_{ab}=\stackrel{\rm t}{N}_{abc}\theta^c} +\object{Weyl Nonmetricity 1-form NW.a.b} +{\stackrel{\rm w}{N}_{ab}=\stackrel{\rm w}{N}_{abc}\theta^c} + +We have also two auxiliary 1-forms +\object{Weyl Vector NNW}{\stackrel{\rm w}{N}} +\object{Nonmetricity Trace NNT}{\stackrel{\rm t}{N}} + +They are computed according to the following formulas +\begin{equation} +\stackrel{\rm w}{N} = N^a{}_a +\end{equation} +\begin{equation} +\stackrel{\rm t}{N} = \theta^a\,\partial^b\ipr N_{ab} +- \frac{1}{d} \stackrel{\rm w}{N} +\end{equation} +\begin{equation} +\stackrel{\rm w}{N}_{ab} = \frac{1}{d}g_{ab}\stackrel{\rm w}{N} +\end{equation} +\begin{equation} +\stackrel{\rm t}{N}_{ab}=\frac{d}{(d-1)(d+2)}\left[ +\theta_b\partial_a\ipr\stackrel{\rm t}{N} ++\theta_a\partial_b\ipr\stackrel{\rm t}{N} +-\frac{2}{d} g_{ab} \stackrel{\rm t}{N}\right] +\end{equation} +\begin{equation} +\stackrel{\rm a}{N}_{ab}=\frac{1}{3}\left[ +\partial_a\ipr(\theta^m\wedge\stackrel{0}{N}_{bm}) ++\partial_b\ipr(\theta^m\wedge\stackrel{0}{N}_{am})\right] +\end{equation} +where +\[\stackrel{\rm 0}{N}_{ab}= +N_{abc} +-\stackrel{\rm t}{N}_{abc} +-\stackrel{\rm w}{N}_{abc} +\] +And finally +\begin{equation} +\stackrel{\rm c}{N}_{ab}= +N_{abc} +-\stackrel{\rm a}{N}_{abc} +-\stackrel{\rm t}{N}_{abc} +-\stackrel{\rm w}{N}_{abc} +\end{equation} + +\section{Newman-Penrose Formalism} + +The method of spinorial differential forms described in the +previous sections are essentially equivalent to the well +known Newman-Penrose formalism but for the sake of convenience +\grg\ has complete set of macro objects which allows to +write the Newman-Penrose equations in +traditional notation. All these objects refer (up to some sign +and 1/2 factors) to other \grg\ built-in objects. + +In this section upper sign corresponds to the +signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and lower one to the +signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. +\seethis{See \pref{spinors}.} +The frame must be null as explained in section \ref{spinors}. + +For the Newman-Penrose formalism we use notation and conventions +of the book \emph{Exact Solutions of the Einstein Field Equations} +by D. Kramer, H. Stephani, M. MacCallum and E. Herlt, ed. +E. Schmutzer (Berlin, 1980). We denote this book as ESEFE. + +We chose the relationships between NP null tetrad and \grg\ null +frame as follows +\begin{equation} +l^\mu=h^\mu_0,\quad +k^\mu=h^\mu_1,\quad +\overline{m}\!{}^\mu=h^\mu_2,\quad +m^\mu=h^\mu_3 +\end{equation} + +The NP vector operators are just the components of the +vector frame $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +partial +\end_layout + +\end_inset + +a +\begin_inset Formula $ +\begin{eqnarray} +\mbox{\tt DD}&=& D =\partial_1 \\ +\mbox{\tt DT}&=& \Delta=\partial_0 \\ +\mbox{\tt du}&=& \delta=\partial_3 \\ +\mbox{\tt dd}&=& \overline\delta=\partial_2 +\end{eqnarray} + +The spin coefficient are the components of the connection +1-form +\object{SPCOEF.AB.c}{ \omega_{AB\,c}=\partial_c\ipr\omega_{AB}} +or in the NP notation +\begin{eqnarray} +\mbox{\tt alphanp }&=& \alpha =\pm\omega_{(1)2} \\ +\mbox{\tt betanp }&=& \beta =\pm\omega_{(1)3} \\ +\mbox{\tt gammanp }&=& \gamma =\pm\omega_{(1)0} \\ +\mbox{\tt epsilonnp }&=& \epsilon =\pm\omega_{(1)1} \\ +\mbox{\tt kappanp }&=& \kappa =\pm\omega_{(0)1} \\ +\mbox{\tt rhonp }&=& \rho =\pm\omega_{(0)2} \\ +\mbox{\tt sigmanp }&=& \sigma =\pm\omega_{(0)3} \\ +\mbox{\tt taunp }&=& \tau =\pm\omega_{(0)0} \\ +\mbox{\tt munp }&=& \mu =\pm\omega_{(2)3} \\ +\mbox{\tt nunp }&=& \nu =\pm\omega_{(2)0} \\ +\mbox{\tt lambdanp }&=& \lambda =\pm\omega_{(2)2} \\ +\mbox{\tt pinp }&=& \pi =\pm\omega_{(2)1} \\ +\end{eqnarray} +where the first index of the +quantity $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +(AB)c +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is included inn parentheses to remind +that it is summed spinorial index. + +Finally for the curvature we have +\object{PHINP.AB.CD\cc }{ +\Phi_{AB\dot{C}\dot{D}} = \pm\frac{1}{2}C_{AB\dot C\dot D} } +\object{PSINP.ABCD }{\Psi_{ABCD}=C_{ABCD}} +the conventions for the scalar curvature $ +\end_inset + +R +\begin_inset Formula $ in ESEFE and +in \grg\ are the same. + +For the signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ the Newman-Penrose equations for +the quantities introduced above can be found in section 7.1 of ESEFE. +For other signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ one must alter the sign of +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Psi +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ABCD +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Phi +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +C +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +dot +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +D +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + +R +\begin_inset Formula $ in Eqs. (7.28)--(7.45). + +\section{Electromagnetic Field} + +Formulas in this section are valid only in spaces +with the signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,…,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,…,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. +The sign factor $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + + +\begin_inset Formula $ in the expressions below is +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + +=- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + diag +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +0 +\begin_inset Formula $ ($ +\end_inset + ++1 +\begin_inset Formula $ for the first signature and $ +\end_inset + +-1 +\begin_inset Formula $ +for the second). + +Let us introduce the +\object{EM Potential A}{A=A_\mu dx^\mu} +and the +\object{Current 1-form J}{J=j_\mu dx^\mu} + +The EM strength tensor +$ +\end_inset + +F +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +alpha +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +beta +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +partial +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +alpha +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +beta +\end_layout + +\end_inset + +- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +partial +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +beta +\end_layout + +\end_inset + + A +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +alpha +\end_layout + +\end_inset + + +\begin_inset Formula $ +\object{EM Tensor FT.a.b}{F_{ab}= +\partial_b\ipr\partial_a\ipr F} +where $ +\end_inset + +F +\begin_inset Formula $ is the +\object{EM 2-form FF}{F} +which can be found {\tt From EM potential} +\begin{equation} +F=dA +\end{equation} +or {\tt From EM tensor} +\begin{equation} +F = \frac{1}{2}F_{ab}\,S^{ab} +\end{equation} + +The EM action $ +\end_inset + +d +\begin_inset Formula $-form +\object{EM Action EMACT}{L_{\rm EM}= +-\frac{1}{8\pi}\,F\wedge *F} + +The {\tt Maxwell Equations} +\object{First Maxwell Equation MWFq}{d*F=-4\pi\sigma\,(-1)^{d}\,*J} +\object{Second Maxwell Equation MWSq}{dF=0} + +The current must satisfy the +\object{Continuity Equation COq}{d*J=0} + +The +\object{EM Energy-Momentum Tensor TEM.a.b}{T_{ab}^{\rm EM}} +is given by the equation +\begin{equation} +T^{\rm EM}_{ab} = \frac{\sigma}{4\pi} +F_{am}F_b{}^m +s\sigma\,g_{ab}\,*L_{\rm EM} +\end{equation} + +The rest of the section is valid in the dimension 4 only. + +In 4 dimensions the tensor $ +\end_inset + +F +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and its dual +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +stackrel +\end_layout + +\end_inset + +* +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +F +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +frac +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +1 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +2 +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +cal +\end_layout + +\end_inset + + E +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +mn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +F +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +mn +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +are expressed via usual 3-dimensional vectors $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +vec +\end_layout + +\end_inset + + E +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +vec +\end_layout + +\end_inset + + H +\begin_inset Formula $ +\begin{eqnarray} +F_{ab}&=&-\sigma\left(\begin{array}{rrr} +E_1&E_2&E_3\\ +&-H_3&H_2\\ +&&-H_1\end{array}\right)\\[1.5mm] +\stackrel{*}{F}_{ab}&=&\sigma\left(\begin{array}{rrr} +H_1&H_2&H_3\\ +&E_3&-E_2\\ +&&E_1\end{array}\right) +\end{eqnarray} +Similarly for the current we have +\begin{equation} +J=\sigma(-\rho dt + \vec j\,d\vec x) +\end{equation} + +The {\tt EM scalars} +\object{First EM Scalar SCF}{I_1=\frac12F_{ab}F^{ab} +={\vec H}^2-{\vec E}^2} +\object{Second EM Scalar SCS}{I_2=\frac12\stackrel{*}{F}_{ab}F^{ab} +=2\vec E\cdot\vec H} +can be obtained as follows by {\tt Standard way} +\begin{equation} +I_1 = -*(F\wedge*F) +\end{equation} +\begin{equation} +I_2 = *(F\wedge F) +\end{equation} + +The +\object{Complex EM 2-form FFU}{\Phi} +can be found {\tt From EM 2-form} +\begin{equation} +\Phi=F-i*F +\end{equation} +or {\tt From EM Spinor} +\begin{equation} +\Phi = 2\Phi_{AB}\,S^{AB} +\end{equation} + +The 2-form $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Phi +\end_layout + +\end_inset + + +\begin_inset Formula $ must obey the +\object{Selfduality Equation SDq.AB\cc}{\Phi\wedge S_{\dot A\dot B}} +and gives rise to the +\object{Complex Maxwell Equation MWUq}{d\Phi=-4i\sigma\pi\,*J} + +The EM 2-form $ +\end_inset + +F +\begin_inset Formula $ can be restored {\tt From Complex EM 2-form} +\begin{equation} +F=\frac{1}{2}(\Phi+\overline\Phi) +\end{equation} + +The symmetric +\object{Undotted EM Spinor FIU.AB}{\Phi_{AB}} +is the spinorial analog of the tensor $ +\end_inset + +F +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +\begin{equation} + F_{ab} \tsst \epsilon_{AB} \Phi_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \Phi_{AB} +\end{equation} +It can be obtained either {\tt From complex EM 2-form} +\begin{equation} +\Phi_{AB} = -\frac{i}{2}*(\Phi\wedge S_{AB}) +\end{equation} +of {\tt From EM 2-form} +\begin{equation} +\Phi_{AB} = -i*(F\wedge S_{AB}) +\end{equation} + +The +\object{Complex EM Scalar SCU}{\iota=I_1-iI_2} +can be found {\tt From EM Spinor} +\begin{equation} +\iota = 2\Phi_{AB}\Phi^{AB} +\end{equation} +or {\tt From Complex EM 2-form} +\begin{equation} +\iota = -\frac{i}{2} *(\Phi\wedge\Phi) +\end{equation} + +Finally we have the +\object{EM Energy-Momentum Spinor TEMS.AB.CD\cc} +{T^{\rm EM}_{AB\dot A\dot B}=\frac{1}{2\pi}\Phi_{AB}\Phi_{\dot A\dot B}} + + +\section{Dirac Field} + +In this section upper sign corresponds to the +signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and lower one to the +signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +The four component Dirac spinor consists of two 1-index spinors +\begin{equation} +\psi=\left(\begin{array}{c}\phi^A\\ \chi_{\dot A}\end{array}\right),\ \ +\overline\psi=\left(\chi_A\ \ \phi^{\dot A}\right) +\end{equation} +Thus we have the {\tt Dirac spinor} as the union of two objects +\object{Phi Spinor PHI.A}{\phi_A} +\object{Chi Spinor CHI.B}{\chi_B} + +The gamma-matrices are expressed via sigma-matrices as follows +\begin{equation} +\gamma^m=\sqrt2\left(\begin{array}{cc} +0&\sigma^{mA\dot B}\\ \sigma^m\!{}_{B\dot A}&0\end{array}\right) +\end{equation} + +Dirac field action 4-form +\begin{eqnarray} +&&\mbox{\tt Dirac Action 4-form DACT}=L_{\rm D}=\nonumber\\[1mm] +&&\quad=\left[\frac{i}2(\overline\psi\gamma^a +(\nabla_a+ieA_a)\psi-(\nabla_a-ieA_a)\overline\psi\gamma^a\psi) +-m_{\rm D}\overline\psi\psi\right]\upsilon +\end{eqnarray} + +The {\tt Standard way} to compute this quantity is +\begin{eqnarray} +L_{\rm D} &=& -\frac{i}{\sqrt2}\left[ +\phi_{\dot A}\theta^{A\dot A}\!\wedge*(D+ieA)\phi_A-{\rm c.c.} +-\chi_{\dot A} \theta^{A\dot A}\!\wedge*(D-ieA)\chi_A -{\rm c.c.}\right]- +\nonumber\\[1mm]&&\qquad\qquad\quad +-m_{\rm D}\left(\phi^A\chi_A+{\rm c.c.}\right)\upsilon +\end{eqnarray} + +The {\tt Dirac equation} is +\object{Phi Dirac Equation DPq.A\cc}{ +i\sqrt2\partial_{B\dot A}\ipr(D+ieA-\frac12Q)\phi^B-m_{\rm D}\chi_{\dot A}=0} +\object{Chi Dirac Equation DCq.A\cc}{ +i\sqrt2\partial_{B\dot A}\ipr(D-ieA-\frac12Q)\chi^B-m_{\rm D}\phi_{\dot A}=0} +where $ +\end_inset + +Q +\begin_inset Formula $ is the torsion trace 1-form. Notice that terms with the +electromagnetic field $ +\end_inset + +eA +\begin_inset Formula $ are included in equations iff +the value of $ +\end_inset + +A +\begin_inset Formula $ is defined. The unit charge $ +\end_inset + +e +\begin_inset Formula $ is given by the +constant \comm{ECONST}. + +The current 1-form can be computed {\tt From Dirac Spinor} +\begin{equation} +J=\mp\sqrt2e(\phi_A\phi_{\dot A}+\chi_A\chi_{\dot A})\theta^{A\dot A} +\end{equation} + +The symmetrized +\object{Dirac Energy-Momentum Tensor TDI.a.b}{T^{\rm D}_{ab}} +can be obtained as follows +\begin{eqnarray} +T^{\rm D}_{ab}&=& +*(\theta_{(a}\wedge T^{\rm D}_{b)})\nonumber\\[1mm] +T^{\rm D}_a&=&\mp\frac{i}{\sqrt2}\Big[ +*\theta^{A\dot A}\partial_a\ipr(D+ieA)\phi_A\phi_{\dot A} +-{\rm c.c.}\nonumber\\ +&&\qquad-*\theta^{A\dot A}\partial_a\ipr(D-ieA)\chi_A\chi_{\dot A} +-{\rm c.c.}\Big] +\pm\partial_a\ipr L_{\rm D} +\end{eqnarray} + +The +\object{Undotted Dirac Spin 3-Form SPDIU.AB}{s^{\rm D}_{AB}} +\begin{equation} +s^{\rm D}_{AB}=\frac{i}{2\sqrt2} +\left(*\theta_{(A|\dot A}\phi_{B)}\phi^{\dot A} +-*\theta_{(A|\dot A}\chi_{B)}\chi^{\dot A}\right) +\end{equation} + +The Dirac field mass $ +\end_inset + +m +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + D +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is given by the constant +\comm{DMASS}. + + +\section{Scalar Field} + +Formulas in this section are valid in any dimension +with the signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,…,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,…,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. +The sign factor $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + + +\begin_inset Formula $ is $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + +=- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + diag +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +0 +\begin_inset Formula $ +($ +\end_inset + ++1 +\begin_inset Formula $ for the first signature and $ +\end_inset + +-1 +\begin_inset Formula $ for the second). + +The scalar field +\object{Scalar Field FI}{\phi} + +The minimal scalar field action $ +\end_inset + +d +\begin_inset Formula $-form +\object{Minimal Scalar Action SACTMIN}{ +L_{\rm Smin}= +-\frac{1}{2}\left[\sigma(\partial_\alpha\phi)^2+ +m_{\rm s}^2 \phi^2\right]\upsilon} + +The nonminimal scalar field action +\object{Scalar Action SACT}{ +L_{\rm S}= +-\frac{1}{2}\left[\sigma(\partial_\alpha\phi)^2+ +(m_{\rm s}^2+a_0R) \phi^2\right]\upsilon} + +The scalar field equation +\object{Scalar Equation SCq} +{s\sigma(-1)^d*d*d\phi-(m_{\rm s}^2+a_0R)\phi=0} +which gives +\[-\sigma\rim{\nabla}{}^\pi\rim{\nabla}_\pi\phi-(m_{\rm s}^2+a_0R)\phi=0 +\] + +The minimal energy-momentum tensor is +\begin{eqnarray} +&&\mbox{\tt Minimal Scalar Energy-Momentum Tensor TSCLMIN.a.b} +=T^{\rm Smin}_{ab}= \nonumber\\ +&&\qquad\qquad=\partial_a\phi\partial_b\phi+s\sigma\,g_{ab} +*L_{\rm Smin} +\end{eqnarray} +The nonminimal part of the scalar field energy-momentum +\seethis{See pages \pageref{graveq}\ and \pageref{metreq}.} +tensor can be taken into account in the left-hand side +of gravitational equations. + +The scalar field mass $ +\end_inset + +m +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ are given by the +constant {\tt SMASS}. The nonminimal interaction +terms are included iff the switch \comm{NONMIN} \swind{NONMIN} +is turned on and the value of nonminimal interaction constant +$ +\end_inset + +a0 +\begin_inset Formula $ is determined by the object +\object{A-Constants ACONST.i2}{a_i} +The default value of $ +\end_inset + +a0 +\begin_inset Formula $ is the constant \comm{AC0}. + +\section{Yang-Mills Field} + +Formulas in this section are valid in any dimension +with the signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,…,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,…,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. +The sign factor $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + + +\begin_inset Formula $ in the expressions below is +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + +=- +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + diag +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +0 +\begin_inset Formula $ ($ +\end_inset + ++1 +\begin_inset Formula $ for the first signature and $ +\end_inset + +-1 +\begin_inset Formula $ +for the second). The indices $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + + i,j,k,l,m,n +\begin_inset Formula $ +are the internal space Yang-Mills indices and we a +assume that the internal Yang-Mills metric is $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +delta +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ij +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +The Yang-Mills potential 1-form +\object{YM Potential AYM.i9}{A^i=A^i_\mu dx^\mu} + +The structural constants +\object{Structural Constants SCONST.i9.j9.k9}{c^i{}_{jk}=c^i{}_{[jk]}} + +The Yang-Mills strength 2-form +\object{YM 2-form FFYM.i9}{F^i} +and strength tensor +\object{YM Tensor FTYM.i9.a.b}{F^i{}_{ab}} + +The $ +\end_inset + +Fi +\begin_inset Formula $ can be computed {\tt From YM potential} +\begin{equation} +F^i = dA^i + \frac12 c^i{}_{jk} \, A^j\wedge A^k +\end{equation} +or {\tt From YM tensor} +\begin{equation} +F^i = \frac12 F^i{}_{ab}\, S^{ab} +\end{equation} + +The {\tt Standard way} to find Yang-Mills strength tensor is +\begin{equation} +F^i{}_{ab}=\partial_b\ipr\partial_a\ipr F^i +\end{equation} + +The Yang-Mills action $ +\end_inset + +d +\begin_inset Formula $-form +\object{YM Action YMACT}{L_{\rm YM}= +-\frac{1}{8\pi}F^i\wedge*F_i} + +The {\tt YM Equations} +\object{First YM Equation YMFq.i9}{d*F^i + c^i{}_{jk} \, A^j\wedge *F^k=0} +\object{Second YM Equation YMSq.i9}{dF^i + c^i{}_{jk} \, A^j\wedge F^k=0} + +The energy-momentum tensor +\object{YM Energy-Momentum Tensor TYM.a.b} +{\frac{\sigma}{4\pi}F^i{}_{am}F^i{}_b{}^m + s\sigma\,g_{ab}\, +*L_{\rm YM}} + + +\section{Geodesics} + +The geodesic equation +\object{Geodesic Equation GEOq\^m}{ +\frac{d^2x^\mu}{dt^2}+\{^\mu_{\pi\tau}\} +\frac{dx^\pi}{dt}\frac{dx^\tau}{dt}=0} +Here the parameter $ +\end_inset + +t +\begin_inset Formula $ must be declared by the +\seethis{See page \pageref{affpar}.} +\cmdind{Affine Parameter} +{\tt Affine parameter} declaration. + +\section{Null Congruence and Optical Scalars} + +Let us consider the congruence defined by the vector field +$ +\end_inset + +k +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +alpha +\end_layout + +\end_inset + + +\begin_inset Formula $ +\object{Congruence KV}{k=k^\mu\partial_\mu} + +This congruence is null iff +\object{Null Congruence Condition NCo}{k\cdot k=0} +holds. + +The congruence is geodesic iff the condition +\object{Geodesics Congruence Condition GCo'a}{k^\mu\rim{\nabla}_\mu k^a=0} +is fulfilled. + +For the null geodesic congruence one can calculate the +{\tt Optical scalars} +\object{Congruence Expansion thetaO}{\theta= +\frac{1}{2}\rim{\nabla}{}^\pi k_\pi} +\object{Congruence Squared Rotation omegaSQO}{\omega^2= +\frac{1}{2}(\rim{\nabla}_{[\alpha}k_{\beta]})^2} +\object{Congruence Squared Shear sigmaSQO}{\sigma\overline\sigma= +\frac{1}{2}\left[ (\rim{\nabla}_{(\alpha}k_{\beta)})^2 +-2\theta^2\right]} + +\section{Timelike Congruences and Kinematics} + +Let us consider the congruence determined by the velocity +vector $ +\end_inset + +u +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +alpha +\end_layout + +\end_inset + + +\begin_inset Formula $ +\object{Velocity UU'a}{u^a} +\object{Velocity Vector UV}{u=u^a\partial_a} + +The velocity vector must be normalized and the quantity +\object{Velocity Square USQ}{u^2=u\cdot u} +must be constant but nonzero. + +If the frame metric coincides with its default +diagonal value \seethis{See \pref{defaultmetric}.} +$ +\end_inset + +g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + += +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + diag +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +(-1,…) +\begin_inset Formula $ +then {\tt By default} we have for the velocity +\begin{equation} +u^a=(1,0,\dots,0) +\end{equation} +which means that the congruence is comoving in the given frame. + +In general case the velocity can be obtained +{\tt From velocity vector} +\begin{equation} +u^a=u\ipr \theta^a +\end{equation} + +We introduce the auxiliary object +\object{Projector PR'a.b}{P^a{}_b= +\delta^a_b-\frac{1}{u^2}u^an_b} + +The following four quantities called {\tt Kinematics} +comprise the complete set of the congruence characteristics +\object{Acceleration accU'a}{A^a=\rim{\nabla}_uu^a} +\object{Vorticity omegaU.a.b}{\omega_{ab}= +P^m{}_aP^n{}_b \rim{\nabla}_{[m}u_{n]}} +\object{Volume Expansion thetaU}{\Theta=\rim{\nabla}_au^a} +\object{Shear sigmaU.a.b}{ +P^m{}_aP^n{}_b \rim{\nabla}_{(m}u_{n)}- +\frac{1}{(d-1)}P_{ab}\Theta} + + +\section{Ideal And Spin Fluid} + + +The ideal fluid is characterized by the +\object{Pressure PRES}{p} +and +\object{Energy Density ENER}{\varepsilon} + +The ideal fluid energy-momentum tensor is +\begin{eqnarray} +&&\mbox{\tt Ideal Fluid Energy-Momentum Tensor TIFL.a.b}= +T^{\rm IF}_{ab} = \nonumber\\ +&&\qquad\qquad=(\varepsilon+p)u_a u_b - u^2p g_{ab} +\end{eqnarray} + +The rest of the section requires the nonmetricity be zero +(\comm{NONMETR} is off). + +In addition spin-fluid is characterized by +\object{Spin Density SPFLT.a.b }{S^{\rm SF}_{ab}=S^{\rm SF}_{[ab]}} +or equivalently by +\object{Spin Density 2-form SPFL }{S^{\rm SF}} + +The spin 2-form can be obtained {\tt From spin density} +\begin{equation} +S^{\rm SF}=\frac{1}{2}S^{\rm SF}_{ab} \theta^a\wedge\theta^a +\end{equation} +and $ +\end_inset + +s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is determined {\tt From spin density 2-form} +\begin{equation} +S^{\rm SF}_{ab}= \partial_b\ipr\partial_a\ipr S^{\rm SF} +\end{equation} + +The spin density must satisfy the Frenkel condition +\object{Frenkel Condition FCo}{u\ipr S^{\rm SF}=0} + +The spin fluid energy-momentum tensor is +\begin{eqnarray} +&&\mbox{\tt Spin Fluid Energy-Momentum Tensor TSFL.a.b}=T^{\rm SF}_{ab}= +\nonumber\\ +&&\qquad\qquad=(\varepsilon+p)u_a u_b - u^2p g_{ab}+\Delta_{(ab)} +\end{eqnarray} +where +\begin{equation} +\Delta_{ab}=-2(g^{cd}+u^{-2}\,u^cu^d) \nabla_c S^{\rm SF}_{(ab)d} +\end{equation} +\begin{equation} +s^{\rm SF}_{abc}=u_a\,S^{\rm SF}_{bc} +\end{equation} +if torsion is zero (\comm{TORSION} off) and +\begin{equation} +\Delta_{ab}=2u^{-2}\,u_au^d\,\nabla_u S^{\rm SF}_{bd} +\end{equation} +if torsion is nonzero (\comm{TORSION} on). + +Notice that the energy-momentum \seethis{See \pref{tsym}.} +tensor $ +\end_inset + +T +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + SF +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is symmetrized. + +Finally yet another representation for the spin +is the undotted spin 3-form +\object{Undotted Fluid Spin 3-form SPFLU.AB }{s^{\rm SF}_{AB}} +which is given by the standard spinor $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +tsst +\end_layout + +\end_inset + + +\begin_inset Formula $ tensor correspondence rules +\begin{equation} + s^{\rm SF}_{mab}\,*\theta^m \tsst \epsilon_{AB} s^{\rm SF}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}s^{\rm SF}_{AB} +\end{equation} +according to Eq. (\ref{asys}). \seethis{See \pref{asys}.} +This quantity is used in the right-hand side of gravitational equations. + +\section{Total Energy-Momentum And Spin} +\label{totalc} + +\enlargethispage{4mm} + + +The total energy-momentum tensor +\object{Total Energy-Momentum Tensor TENMOM.a.b}{T_{ab}} +and the total undotted spin 3-form \seethis{See pages \pageref{graveq}\ and \pageref{metreq}.} +\object{Total Undotted Spin 3-form SPINU.AB}{s_{AB}} +play the role of sources in the right-hand side of the +gravitational equations. + +The expression for these quantities read +\begin{equation} +T_{ab} = +T^{\rm D}_{ab}+ +T^{\rm EM}_{ab}+ +T^{\rm YM}_{ab}+ +T^{\rm Smin}_{ab}+ +T^{\rm IF}_{ab}+ +T^{\rm SF}_{ab} \label{b1} +\end{equation} +\begin{equation} +s_{AB} = s_{AB}^{\rm D} + s_{AB}^{\rm SF} \label{b2} +\end{equation} +When $ +\end_inset + +T +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + +s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ are calculated \grg\ does not tries to find value +of all objects in the right-hand side of Eqs. (\ref{b1}), (\ref{b2}) +instead it adds only the quantities whose value are currently +defined. In particular if none of above tensors and spinors are +defined then $ +\end_inset + +T +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +=s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + +=0 +\begin_inset Formula $. + +Notice that $ +\end_inset + +T +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and all tensors in the right-hand side +of Eq. (\ref{b1}) are symmetric. +\seethis{See \pref{tsym}.} +They are the symmetric parts of the canonical energy-momentum tensors. + +In addition we introduce the +\object{Total Energy-Momentum Trace TENMOMT}{T=T^a{}_a} +and the spinor +\object{Total Energy-Momentum Spinor TENMOMS.AB.CD\cc}{T_{AB\dot C\dot D}} +is a spinorial equivalent of the traceless part of $ +\end_inset + +T +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ +\begin{equation} +T_{ab}-\frac{1}{4}g_{ab}T \tsst T_{AB\dot A\dot B} +\end{equation} + + +\section{Einstein Equations} + +The Einstein equation +\object{Einstein Equation EEq.a.b} +{R_{ab}-\frac{1}{2}g_{ab}R +\Lambda R =8\pi G\, T_{ab}} + +And the {\tt Spinor Einstein equations} +\object{Traceless Einstein Equation CEEq.AB.CD\cc}{ +C_{AB\dot C\dot D} = 8\pi G\, T_{AB\dot C\dot D}} +\object{Trace of Einstein Equation TEEq} +{R-4\Lambda = -8\pi G\, T} + +The cosmological constant is included in these equations +iff the switch \comm{CCONST} is turned on \swind{CCONST} +and its value is given by the constant \comm{CCONST}. +The gravitational constant $ +\end_inset + +G +\begin_inset Formula $ is given by the constant \comm{GCONST}. + + +\section{Gravitational Equations in Space With Torsion} + +Equations in this section are valid in dimension $ +\end_inset + +d=4 +\begin_inset Formula $ +with the signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ only. +The $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + +=1 +\begin_inset Formula $ for the first signature and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + +=-1 +\begin_inset Formula $ +for the second. The nonmetricity must be zero and the +switch \comm{NONMETR} turned off. + +Let us consider the action +\begin{equation} +S=\int\left[\frac{\sigma}{16\pi G}L_{\rm g} ++L_{\rm m}\right] +\end{equation} +where +\object{Action LACT}{L_{\rm g}=\upsilon\,{\cal L}_{\rm g}} +is the gravitational action 4-form and +\begin{equation} +L_{\rm m} = \upsilon\,{\cal L}_{\rm m} +\end{equation} +is the matter action 4-form. + +Let us define the following variational derivatives +\begin{equation} +Z^\mu{}_{a} = \frac{1}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta h^a_\mu} +,\qquad +t^\mu{}_{a} = \frac{\sigma}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta h^a_\mu} +\end{equation} +\begin{equation} +V^\mu{}_{ab} = \frac{1}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta \omega^{ab}{}_\mu} +,\qquad +s^\mu{}_{ab} = \frac{\sigma}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta \omega^{ab}{}_\mu} +\end{equation} +Then the gravitational equations reads +\begin{eqnarray} +Z^\mu{}_a &=& -16\pi G\,t^\mu{}_a \label{zma} \\[2mm] +V^\mu{}_{ab} &=& -16\pi G\,s^\mu{}_{ab} \label{vab} +\end{eqnarray} +Here the first equation is an analog of Einstein equation +and has the canonical nonsymmetric energy-momentum +tensor $ +\end_inset + +t +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mu +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset Formula $ as a source. The source in the second +equation is the spin tensor $ +\end_inset + +s +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +mu +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. + +Now we rewrite these equation in other equivalent form. +First let us define the following 3-forms +\begin{equation} +Z_a = Z^m{}_a\,*\theta_m,\qquad t_a = t^m{}_a\,*\theta_m +\end{equation} +\begin{equation} +V_{ab} = V^m{}_{ab}\,*\theta_m,\qquad s_{ab} = s^m{}_{ab}\,*\theta_m +\end{equation} +Notice that Eq. (\ref{zma}) is not symmetric but \label{tsym} +the antisymmetric part of this equation is expressed via second +Eq. (\ref{vab}) due to Bianchi identity. Therefore only the +symmetric part of Eq. (\ref{zma}) is essential. +Eq. (\ref{vab}) is +antisymmetric and we can consider its spinorial analog +using the standard relations +\begin{eqnarray} +V_{ab} &\tsst& V_{A\dot AB\dot B}= +\epsilon_{AB} V_{\dot A\dot B} + \epsilon_{\dot A\dot B}V_{AB} \\ +s_{ab} &\tsst& s_{A\dot AB\dot B}= +\epsilon_{AB} s_{\dot A\dot B} + \epsilon_{\dot A\dot B}s_{AB} +\end{eqnarray} \seethis{See \pref{asys}.} + +Finally we define the {\tt Gravitational equations} in the form \label{graveq} +\object{Metric Equation METRq.a.b}{-\frac12Z_{(ab)}=8\pi G\,T_{ab}} +\object{Torsion Equation TORSq.AB}{V_{AB}=-16\pi G\,s_{AB}} +where the currents in the right-hand side of equations are +\seethis{See \pref{totalc}.} +\object{Total Energy-Momentum Tensor TENMOM.a.b}{T_{ab}=t_{(ab)}} +\object{Total Undotted Spin 3-form SPINU.AB}{s_{AB}} + +Now let us consider the equations which are used in \grg\ to +compute the left-hand side of the gravitational equations +$ +\end_inset + +Z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +(ab) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + +V +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. We have to emphasize that we use +\seethis{See \pref{spinors}.} +spinors and all restrictions imposed by the spinorial formalism +must be fulfilled. + +We consider the Lagrangian which is an arbitrary algebraic function +of the curvature and torsion tensors +\begin{equation} +{\cal L}_{\rm g} = {\cal L}_{\rm g}(R_{abcd},Q_{abc}) +\end{equation} +No derivatives of the torsion or curvature are permitted. +For such a Lagrangian we define so called curvature and torsion +momentums +\begin{equation} +\widetilde{R}{}^{abcd} = +2\frac{\partial{\cal L}_{\rm g}(R,Q)}{\partial R_{abcd}},\qquad +\widetilde{Q}{}^{abc} = +2\frac{\partial{\cal L}_{\rm g}(R,Q)}{\partial Q_{abc}},\qquad +\end{equation} + +The corresponding objects are +\object{Undotted Curvature Momentum POMEGAU.AB}{\widetilde{\Omega}_{AB}} +\object{Torsion Momentum PTHETA'a}{\widetilde{\Theta}{}^a} +where +\begin{eqnarray} +\widetilde{\Omega}_{ab} &=& \frac12 \widetilde{R}_{abcd}\,S^{cd} \\[1mm] +\widetilde{\Theta}{}^a &=& \frac12 \widetilde{Q}{}^a{}_{cd}\,S^{cd} +\end{eqnarray} +and +\begin{equation} +\widetilde{\Omega}_{ab} \tsst \widetilde{\Omega}_{A\dot AB\dot B}= +\epsilon_{AB} \widetilde{\Omega}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}\widetilde{\Omega}_{AB} +\end{equation} + +If value of three objects $ +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ ({\tt Action}), +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ ({\tt Undotted curvature momentum}) +and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Theta +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset Formula $ are specified then the +{\tt Gravitational equations} can be calculated using equations +({\tt Standard way}) +\begin{eqnarray} +Z_{(ab)} &=& *(\theta_{(a}\wedge Z_{b)}),\nonumber\\[1mm] +Z_a &=& D\widetilde{\Theta}_a + + (\partial_a\ipr\Theta^b)\wedge\widetilde{\Theta}_b + +2(\partial_a\ipr\Omega^{MN})\wedge\widetilde{\Omega}_{MN} +\nonumber\\ +&& + {\rm c.c.}-\partial_a L_{\rm g} +\end{eqnarray} +\begin{eqnarray} +&&V_{AB} = -D\widetilde{\Omega}_{AB} - \widetilde{\Theta}_{AB},\nonumber\\[1mm] +&& +\theta_{[a}\wedge\widetilde{\Theta}_{b]} \tsst +\epsilon_{AB} \widetilde{\Theta}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}\widetilde{\Theta}_{AB} +\end{eqnarray} + +Since gravitational equations are computed in the +spinorial formalism with the standard null frame +\seethis{See pages \pageref{spinors}\ and \pageref{spinors1}.} +the metric equation is complex and components $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +02 +\begin_inset Formula $, +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +12 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +22 +\begin_inset Formula $ are conjugated to $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +03 +\begin_inset Formula $. +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +13 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +33 +\begin_inset Formula $. Since these components are not independent +For the sake of efficiency by default \grg\ computes only +the $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +00 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +01 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +02 +\begin_inset Formula $, +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +11 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +12 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +22 +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +23 +\begin_inset Formula $ +components of $ +\end_inset + +Z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +(ab) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ only. +If you want to have all components the switch \comm{FULL} must be +turned on. \swind{FULL} + +These equations allows one to compute field equations for +gravity theory with an arbitrary Lagrangian. +But the value of three quantities $ +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $, +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Theta +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset Formula $ +must be specified by the user. In addition \grg\ has built-in +formulas for the most general quadratic in torsion and curvature +Lagrangian. The {\tt Standard way} for $ +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $, +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Theta +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{} +\end_layout + +\end_inset + +a +\begin_inset Formula $ is \label{thetau} +\begin{eqnarray} +\widetilde{\Theta}{}^a &=& +i\mu_1 (\stackrel{\scriptscriptstyle\rm c}{\vartheta}{}^a -{\rm c.c.}) ++i\mu_2 (\stackrel{\scriptscriptstyle\rm t}{\vartheta}{}^a -{\rm c.c.}) ++i\mu_3 (\stackrel{\scriptscriptstyle\rm a}{\vartheta}\!{}^a -{\rm c.c.}), \\[2mm] +\widetilde{\Omega}_{AB} &=& +i(\lambda_0-\sigma\,8\pi G\, a_0\phi^2)\, S_{AB} \nonumber\\&& ++i\lambda_1 \OO{w}_{AB} +-i\lambda_2 \OO{c}_{AB} ++i\lambda_3 \OO{r}_{AB} \nonumber\\&& ++i\lambda_4 \OO{a}_{AB} +-i\lambda_5 \OO{b}_{AB} ++i\lambda_6 \OO{d}_{AB} , \\[2mm] +L_{\rm g} &=& (-2\Lambda +\frac{1}{2}\lambda_0R +-\sigma\,4\pi G a_0 \phi^2 R) \upsilon ++ \Omega^{AB}\wedge\widetilde{\Omega}_{AB} + {\rm c.c.} \nonumber\\&& ++ \frac{1}{2} \Theta^a\wedge\widetilde{\Theta}_a +\end{eqnarray} + +The cosmological term $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Lambda +\end_layout + +\end_inset + + +\begin_inset Formula $ is included into +equations iff the switch \comm{CCONST} is turned on \swinda{CCONST} +and the value of $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Lambda +\end_layout + +\end_inset + + +\begin_inset Formula $ is given by the constant \comm{CCONST}. +The term with the scalar field $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +phi +\end_layout + +\end_inset + + +\begin_inset Formula $ is included into +equations iff the switch \comm{NONMIN} is on. \swinda{NONMIN} +The gravitational constant $ +\end_inset + +G +\begin_inset Formula $ is given by the constant \comm{GCONST}. +The parameters of the quadratic Lagrangian are given by the objects +\object{L-Constants LCONST.i6}{\lambda_i} +\object{M-Constants MCONST.i3}{\mu_i} +\object{A-Constants ACONST.i2}{a_i} +The default value of these objects ({\tt Standard way}) is +\begin{eqnarray} +\lambda_i &=& (\mbox{\tt LC0},\mbox{\tt LC1},\mbox{\tt LC2},\mbox{\tt LC3},\mbox{\tt LC4},\mbox{\tt LC5},\mbox{\tt LC6}), \\ +\mu_i &=& (0,\mbox{\tt MC1},\mbox{\tt MC2},\mbox{\tt MC32}), \\ +a_i &=& (\mbox{\tt AC0},0,0) +\end{eqnarray} + +\section{Gravitational Equations in Riemann Space} + +Equations in this section are valid in dimension $ +\end_inset + +d=4 +\begin_inset Formula $ +with the signature $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(-,+,+,+) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +(+,-,-,-) +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ only. +The $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + +=1 +\begin_inset Formula $ for the first signature and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +sigma +\end_layout + +\end_inset + +=-1 +\begin_inset Formula $ +for the second. The nonmetricity and torsion must be zero and the +switches \comm{NONMETR} and \comm{TORSION} must be turned off. + +Let us consider the action +\begin{equation} +S=\int\left[\frac{\sigma}{16\pi G}L_{\rm g} ++L_{\rm m}\right] +\end{equation} +where +\object{Action LACT}{L_{\rm g}=\upsilon\,{\cal L}_{\rm g}} +is the gravitational action 4-form and +\begin{equation} +L_{\rm m} = \upsilon\,{\cal L}_{\rm m} +\end{equation} +is the matter action 4-form. + +Let us define the following variational derivatives +\begin{equation} +Z^\mu{}_{a} = \frac{1}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta h^a_\mu} +,\qquad +T^\mu{}_{a} = \frac{\sigma}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta h^a_\mu} +\end{equation} +Then the {\tt Metric equation} is \label{metreq} +\object{Metric Equation METRq.a.b}{-\frac12Z_{ab}=8\pi G\,T_{ab}} +Notice that $ +\end_inset + +Z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and $ +\end_inset + +T +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ are automatically symmetric. + +Let us define 3-form +\begin{equation} +Z_a = Z^m{}_a\,*\theta_m,\qquad t_a = t^m{}_a\,*\theta_m +\end{equation} + +Now we consider the equations which are used in \grg\ to +compute the left-hand side of the metric equation +$ +\end_inset + +Z +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +ab +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $. We have to emphasize that we use +spinors and all restrictions imposed by the spinorial formalism +\seethis{See pages \pageref{spinors}\ or \pageref{spinors1}.} +must be fulfilled. + +We consider the Lagrangian which is an arbitrary algebraic function +of the curvature tensor +\begin{equation} +{\cal L}_{\rm g} = {\cal L}_{\rm g}(R_{abcd}) +\end{equation} +No derivatives of the curvature are permitted. +For such a Lagrangian we define so called curvature momentum +\begin{equation} +\widetilde{R}{}^{abcd} = +2\frac{\partial{\cal L}_{\rm g}(R)}{\partial R_{abcd}} +\end{equation} + +The corresponding \grg\ built-in object is +\object{Undotted Curvature Momentum POMEGAU.AB}{\widetilde{\Omega}_{AB}} +where +\begin{eqnarray} +\widetilde{\Omega}_{ab} &=& \frac12 \widetilde{R}_{abcd}\,S^{cd} \\[1mm] +\end{eqnarray} +and +\begin{equation} +\widetilde{\Omega}_{ab} \tsst \widetilde{\Omega}_{A\dot AB\dot B}= +\epsilon_{AB} \widetilde{\Omega}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}\widetilde{\Omega}_{AB} +\end{equation} + +If value of the objects $ +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ ({\tt Action}) and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ ({\tt Undotted curvature momentum}) is specified +then the {\tt Metric equation} can be calculated using equations +({\tt Standard way}) +\begin{eqnarray} +Z_{ab} &=& *(\theta_{(a}\wedge Z_{b)}),\nonumber\\[1mm] +Z_a &=& D [ +2\partial_m\ipr D\widetilde{\Omega}_a{}^{m} +-{\frac{1}{2}}\theta_a\!\wedge +(\partial_m\ipr\partial_n\ipr D\widetilde{\Omega}{}^{mn})] +\nonumber\\&& + +2(\partial_a\ipr\Omega^{MN})\wedge\widetilde{\Omega}_{MN} + + {\rm c.c.}-\partial_a L_{\rm g} +\end{eqnarray} + +Since gravitational equations are computed in the +spinorial formalism with the standard null frame +\seethis{See \pref{spinors}\ or \pref{spinors1}.} +the metric equation is complex and components $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +02 +\begin_inset Formula $, +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +12 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +22 +\begin_inset Formula $ are conjugated to $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +03 +\begin_inset Formula $, +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +13 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +33 +\begin_inset Formula $. +For the sake of efficiency by default \grg\ computes only +the components $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +00 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +01 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +02 +\begin_inset Formula $, +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +11 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +12 +\begin_inset Formula $, $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +22 +\begin_inset Formula $ and $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +scriptstyle +\end_layout + +\end_inset + +23 +\begin_inset Formula $ +only. If you want to have all components the switch \comm{FULL} must be +turned on. \swinda{FULL} + +These equations allows one to compute field equations for +gravity theory with an arbitrary Lagrangian. +But the value of three quantities $ +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ must be specified by user. +In addition \grg\ has built-in +formulas for the most general quadratic in the curvature +Lagrangian. The {\tt Standard way} for $ +\end_inset + +L +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +rm +\end_layout + +\end_inset + + g +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ and +$ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +widetilde +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Omega +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +{ +\end_layout + +\end_inset + +AB +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset Formula $ is +\begin{eqnarray} +\widetilde{\Omega}_{AB} &=& +i(\lambda_0-\sigma8\pi G\, a_0\phi^2)\, S_{AB} \nonumber\\&& ++i\lambda_1 \OO{w}_{AB} +-i\lambda_2 \OO{c}_{AB} ++i\lambda_3 \OO{r}_{AB}, \\[2mm] +L_{\rm g} &=& (-2\Lambda +{\frac{1}{2}}\lambda_0R +-\sigma4\pi G a_0 \phi^2 R) \upsilon ++ \Omega^{AB}\wedge\widetilde{\Omega}_{AB} + {\rm c.c.} +\end{eqnarray} + +The cosmological term is included into +equations iff the switch \comm{CCONST} is on \swinda{CCONST} +and the value of $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +Lambda +\end_layout + +\end_inset + + +\begin_inset Formula $ is given by the constant \comm{CCONST}. +The term with the scalar field $ +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + +\backslash +phi +\end_layout + +\end_inset + + +\begin_inset Formula $ is included into +equations iff the switch \comm{NONMIN} is on. \swinda{NONMIN} +The gravitational constant $ +\end_inset + +G +\begin_inset Formula $ is given by the constant \comm{GCONST}. +The parameters of the quadratic lagrangian are given by the object +\object{L-Constants LCONST.i6}{\lambda_i} +\object{A-Constants ACONST.i2}{a_i} +The default value of these objects ({\tt Standard way}) is +\begin{eqnarray} +\lambda_i &=& (\mbox{\tt LC0},\mbox{\tt LC1},\mbox{\tt LC2},\mbox{\tt LC3},\mbox{\tt LC4},\mbox{\tt LC5},\mbox{\tt LC6}), \\ +a_i &=& (\mbox{\tt AC0},0,0) +\end{eqnarray} + + + +\appendix + +\chapter{\grg\ Switches}\vspace*{-6mm} +\index{Switches} + +\tabcolsep=1.5mm + +\begin{tabular}{|c|c|l|c|} +\hline +Switch & Default &\qquad Description & See \\ + & State & & page\\ +\hline +\tt AEVAL & Off & Use {\tt AEVAL} instead of {\tt REVAL}. &\pageref{AEVAL}\\ +\tt WRS & On & Re-simplify object before printing. &\pageref{WRS}\\ +\tt WMATR & Off & Write 2-index objects in matrix form. &\pageref{WMATR}\\ +\tt TORSION & Off & Torsion. &\pageref{TORSION}\\ +\tt NONMETR & Off & Nonmetricity. &\pageref{NONMETR}\\ +\tt UNLCORD & On & Save coordinates in {\tt Unload}. &\pageref{UNLCORD}\\ +\tt AUTO & On & Automatic object calculation in expressions. &\pageref{AUTO}\\ +\tt TRACE & On & Trace the calculation process. &\pageref{TRACE}\\ +\tt SHOWCOMMANDS & Off & Show compound command expansion. &\pageref{SHOWCOMMANDS}\\ +\tt EXPANDSYM & Off & Enable {\tt Sy Asy Cy} in expressions &\pageref{EXPANDSYM}\\ +\tt DFPCOMMUTE & On & Commutativity of {\tt DFP} derivatives. &\pageref{DFPCOMMUTE}\\ +\tt NONMIN & Off & Nonminimal interaction for scalar field. &\pageref{NONMIN}\\ +\tt NOFREEVARS & Off & Prohibit free variables in {\tt Print}. &\pageref{NOFREEVARS}\\ +\tt CCONST & Off & Include cosmological constant in equations. &\pageref{CCONST}\\ +\tt FULL & Off & Number of components in {\tt Metric Equation}. &\pageref{FULL}\\ +\tt LATEX & Off & \LaTeX\ output mode. &\pageref{LATEX}\\ +\tt GRG & Off & \grg\ output mode. &\pageref{GRG}\\ +\tt REDUCE & Off & \reduce\ output mode. &\pageref{REDUCE}\\ +\tt MAPLE & Off & {\sc Maple} output mode. &\pageref{MAPLE}\\ +\tt MATH & Off & {\sc Mathematica} output mode. &\pageref{MATH}\\ +\tt MACSYMA & Off & {\sc Macsyma} output mode. &\pageref{MACSYMA}\\ +\tt DFINDEXED & Off & Print {\tt DF} in index notation. &\pageref{DFINDEXED}\\ +\tt BATCH & Off & Batch mode. &\pageref{BATCH}\\ +\tt HOLONOMIC & On & Keep frame holonomic. &\pageref{HOLONOMIC}\\ +\tt SHOWEXPR & Off & Print expressions during algebraic &\pageref{SHOWEXPR}\\ +\tt & & classification. &\\ +\hline +\end{tabular} + +\chapter{Macro Objects} +\index{Macro Objects} + +Macro objects can be used in expression, in {\tt Write} and +{\tt Show} commands but not in the {\tt Find} command. +The notation for indices is the same as in the {\tt New Object} +declaration (see page \pageref{indices}). + +\begin{center} + +\section{Dimension and Signature} + +\begin{tabular}{|l|l|} +\hline +\tt dim & Dimension $d$ \\ +\hline +\tt sdiag.idim & {\tt sdiag(\parm{n})} is the $n$'th element of the \\ + & signature diag($-1,+1$\dots) \\ +\hline +\tt sign & Product of the signature specification \\ +\tt sgnt & elements $\prod_{n=0}^{d-1}\mbox{\tt sdiag(}n\mbox{\tt)}$ \\[1mm] +\hline +\tt mpsgn & {\tt sdiag(0)} \\ +\tt pmsgn & {\tt -sdiag(0)} \\ +\hline +\end{tabular} + +\section{Metric and Frame} + +\begin{tabular}{|l|l|} +\hline +\tt x\^m & $m$'th coordinate \\ +\tt X\^m & \\ +\hline +\tt h'a\_m & Frame coefficients \\ +\tt hi.a\^m & \\ +\hline +\tt g\_m\_n & Holonomic metric \\ +\tt gi\^m\^n & \\ +\hline +\end{tabular} + +\section{Delta and Epsilon Symbols} + +\begin{tabular}{|l|l|} +\hline +\tt del'a.b & Delta symbols \\ +\tt delh\^m\_n & \\ +\hline +\tt eps.a.b.c.d & Totally antisymmetric symbols \\ +\tt epsi'a'b'c'd & (number of indices depend on $d$) \\ +\tt epsh\_m\_n\_p\_q & \\ +\tt epsih\^m\^n\^p\^q & \\ +\hline +\end{tabular} + +\section{Spinors} + +\begin{tabular}{|l|l|} +\hline +\tt DEL'A.B & Delta symbol \\ +\hline +\tt EPS.A.B & Spinorial metric \\ +\tt EPSI'A'B & \\ +\hline +\tt sigma'a.A.B\cc & Sigma matrices \\ +\tt sigmai.a'A'B\cc & \\ +\hline +\tt cci.i3 & Frame index conjugation in standard null frame \\ + & {\tt cci(0)=0}\ {\tt cci(1)=1}\ {\tt cci(2)=3}\ {\tt cci(3)=2} \\ +\hline +\end{tabular} + +\section{Connection Coefficients} + +\begin{tabular}{|l|l|} +\hline +\tt CHR\^m\_n\_p & Christoffel symbols $\{{}^\mu_{\nu\pi}\}$ \\ +\tt CHRF\_m\_n\_p & and $[{}_{\mu},_{\nu\pi}]$ \\ +\tt CHRT\_m & Christoffel symbol trace $\{{}^\pi_{\pi\mu}\}$ \\ +\hline +\tt SPCOEF.AB.c & Spin coefficients $\omega_{AB\,c}$ \\ +\hline +\end{tabular} + +\section{NP Formalism} + +\begin{tabular}{|l|c|} +\hline +\tt PHINP.AB.CD~ & $\Phi_{AB\dot{c}\dot{D}}$ \\ +\tt PSINP.ABCD & $\Psi_{ABCD}$ \\ +\hline +\tt alphanp & $\alpha$ \\ +\tt betanp & $\beta$ \\ +\tt gammanp & $\gamma$ \\ +\tt epsilonnp & $\epsilon$ \\ +\tt kappanp & $\kappa$ \\ +\tt rhonp & $\rho$ \\ +\tt sigmanp & $\sigma$ \\ +\tt taunp & $\tau$ \\ +\tt munp & $\mu$ \\ +\tt nunp & $\nu$ \\ +\tt lambdanp & $\lambda$ \\ +\tt pinp & $\pi$ \\ +\hline +\tt DD & $D$ \\ +\tt DT & $\Delta$ \\ +\tt du & $\delta$ \\ +\tt dd & $\overline\delta$ \\ +\hline +\end{tabular} + +\end{center} + +\chapter{Objects} + +Here we present the complete list of built-in objects +with names and identifiers. +The notation for indices is the same as in the +{\tt New Object} declaration (see page \pageref{indices}). +Some names (group names) refer to a set of objects. +For example the group name {\tt Spinorial S - forms} below +denotes {\tt SU.AB} and {\tt SD.AB\cc} + +\begin{center} + + +\section{Metric, Frame, Basis, Volume \dots} +\begin{tabular}{|l|l|}\hline +\tt Frame &\tt T'a\\ +\tt Vector Frame &\tt D.a\\ +\hline +\tt Metric &\tt G.a.b\\ +\tt Inverse Metric &\tt GI'a'b\\ +\tt Det of Metric &\tt detG\\ +\tt Det of Holonomic Metric &\tt detg\\ +\tt Sqrt Det of Metric &\tt sdetG\\ +\hline +\tt Volume &\tt VOL\\ +\hline +\tt Basis &\tt b'idim \\ +\tt Vector Basis &\tt e.idim \\ +\hline +\tt S-forms &\tt S'a'b\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial S-forms} \\ +\tt Undotted S-forms &\tt SU.AB\\ +\tt Dotted S-forms &\tt SD.AB\cc\\ +\hline\end{tabular} + +\section{Rotation Matrices} +\begin{tabular}{|l|l|}\hline +\tt Frame Transformation &\tt L'a.b \\ +\tt Spinorial Transformation &\tt LS.A'B \\ +\hline\end{tabular} + +\section{Connection and related objects} +\begin{tabular}{|l|l|}\hline +\tt Frame Connection &\tt omega'a.b\\ +\tt Holonomic Connection &\tt GAMMA\^m\_n\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial Connection}\\ +\tt Undotted Connection &\tt omegau.AB\\ +\tt Dotted Connection &\tt omegad.AB\cc\\ +\hline +\tt Riemann Frame Connection &\tt romega'a.b\\ +\tt Riemann Holonomic Connection &\tt RGAMMA\^m\_n\\ +\hline +\multicolumn{2}{|c|}{\tt Riemann Spinorial Connection}\\ +\tt Riemann Undotted Connection &\tt romegau.AB\\ +\tt Riemann Dotted Connection &\tt romegad.AB\cc\\ +\hline +\tt Connection Defect &\tt K'a.b\\ +\hline\end{tabular} + +\section{Torsion} +\begin{tabular}{|l|l|}\hline +\tt Torsion &\tt THETA'a\\ +\tt Contorsion &\tt KQ'a.b\\ +\tt Torsion Trace 1-form &\tt QQ\\ +\tt Antisymmetric Torsion 3-form &\tt QQA\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial Contorsion}\\ +\tt Undotted Contorsion &\tt KU.AB\\ +\tt Dotted Contorsion &\tt KD.AB\cc\\ +\hline +\multicolumn{2}{|c|}{\tt Torsion Spinors }\\ +\multicolumn{2}{|c|}{\tt Torsion Components }\\ +\tt Torsion Trace &\tt QT'a\\ +\tt Torsion Pseudo Trace &\tt QP'a\\ +\tt Traceless Torsion Spinor &\tt QC.ABC.D\cc\\ +\hline +\multicolumn{2}{|c|}{\tt Torsion 2-forms}\\ +\tt Traceless Torsion 2-form &\tt THQC'a\\ +\tt Torsion Trace 2-form &\tt THQT'a\\ +\tt Antisymmetric Torsion 2-form &\tt THQA'a\\ +\hline +\multicolumn{2}{|c|}{\tt Undotted Torsion 2-forms}\\ +\tt Undotted Torsion Trace 2-form &\tt THQTU'a\\ +\tt Undotted Antisymmetric Torsion 2-form &\tt THQAU'a\\ +\tt Undotted Traceless Torsion 2-form &\tt THQCU'a\\ +\hline\end{tabular} + + +\section{Curvature} + +\label{curspincoll} +\begin{tabular}{|l|l|}\hline +\tt Curvature &\tt OMEGA'a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial Curvature}\\ +\tt Undotted Curvature &\tt OMEGAU.AB\\ +\tt Dotted Curvature &\tt OMEGAD.AB\cc\\ +\hline +\tt Riemann Tensor &\tt RIM'a.b.c.d\\ +\tt Ricci Tensor &\tt RIC.a.b\\ +\tt A-Ricci Tensor &\tt RICA.a.b\\ +\tt S-Ricci Tensor &\tt RICS.a.b\\ +\tt Homothetic Curvature &\tt OMEGAH\\ +\tt Einstein Tensor &\tt GT.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Curvature Spinors}\\ +\multicolumn{2}{|c|}{\tt Curvature Components}\\ +\tt Weyl Spinor &\tt RW.ABCD\\ +\tt Traceless Ricci Spinor &\tt RC.AB.CD\cc\\ +\tt Scalar Curvature &\tt RR\\ +\tt Ricanti Spinor &\tt RA.AB\\ +\tt Traceless Deviation Spinor &\tt RB.AB.CD\cc\\ +\tt Scalar Deviation &\tt RD\\ +\hline +\multicolumn{2}{|c|}{\tt Undotted Curvature 2-forms}\\ +\tt Undotted Weyl 2-form &\tt OMWU.AB \\ +\tt Undotted Traceless Ricci 2-form &\tt OMCU.AB \\ +\tt Undotted Scalar Curvature 2-form &\tt OMRU.AB \\ +\tt Undotted Ricanti 2-form &\tt OMAU.AB \\ +\tt Undotted Traceless Deviation 2-form &\tt OMBU.AB \\ +\tt Undotted Scalar Deviation 2-form &\tt OMDU.AB \\ +\hline +\multicolumn{2}{|c|}{\tt Curvature 2-forms}\\ +\tt Weyl 2-form &\tt OMW.a.b \\ +\tt Traceless Ricci 2-form &\tt OMC.a.b \\ +\tt Scalar Curvature 2-form &\tt OMR.a.b \\ +\tt Ricanti 2-form &\tt OMA.a.b \\ +\tt Traceless Deviation 2-form &\tt OMB.a.b \\ +\tt Antisymmetric Curvature 2-form &\tt OMD.a.b \\ +\tt Homothetic Curvature 2-form &\tt OSH.a.b \\ +\tt Antisymmetric S-Ricci 2-form &\tt OSA.a.b \\ +\tt Traceless S-Ricci 2-form &\tt OSC.a.b \\ +\tt Antisymmetric S-Curvature 2-form &\tt OSV.a.b \\ +\tt Symmetric S-Curvature 2-form &\tt OSU.a.b \\ +\hline +\end{tabular} + + +\section{Nonmetricity} +\begin{tabular}{|l|l|}\hline +\tt Nonmetricity &\tt N.a.b\\ +\tt Nonmetricity Defect &\tt KN'a.b\\ +\tt Weyl Vector &\tt NNW\\ +\tt Nonmetricity Trace &\tt NNT\\ +\hline +\multicolumn{2}{|c|}{\tt Nonmetricity 1-forms}\\ +\tt Symmetric Nonmetricity 1-form &\tt NC.a.b\\ +\tt Antisymmetric Nonmetricity 1-form &\tt NA.a.b\\ +\tt Nonmetricity Trace 1-form &\tt NT.a.b\\ +\tt Weyl Nonmetricity 1-form &\tt NW.a.b\\ +\hline\end{tabular} + + +\section{EM field} +\begin{tabular}{|l|l|}\hline +\tt EM Potential &\tt A\\ +\tt Current 1-form &\tt J\\ +\tt EM Action &\tt EMACT\\ +\tt EM 2-form &\tt FF\\ +\tt EM Tensor &\tt FT.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Maxwell Equations}\\ +\tt First Maxwell Equation &\tt MWFq\\ +\tt Second Maxwell Equation &\tt MWSq\\ +\hline +\tt Continuity Equation &\tt COq\\ +\tt EM Energy-Momentum Tensor &\tt TEM.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt EM Scalars}\\ +\tt First EM Scalar &\tt SCF\\ +\tt Second EM Scalar &\tt SCS\\ +\hline +\tt Selfduality Equation &\tt SDq.AB\cc\\ +\tt Complex EM 2-form &\tt FFU\\ +\tt Complex Maxwell Equation &\tt MWUq\\ +\tt Undotted EM Spinor &\tt FIU.AB\\ +\tt Complex EM Scalar &\tt SCU\\ +\tt EM Energy-Momentum Spinor &\tt TEMS.AB.CD\cc\\ +\hline\end{tabular} + +\section{Scalar field} +\begin{tabular}{|l|l|}\hline +\tt Scalar Equation &\tt SCq\\ +\tt Scalar Field &\tt FI\\ +\tt Scalar Action &\tt SACT\\ +\tt Minimal Scalar Action &\tt SACTMIN\\ +\tt Minimal Scalar Energy-Momentum Tensor &\tt TSCLMIN.a.b\\ +\hline\end{tabular} + + +\section{YM field} +\begin{tabular}{|l|l|}\hline +\tt YM Potential &\tt AYM.i9\\ +\tt Structural Constants &\tt SCONST.i9.j9.k9\\ +\tt YM Action &\tt YMACT\\ +\tt YM 2-form &\tt FFYM.i9\\ +\tt YM Tensor &\tt FTYM.i9.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt YM Equations}\\ +\tt First YM Equation &\tt YMFq.i9\\ +\tt Second YM Equation &\tt YMSq.i9\\ +\hline +\tt YM Energy-Momentum Tensor &\tt TYM.a.b\\ +\hline\end{tabular} + +\section{Dirac field} +\begin{tabular}{|l|l|}\hline +\multicolumn{2}{|c|}{\tt Dirac Spinor}\\ +\tt Phi Spinor &\tt PHI.A\\ +\tt Chi Spinor &\tt CHI.B\\ +\hline +\tt Dirac Action 4-form &\tt DACT\\ +\tt Undotted Dirac Spin 3-Form &\tt SPDIU.AB\\ +\tt Dirac Energy-Momentum Tensor &\tt TDI.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Dirac Equation}\\ +\tt Phi Dirac Equation &\tt DPq.A\cc\\ +\tt Chi Dirac Equation &\tt DCq.A\cc\\ +\hline\end{tabular} + +\section{Geodesics} +\begin{tabular}{|l|l|}\hline +\tt Geodesic Equation &\tt GEOq\^m\\ +\hline\end{tabular} + +\section{Null Congruence} +\begin{tabular}{|l|l|}\hline +\tt Congruence &\tt KV\\ +\tt Null Congruence Condition &\tt NCo\\ +\tt Geodesics Congruence Condition&\tt GCo'a\\ +\hline +\multicolumn{2}{|c|}{\tt Optical Scalars}\\ +\tt Congruence Expansion &\tt thetaO\\ +\tt Congruence Squared Rotation &\tt omegaSQO\\ +\tt Congruence Squared Shear &\tt sigmaSQO\\ +\hline\end{tabular} + +\section{Kinematics} +\begin{tabular}{|l|l|}\hline +\tt Velocity Vector &\tt UV\\ +\tt Velocity &\tt UU'a\\ +\tt Velocity Square &\tt USQ\\ +\tt Projector &\tt PR'a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Kinematics}\\ +\tt Acceleration &\tt accU'a\\ +\tt Vorticity &\tt omegaU.a.b\\ +\tt Volume Expansion &\tt thetaU\\ +\tt Shear &\tt sigmaU.a.b\\ +\hline\end{tabular} + +\section{Ideal and Spin Fluid} +\begin{tabular}{|l|l|}\hline +\tt Pressure &\tt PRES\\ +\tt Energy Density &\tt ENER\\ +\tt Ideal Fluid Energy-Momentum Tensor &\tt TIFL.a.b\\ +\hline +\tt Spin Fluid Energy-Momentum Tensor &\tt TSFL.a.b \\ +\tt Spin Density &\tt SPFLT.a.b \\ +\tt Spin Density 2-form &\tt SPFL \\ +\tt Undotted Fluid Spin 3-form &\tt SPFLU.AB \\ +\tt Frenkel Condition &\tt FCo \\ +\hline\end{tabular} + +\section{Total Energy-Momentum and Spin} +\begin{tabular}{|l|l|}\hline +\tt Total Energy-Momentum Tensor &\tt TENMOM.a.b\\ +\tt Total Energy-Momentum Spinor &\tt TENMOMS.AB.CD\cc\\ +\tt Total Energy-Momentum Trace &\tt TENMOMT\\ +\tt Total Undotted Spin 3-form &\tt SPINU.AB\\ +\hline\end{tabular} + +\section{Einstein Equations} +\begin{tabular}{|l|l|}\hline +\tt Einstein Equation &\tt EEq.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Spinor Einstein Equations}\\ +\tt Traceless Einstein Equation &\tt CEEq.AB.CD\cc\\ +\tt Trace of Einstein Equation &\tt TEEq\\ +\hline\end{tabular} + +\section{Constants} +\begin{tabular}{|l|l|}\hline +\tt A-Constants &\tt ACONST.i2\\ +\tt L-Constants &\tt LCONST.i6\\ +\tt M-Constants &\tt MCONST.i3\\ +\hline\end{tabular} + +\section{Gravitational Equations} +\begin{tabular}{|l|l|}\hline +\tt Action &\tt LACT\\ +\tt Undotted Curvature Momentum &\tt POMEGAU.AB\\ +\tt Torsion Momentum &\tt PTHETA'a\\ +\hline +\multicolumn{2}{|c|}{\tt Gravitational Equations}\\ +\tt Metric Equation &\tt METRq.a.b\\ +\tt Torsion Equation &\tt TORSq.AB\\ +\hline\end{tabular} + +\end{center} + + +\chapter{Standard Synonymy} +\index{Synonymy} + +Below we present the default synonymy as it is defined in the +global configuration file. See section \ref{tuning} to find out +how to change the default synonymy or define a new one. + +\begin{verbatim} + Affine Aff + Anholonomic Nonholonomic AMode ABasis + Antisymmetric Asy + Change Transform + Classify Class + Components Comp + Connection Con + Constants Const Constant + Coordinates Cord + Curvature Cur + Dimension Dim + Dotted Do + Equation Equations Eq + Erase Delete Del + Evaluate Eval Simplify + Find F Calculate Calc + Form Forms + Functions Fun Function + Generic Gen + Gravitational Gravity Gravitation Grav + Holonomic HMode HBasis + Inverse Inv + Load Restore + Next N + Normalize Normal + Object Obj + Output Out + Parameter Par + Rotation Rot + Scalar Scal + Show ? + Signature Sig + Solutions Solution Sol + Spinor Spin Spinorial Sp + standardlisp lisp + Switch Sw + Symmetries Sym Symmetric + Tensor Tensors Tens + Torsion Tors + Transformation Trans + Undotted Un + Unload Save + Vector Vec + Write W + Zero Nullify +\end{verbatim} + + +\makeatletter +\if@openright\cleardoublepage\else\clearpage\fi +\makeatother +\thispagestyle{empty} +\def\indexname{INDEX} +\printindex + +{document} + +%======== End of grg32.tex ==============================================% + +$ +\end_inset + + + +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\family typewriter + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout +} +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document ADDED doc/grg32.pdf Index: doc/grg32.pdf ================================================================== --- /dev/null +++ doc/grg32.pdf cannot compute difference between binary files ADDED doc/grg32.tex Index: doc/grg32.tex ================================================================== --- /dev/null +++ doc/grg32.tex @@ -0,0 +1,7015 @@ +%==========================================================================% +% GRG 3.2 Manual (C) 1988-97 Vadim V. Zhytnikov % +%==========================================================================% +% LaTeX 2e and MakeIndex are required to pront this document: % +% % +% latex grg32 % +% latex grg32 % +% latex grg32 % +% makeindex grg32 % +% latex grg32 % +% % +% If you do not have MakeIndex just omit two last steps. % +% The document is intended for two-side printing. % +%==========================================================================% + +\documentclass[twoside,openright]{report} + +\oddsidemargin=1.5cm +\evensidemargin=1.3cm + +%%% This is for PS fonts and dvips driver +%\usepackage{mathptm} +%\usepackage{palatino} +%\renewcommand{\bfdefault}{b} +%\newcommand{\grgtt}{\bfseries\ttfamily} +%\usepackage[dvips]{color} +%\definecolor{shade}{gray}{.9} +%\newcommand{\shadedbox}[1]{\fcolorbox{black}{shade}{#1}} +%%% This is for CM fonts +\newcommand{\grgtt}{\ttfamily} +\renewcommand{\ttdefault}{cmtt} +\newcommand{\shadedbox}[1]{\fbox{#1}} +%%% + + +%\usepackage{calrsfs} % rsfs for mathcal + +%%% +\makeatletter +\let\@afterindentfalse\@afterindenttrue +\@afterindenttrue +\makeatother +%%% + +%%% +\usepackage{makeidx} +\makeindex +\newcommand{\cmdind}[1]{\index{Commands!\comm{#1}}\index{#1@\comm{#1} (command)}} +\newcommand{\cmdindx}[2]{\index{Commands!\comm{#1}}\index{#1@\comm{#1} (command)!\comm{#2}}} +\newcommand{\swind}[1]{\index{Switches!\comm{#1}}% +\index{#1@\comm{#1} (switch)}% +\label{#1}} +\newcommand{\swinda}[1]{\index{Switches!\comm{#1}}% +\index{#1@\comm{#1} (switch)}} +%%% + +%%% +\newcommand{\rim}[1]{\stackrel{\scriptscriptstyle\{\}}{#1}\!} +%%% + +%%% +\newcommand{\object}[2]{% +\begin{equation} +\mbox{\comm{#1}} =\ #2 +\end{equation}} +\newcommand{\tsst}{\longleftrightarrow} +\newcommand{\vv}{\vphantom{\rule{5mm}{5mm}}} +\newcommand{\RR}[1]{\stackrel{\rm #1}{R}\!{}} +\newcommand{\OO}[1]{\stackrel{\rm #1}{\Omega}\!{}} +%%% + +%%% +\newcommand{\ipr}{\rule{1.8mm}{.1mm}\rule{.1mm}{2.2mm}\,} % _| int. product +%%% + +%%% +\newcommand{\spref}[1]{section \ref{#1} on page \pageref{#1}} +\newcommand{\pref}[1]{page \pageref{#1}} +%%% + +%%% +\newcommand{\seethis}[1]{\marginpar{\footnotesize\it #1}} +\newcommand{\rseethis}[1]{ +\reversemarginpar +\marginpar{\footnotesize\it #1} +\normalmarginpar} +\newcommand{\important}[1]{\marginpar{\itshape\bfseries\fbox{\ !\ } #1}} +%%% + +%%% Footnotes simbol ... +\renewcommand{\thefootnote}{\fnsymbol{footnote}} % + ++ etc for footnotes +\makeatletter +\def\@fnsymbol#1{\ensuremath{\ifcase#1\or \dagger\or \ddagger\or + \mathchar "278\or \mathchar "27B\or \|\or *\or **\or \dagger\dagger + \or \ddagger\ddagger \else\@ctrerr\fi}} +\makeatother +%%% + +%%% Page layout ... +\textheight=180mm +\textwidth=120mm +%\marginparsep=2mm +%\marginparwidth=28mm +\marginparsep=5mm +\marginparwidth=25mm +\parindent=6mm +\parskip=1.2mm plus 1mm minus 1mm +%%% +\newlength{\myparindent} +\myparindent=\parindent + +%%% My own \tt font ... +\makeatletter +\def\verbatim@font{\grgtt} +\makeatother +\renewcommand{\tt}{\grgtt} +%%% + +%%% +%%% Special symbols ... +\def\^{{\tt \char'136}} %%% \^ is ^ +\def\_{{\tt \char'137}} %%% \_ is _ +\newcommand{\w}{{\tt \char'057 \char'134}} %%% \w is /\ +\newcommand{\bs}{{\tt \char'134}} %%% \bs is \ +\newcommand{\ul}{{\tt \char'137}} %%% \ul is _ +\newcommand{\dd}{{\tt \char'043}} %%% \dd is # +\newcommand{\cc}{{\tt \char'176}} %%% \cc is ~ +\newcommand{\ip}{{\tt \char'137 \char'174}} %%% \ip is _| +\newcommand{\ii}{{\tt \char'174}} %%% \ii is | +\newcommand{\udr}{\mbox{$\Updownarrow$}} +%%% + +%%% \grg GRG logo ... +\newcommand{\grg}{{\sc GRG}} +\newcommand{\reduce}{{\sc Reduce}} +\newcommand{\maple}{{\sc Maple}} +\newcommand{\macsyma}{{\sc Macsyma}} +\newcommand{\mathematica}{{\sc Mathematica}} + +%%% \marg ... +\newcommand{\marg}[1]{\marginpar{\tiny#1}} + +%%% \command{...} commands in (shaded) box +\def\mynewline{\ifvmode \relax \else + \unskip\nobreak\hfil\break\fi} +\newcommand{\command}[1]{\vspace{1.2mm}\mynewline\hspace*{6mm}% +\shadedbox{\begin{tabular}{l}\tt% +#1 \end{tabular}}\vspace{1.2mm}\newline} +%%% parts of the commands +\newcommand{\file}[1]{{\sf#1}} +\newcommand{\comm}[1]{{\upshape\tt#1}} % \comm short in-line command +\newcommand{\parm}[1]{{\sf\slshape#1\/}} % \parm command parameter +\newcommand{\opt}[1]{{\rm[}#1{\rm]}} % \opt optional part of command +\newcommand{\user}[1]{{\bfseries\ttfamily#1}} % \user user input +\newcommand{\rpt}[1]{#1{\rm[}{\tt,}#1{\rm\dots}{\rm]}} % \rpt repetition + + +\def\closerule{\rule{.1mm}{1mm}\rule{119.8mm}{.1mm}} +\def\openrule{\rule{.1mm}{1mm}\rule[1mm]{119.8mm}{.1mm}} + +%%% \begin{slisting} ... \end{slisting} small font listing with frame +%%% \begin{listing} ... \end{listing} normal font listing without frame +\newcommand{\etrivlistrule} +{\vspace*{-3mm}\endtrivlist{\closerule}\newline} +\makeatletter +\newdimen\allttindent +\allttindent=0mm +\def\docspecials{\do\ \do\$\do\&% + \do\#\do\^\do\^^K\do\_\do\^^A\do\%\do\~} +\def\slisting{\vspace*{-2mm}% +\trivlist \item[]\if@minipage\else\relax\fi +\leftskip\@totalleftmargin \advance\leftskip\allttindent \rightskip\z@ +\parindent\z@\parfillskip\@flushglue\parskip\z@ +\@tempswafalse\openrule \def\par{\if@tempswa\hbox{}\fi\@tempswatrue\@@par} +\obeylines \small\grgtt% + \catcode``=13 \@noligs +\let\do\@makeother \docspecials + \frenchspacing\@vobeyspaces} +\def\listing{\trivlist \item[]\if@minipage\else\relax\fi +\leftskip\@totalleftmargin \advance\leftskip\allttindent \rightskip\z@ +\parindent\z@\parfillskip\@flushglue\parskip\z@ +\@tempswafalse \def\par{\if@tempswa\hbox{}\fi\@tempswatrue\@@par} +\obeylines \grgtt% + \catcode``=13 \@noligs +\let\do\@makeother \docspecials + \frenchspacing\@vobeyspaces} +\let\endslisting=\etrivlistrule +\let\endlisting=\endtrivlist +\makeatother +%%% + +%%% Headings style ... +%\usepackage{fancyheadings} +%%% We just inserat the fancyheadings.sty here literally ... +\makeatletter +% fancyheadings.sty version 1.7 +% Fancy headers and footers. +% Piet van Oostrum, Dept of Computer Science, University of Utrecht +% Padualaan 14, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands +% Telephone: +31-30-531806. piet@cs.ruu.nl (mcvax!sun4nl!ruuinf!piet) +% Sep 16, 1994 +% version 1.4: Correction for use with \reversemargin +% Sep 29, 1994: +% version 1.5: Added the \iftopfloat, \ifbotfloat and \iffloatpage commands +% Oct 4, 1994: +% version 1.6: Reset single spacing in headers/footers for use with +% setspace.sty or doublespace.sty +% Oct 4, 1994: +% version 1.7: changed \let\@mkboth\markboth to +% \def\@mkboth{\protect\markboth} to make it more robust + +\def\lhead{\@ifnextchar[{\@xlhead}{\@ylhead}} +\def\@xlhead[#1]#2{\gdef\@elhead{#1}\gdef\@olhead{#2}} +\def\@ylhead#1{\gdef\@elhead{#1}\gdef\@olhead{#1}} + +\def\chead{\@ifnextchar[{\@xchead}{\@ychead}} +\def\@xchead[#1]#2{\gdef\@echead{#1}\gdef\@ochead{#2}} +\def\@ychead#1{\gdef\@echead{#1}\gdef\@ochead{#1}} + +\def\rhead{\@ifnextchar[{\@xrhead}{\@yrhead}} +\def\@xrhead[#1]#2{\gdef\@erhead{#1}\gdef\@orhead{#2}} +\def\@yrhead#1{\gdef\@erhead{#1}\gdef\@orhead{#1}} + +\def\lfoot{\@ifnextchar[{\@xlfoot}{\@ylfoot}} +\def\@xlfoot[#1]#2{\gdef\@elfoot{#1}\gdef\@olfoot{#2}} +\def\@ylfoot#1{\gdef\@elfoot{#1}\gdef\@olfoot{#1}} + +\def\cfoot{\@ifnextchar[{\@xcfoot}{\@ycfoot}} +\def\@xcfoot[#1]#2{\gdef\@ecfoot{#1}\gdef\@ocfoot{#2}} +\def\@ycfoot#1{\gdef\@ecfoot{#1}\gdef\@ocfoot{#1}} + +\def\rfoot{\@ifnextchar[{\@xrfoot}{\@yrfoot}} +\def\@xrfoot[#1]#2{\gdef\@erfoot{#1}\gdef\@orfoot{#2}} +\def\@yrfoot#1{\gdef\@erfoot{#1}\gdef\@orfoot{#1}} + +\newdimen\headrulewidth +\newdimen\footrulewidth +\newdimen\plainheadrulewidth +\newdimen\plainfootrulewidth +\newdimen\headwidth +\newif\if@fancyplain \@fancyplainfalse +\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi} + +% Command to reset various things in the headers: +% a.o. single spacing (taken from setspace.sty) +% and the catcode of ^^M (so that epsf files in the header work if a +% verbatim crosses a page boundary) +\def\fancy@reset{\restorecr + \def\baselinestretch{1}% + \ifx\undefined\@newbaseline% NFSS not present; 2.09 or 2e + \ifx\@currsize\normalsize\@normalsize\else\@currsize\fi% + \else% NFSS (2.09) present + \@newbaseline% + \fi} + +% Initialization of the head and foot text. + +\headrulewidth 0.4pt +\footrulewidth\z@ +\plainheadrulewidth\z@ +\plainfootrulewidth\z@ + +\lhead[\fancyplain{}{\sl\rightmark}]{\fancyplain{}{\sl\leftmark}} +% i.e. empty on ``plain'' pages \rightmark on even, \leftmark on odd pages +\chead{} +\rhead[\fancyplain{}{\sl\leftmark}]{\fancyplain{}{\sl\rightmark}} +% i.e. empty on ``plain'' pages \leftmark on even, \rightmark on odd pages +\lfoot{} +\cfoot{\rm\thepage} % page number +\rfoot{} + +% Put together a header or footer given the left, center and +% right text, fillers at left and right and a rule. +% The \lap commands put the text into an hbox of zero size, +% so overlapping text does not generate an errormessage. + +\def\@fancyhead#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset\vbox{\hbox +{\rlap{\parbox[b]{\headwidth}{\raggedright#2\strut}}\hfill +\parbox[b]{\headwidth}{\centering#3\strut}\hfill +\llap{\parbox[b]{\headwidth}{\raggedleft#4\strut}}}\headrule}}#5} + + +\def\@fancyfoot#1#2#3#4#5{#1\hbox to\headwidth{\fancy@reset\vbox{\footrule +\hbox{\rlap{\parbox[t]{\headwidth}{\raggedright#2\strut}}\hfill +\parbox[t]{\headwidth}{\centering#3\strut}\hfill +\llap{\parbox[t]{\headwidth}{\raggedleft#4\strut}}}}}#5} + +\def\headrule{{\if@fancyplain\headrulewidth\plainheadrulewidth\fi +\hrule\@height\headrulewidth\@width\headwidth \vskip-\headrulewidth}} + +\def\footrule{{\if@fancyplain\footrulewidth\plainfootrulewidth\fi +\vskip-0.3\normalbaselineskip\vskip-\footrulewidth +\hrule\@width\headwidth\@height\footrulewidth\vskip0.3\normalbaselineskip}} + +\def\ps@fancy{ +\def\@mkboth{\protect\markboth} +\@ifundefined{chapter}{\def\sectionmark##1{\markboth +{\uppercase{\ifnum \c@secnumdepth>\z@ + \thesection\hskip 1em\relax \fi ##1}}{}} +\def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne + \thesubsection\hskip 1em\relax \fi ##1}}} +{\def\chaptermark##1{\markboth {\uppercase{\ifnum \c@secnumdepth>\m@ne + \@chapapp\ \thechapter. \ \fi ##1}}{}} +\def\sectionmark##1{\markright{\uppercase{\ifnum \c@secnumdepth >\z@ + \thesection. \ \fi ##1}}}} +\ps@@fancy +\global\let\ps@fancy\ps@@fancy +\headwidth\textwidth} +\def\ps@fancyplain{\ps@fancy \let\ps@plain\ps@plain@fancy} +\def\ps@plain@fancy{\@fancyplaintrue\ps@@fancy} +\def\ps@@fancy{ +\def\@oddhead{\@fancyhead\@lodd\@olhead\@ochead\@orhead\@rodd} +\def\@oddfoot{\@fancyfoot\@lodd\@olfoot\@ocfoot\@orfoot\@rodd} +\def\@evenhead{\@fancyhead\@rodd\@elhead\@echead\@erhead\@lodd} +\def\@evenfoot{\@fancyfoot\@rodd\@elfoot\@ecfoot\@erfoot\@lodd} +} +\def\@lodd{\if@reversemargin\hss\else\relax\fi} +\def\@rodd{\if@reversemargin\relax\else\hss\fi} + +\let\latex@makecol\@makecol +\def\@makecol{\let\topfloat\@toplist\let\botfloat\@botlist\latex@makecol} +\def\iftopfloat#1#2{\ifx\topfloat\empty #2\else #1\fi} +\def\ifbotfloat#1#2{\ifx\botfloat\empty #2\else #1\fi} +\def\iffloatpage#1#2{\if@fcolmade #1\else #2\fi} +\makeatother +%%% +\pagestyle{fancy} +\addtolength{\headwidth}{\marginparsep} +\addtolength{\headwidth}{\marginparwidth} +\lhead[\bfseries\thepage]{\bfseries\slshape\rightmark} +\chead{} +\rhead[\bfseries\slshape\leftmark]{\bfseries\thepage} +\lfoot{} +\cfoot{} +\rfoot{} +\renewcommand{\uppercase}[1]{#1} +%%% + +%%% Chapter style ... +\makeatletter +\def\@makechapterhead#1{% + \noindent\grgrule\break% + { \hsize=150mm + \parindent \z@ \raggedleft \reset@font + \ifnum \c@secnumdepth >\m@ne + \Large\slshape \@chapapp{} \Huge\bfseries \thechapter + \par + \vskip 20\p@ + \fi + \Huge \bfseries\upshape #1\par + \nobreak + \vskip 40\p@ + }} +\def\@makeschapterhead#1{% + \noindent\grgrule\break% + { \hsize=150mm + \parindent \z@ \raggedleft + \reset@font + \Large\slshape #1\par + \nobreak + \vskip 20\p@ + }} +\renewcommand\chapter{\if@openright\cleardoublepage\else\clearpage\fi + \thispagestyle{empty}% + \global\@topnum\z@ + %\@afterindentfalse + \secdef\@chapter\@schapter} +\makeatother +\renewcommand{\chaptername}{CHAPTER} +\renewcommand{\contentsname}{CONTENTS} +\renewcommand{\appendixname}{APPENDIX} +\newcommand{\grgrule}{\rule{150mm}{.3mm}\relax} +%%% + +%%% Sections ... +%\renewcommand{\thesection}{} +%\renewcommand{\thesubsection}{} +%\renewcommand{\thesubsubsection}{} +\makeatletter +%\renewcommand\section{\@startsection {section}{1}{\z@}% +% {-3.5ex \@plus -1ex \@minus -.2ex}% +% {2.3ex \@plus.2ex}% +% {\normalfont\Large\bfseries}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-3.25ex\@plus -1ex \@minus -.2ex}% + {1.5ex \@plus .2ex}% + {\normalfont\large\slshape\bfseries}} +%\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% +% {-3.25ex\@plus -1ex \@minus -.2ex}% +% {1.5ex \@plus .2ex}% +% {\normalfont\normalsize\bfseries}} +\makeatother +%%% + + + +\begin{document} + + +\begin{titlepage} +\hsize=150mm +\hrulefill +\vspace*{20mm} +\begin{center} +\Huge\bf GRG\\[1mm] +\normalsize Version 3.2 +\end{center} +\begin{center} +\Large Computer Algebra System for\\ +Differential Geometry,\\ +Gravitation and \\ +Field Theory +\vspace*{25mm}\\ +{\Large\itshape\bfseries Vadim V. Zhytnikov}\\ +\vfill +{\normalsize Moscow, 1992--1997 $\bullet$ Chung-Li, 1994} +\end{center} +\hrulefill +\end{titlepage} +\setcounter{page}{0}\thispagestyle{empty} + +\tableofcontents\thispagestyle{empty} + +\chapter{Introduction} + +Calculation of various geometrical and physical quantities and +equations is the usual technical problem which permanently arises +in geometry, field and gravity theory. Numerous indices, +contractions and components make these calculations very tedious +and error-prone. Since this calculus obeys the well defined rules the idea +to automate this kind of problems using computer is quite +natural. Now there are several computer algebra systems such as +\maple, \reduce, \mathematica\ or \macsyma\ which in principle +allow one to do this and it is not so hard +to write a program to calculate, for example, the +curvature tensor or connection. But suppose that we want to +make a non-trivial coordinate transformation or tetrad rotation, +calculate covariant or Lie derivative, compute a complicated +expression with numerous contraction or raise or lower some indices. +All these operations are typical in differential geometry +and field theory but their realization with the help of general +purpose computer algebra systems requires hard programming since +all these systems really know nothing about \emph{covariant properties} +of geometrical quantities. + +The computer algebra system \grg\ is designed in such a way +to make calculation in differential geometry and field theory +as simple and natural as possible. \grg\ is based on the +computer algebra system \reduce\ but \grg\ has its own simple +input language whose commands resembles English phrases. +Working with \grg\ no any knowledge of programming is required. + +\grg\ understands tensors, spinors, vectors, differential forms +and knows all standard operations with these quantities. +Input form for mathematical expressions is very close +to traditional mathematical notation including Einstein summation +rule. \grg\ knows the covariant properties of +these objects, you can easily raise and lower indices, +compute covariant and Lie derivatives, perform +coordinate and frame transformations. +\grg\ works in any dimension and allows one to represent tensor +quantities with respect to holonomic, orthogonal and even +any other arbitrary frame. + +One of the useful features of \grg\ is that it has a large +number of built-in standard field-theory +and geometrical quantities and formulas for their computation. +Thus \grg\ provides ready solutions to many standard problems. + +Another unique feature of \grg\ is that it can export +results of calculations into other computer algebra system. +You can save your data in to the file in the format of +\maple, \mathematica, \macsyma\ or \reduce\ in order to use +this system to proceed analysis of the data. +The \LaTeX\ output format is supported as well. +In addition \grg\ is compatible with \reduce\ graphics +shells providing niece book-quality output with Greek letters, +integral signs etc. + +The main built-in \grg\ capabilities are: +\begin{list}{$\bullet$}{\labelwidth=8mm\leftmargin=10mm} +\item Connection, torsion and nonmetricity. +\item Curvature. +\item Spinorial formalism. +\item Irreducible decomposition of the curvature, torsion, and + nonmetricity in any dimension. +\item Einstein equations. +\item Scalar field with minimal and non-minimal interaction. +\item Electromagnetic field. +\item Yang-Mills field. +\item Dirac spinor field. +\item Geodesic equation. +\item Null congruences and optical scalars. +\item Kinematics for time-like congruences. +\item Ideal and spin fluid. +\item Newman-Penrose formalism. +\item Gravitational equations for the theory with arbitrary + gravitational Lagrangian in Riemann and Riemann-Cartan + spaces. +\end{list} + +I would like to stress that current \grg\ version is +intended for calculations in a concrete coordinate map only. +It cannot operate with tensors as with objects having +abstract symbolic indices. + +This book consist of two main parts. First part +contains detailed description of \grg\ as a programming +system. Second part describes all built-in objects +and formulas for their computation. + + +\chapter{Programming in \grg} + +Throughout the chapter \comm{commands} are printed in +typewriter font. The slanted serif-less font is +used for command \parm{parameters}. +The optional parts of the commands are enclosed in +squared brackets \opt{option} and \rpt{\parm{id}} +stands for one or several repetitions of \parm{id}: +\parm{id} or \comm{\parm{id},\parm{id}} etc. +Examples are separated form the text by horizontal lines +$\stackrel{\rule{0.1mm}{1mm}\rule[1mm]{3mm}{0.1mm}} +{\rule{0.1mm}{1mm}\rule{3mm}{0.1mm}}$ and the user input +can be easily distinguished from the \grg\ output by the prompt +\comm{<-} which precedes every input line. + + +\section{Session, Tasks and Commands} + +To start \grg\ it is necessary to start \reduce\ and +\seethis{ +On some systems you have +to use {\tt\upshape load!\_package grg;}\newline since +{\tt\upshape load} is not defined.\newline +\newline +Sometimes it\newline is better to use two commands\newline +{\tt\upshape load grg32; grg;}\newline +or\newline +{\tt\upshape load grg; grg;}\newline +(See section \ref{configsect} for details.)} +enter the command {\tt load grg;} + +\begin{slisting} +REDUCE 3.5, 15 Oct 93, patched to 15 Jun 95 ... + +1: load grg; + +This is GRG 3.2 release 2 (Feb 9, 1997) ... + +System directory: c:{\bs}reduce{\bs}grg32{\bs} +System variables are upper-cased: E I PI SIN ... +Dimension is 4 with Signature (-,+,+,+) + +<- +\end{slisting} +Symbol \comm{<-} is the \grg\ prompt which shows that +now \grg\ waits for your input. The \grg\ \emph{task} (we prefer +this term instead of usual \emph{program}) consist of the +sequence of commands terminated by semicolon \comm{;}. +Reading the input \grg\ splits it on \emph{atoms}. +There are several types of atoms:\index{Atoms} +\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} +\item The identifier or symbol is a sequence of letters and digits +starting with a letter: +\begin{verbatim} + i I alpha1 beta ABC123D Find +\end{verbatim} +The identifiers in \grg\ may have trailing tilde character \cc. +Any other character may be incorporated in the identifier if +preceded by the exclamation sign:\index{Identifiers} +\begin{verbatim} + beta~ LIMIT!+ +\end{verbatim} +The identifiers in \grg\ play the role of the variables and +functions in mathematical expressions and words in commands. + +\item Integer numbers\index{Numbers} +\begin{verbatim} + 0 123 104341 +\end{verbatim} + +\item String is a sequence of characters enclosed in double quotes\index{Strings} +\begin{verbatim} + "file.txt" "This is a string" "dir *.doc" +\end{verbatim} +The strings in \grg\ are used for file names and operating system +commands. + +\item Nine special two-character atoms +\begin{verbatim} + ** _| /\ |= ~~ .. <= >= -> +\end{verbatim} + +\item Any other characters are considered as single-character atoms. +\end{list} + +The format of \grg\ commands is free. They can span one or several lines +and any number of spaces and tabulations can be inserted between two +neighbor atoms. + +\enlargethispage{3mm} + +The \grg\ session may consist of several independent tasks. +The command\index{Tasks}\cmdind{Quit} +\command{Quit;} +terminates both \grg\ and \reduce\ session and returns the control +to the operating system level. The command\cmdind{Stop} +\command{Stop;} +terminates current \grg\ task and brings +the session control menu:\index{Session control menu} + \begin{slisting} +<- Stop; + + Quit GRG - 0 + Start Task - 1 + Exit to REDUCE - 2 + + Type 0, 1 or 2: +\end{slisting} +\newpage + +\noindent +The option \comm{0} terminates \reduce\ session similarly to the +command \comm{Quit;}. +The choice \comm{1} starts new task by bringing +\grg\ to its initial state: all variables, declarations, substitutions +and results of calculations are cleared and all switches +resume their initial positions.\footnote{Usually +\grg\ does good job by resuming initial state and new task +turns out to be independent of previous ones. But on some +rare occasions the initial state cannot be completely recovered +and it is better to restart \reduce\ and \grg\ completely.} +Finally the option \comm{2} terminates \grg\ task and returns +control to the \reduce\ command level. In this case \grg\ can be +restarted later by the command \comm{grg;}. + +The commands in \grg\ are case insensitive, i.e. command +\comm{Quit;} is equivalent to \comm{quit;} and \comm{QUIT;} etc. +But notice that unlike \reduce\ variables and functions in +mathematical expressions in \grg\ \emph{are case sensitive}. + + +\subsection{Switches} +\index{Switches} + +Switches in \grg\ and \reduce\ are used to control various +system modes of operation. They are denoted by identifiers and +the commands\cmdind{On}\cmdind{Off} +\command{On \rpt{\parm{switch}};\\\tt +Off \rpt{\parm{switch}};} +turns the \parm{switch} on and off respectively. +Any switch defined by \reduce\ is available in \grg\ as well. +In addition \grg\ defines a couple of its own switches. +The full list of \grg\ switches is presented in appendix A. +The command\cmdind{Show Switch}\cmdind{Switch} +\command{\opt{Show} Switch \parm{switch};} +or equivalently +\command{Show \parm{switch};\\\tt +?~\parm{switch};} +prints current \parm{switch} position +\begin{slisting} +<- Show Switch TORSION; +TORSION is Off. +<- On torsion,gcd; +<- switch torsion; +TORSION is On. +<- switch exp; +GCD is On +\end{slisting} +Switches in \grg\ are case insensitive. + +\subsection{Batch File Execution} + +Usually \grg\ works in the interactive mode which +is not always convenient. The command\cmdind{Input}\index{Batch file execution} +\command{\opt{Input} "\parm{file}";} +reads the \parm{file} and executes commands stored in it. +The file names in \grg\ are always denoted by strings and exact +specification of \parm{file} is operating system dependent. +The word \comm{Input} is optional, thus in order to run batch +file it suffices to enter its name \comm{"\parm{file}";}. +The execution of batch file commands can be suspended by the +command\cmdind{Pause} +\command{Pause;} +After this command \grg\ enters the interactive mode. +One can enter one or several commands interactively and then +resume batch file execution by the command\cmdind{Next} +\command{Next;} + +In general no any special end-of-file symbol or command +is required in the \grg\ batch \parm{file} but is necessary +the symbol\index{end-of-file symbol \comm{\$}} +\comm{\$} is recognized by \grg\ as the end-of-file mark. + +If during the batch file execution an error occurs +\grg\ enter interactive mode and ask user +to input the command which is supposed to replace the +erroneous one. After the receiving of \emph{one} command +\grg\ automatically resumes the batch file execution. +The command \comm{Pause;} can be used if it is necessary +to execute \emph{several} commands instead of one. + +The command\cmdind{Output} +\command{Output "\parm{outfile}";} +redirects all \grg\ output into the \parm{outfile}. +The \parm{outfile} can be closed by the equivalent commands +\cmdind{EndO}\cmdind{End of Output} +\command{EndO;\\\tt +End of Output;} + +It is convenient to run long-time \grg\ tasks in background. +The way of doing this depend on the operating system. +For example to execute \grg\ task in background in UNIX it is +necessary to use the following command +\begin{listing} + reduce < task.grg > grg.out & +\end{listing} +Here we assume that the \reduce\ invoking command is \comm{reduce} +and the file \comm{task.grg} contains the \grg\ task commands: +\begin{listing} + load grg; + \parm{grg command}; + \parm{grg command}; + ... + \parm{grg command}; + quit; +\end{listing} +The output of the session will be written into the file \file{grg.out}. + +Since no proper reaction on errors is possible during the +background execution it is good idea to turn the switch +\comm{BATCH} on.\swind{BATCH} This makes \grg\ to terminate +the session immediately in the case of any error. + +\subsection{Operating System Commands} + +The command\cmdind{System} +\command{System "\parm{command}";} +executes the operating system \parm{command}. +The same command without parameters +\command{System;} +temporary suspends \grg\ session and passes the control to the +operating system command level. The details may depend +on the concrete operating system. In particular in UNIX +the command \comm{system;} may fail but UNIX has some +general mechanism for suspending running programs: +you can press \comm{\^Z} to suspend any program and \comm{\%+} +to resume its execution. + + +\subsection{Comments} + +%\reversemarginpar + +The comment commands\cmdind{Comment} +\command{Comment \parm{any text};\\\tt +\% \parm{any text};} +are used to supply additional information to \grg\ tasks +\seethis{See page \pageref{Unload} about the \comm{Unload} command.} +and data saved by the \comm{Unload} command. +The comment can be also attached to the end of any \grg\ command +\command{\parm{grg command} \% \parm{any text};} + +%\normalmarginpar + +\subsection{Timing} + +The command \cmdind{Time}\cmdind{Show Time} +\command{\opt{Show} Time;} +prints time elapsed since the beginning of current \grg\ task +including the percentage of so called garbage collections. +The garbage collection time can be also printed by the +command \cmdind{GC Time}\cmdind{Show GC Time} +\command{\opt{Show} GC Time;} + +If percentage of garbage collections grows and +exceeds say 30\% then memory of your system +is running short and you probably need more RAM. + + +\section{Declarations} + +Any object, variable or function in \grg\ must be declared. +This allows to locate misprints and makes the system more +reliable. Since \grg\ always work in some concrete +coordinate system (map) the coordinate declaration is the +most important one and must be present in every \grg\ task. + +\subsection{Dimension and Signature} + +During installation \grg\ always defines default value of +the dimension and signature.\index{Dimension!default}\index{Signature!default} +\seethis{See \pref{tuning} +to find out how to change the default dimension and signature.} +The information about this default value is printed\index{Dimension}\index{Signature} +upon \grg\ start in the form of the following (or similar) message line: +\begin{slisting} +Dimension is 4 with Signature (-,+,+,+) +\end{slisting} + + +The following command overrides the default dimension and signature\cmdind{Dimension} +\command{Dimension \parm{dim} with \opt{Signature} (\rpt{\parm{pm}});} +where \parm{dim} is the number \comm{2} or greater and \parm{pm} +is \comm{+} or \comm{-}. The \parm{pm} can be preceded or succeeded by +a number which denotes several repetitions of this \parm{pm}. +For example the declarations +\begin{listing} + Dimension 5 with Signature (+,+,-,-,-); + Dimension 5 with (2+,-3); +\end{listing} +are equivalent and defines 5-dimensional space with the +signature ${\rm diag}{\scriptstyle(+1,+1,-1,}$ ${\scriptstyle-1,-1)}$. + +The important point is that the dimension declaration must +be \emph{very first in the task} and goes before any other command. +Current dimension and signature can be printed by the command +\cmdind{Status}\cmdind{Show Status} +\command{\opt{Show} Status;} + + + +\subsection{Coordinates} + +The coordinate declaration command must be present in every +\grg\ task\cmdind{Coordinates} +\command{Coordinates \rpt{\parm{id}};} +Only few commands such as informational commands, other declarations, +switch changing commands may precede the coordinate declaration. +The only way to have a tusk without the coordinate declaration is +to load the file where coordinates where saved by the +\comm{Unload} command.\seethis{See \pref{UnloadLoad} +to find out how to save data and declarations into a file.} +but no any computation can be done before coordinates are +declared. Current coordinate list can be printed by the command\cmdindx{Write}{Coordinates} +\command{Write Coordinates;} + + +\begin{table} +\begin{center}\index{Constants!predefined} +\begin{tabular}{|l|l|} +\hline +\tt E I PI INFINITY & Mathematical constants $e,i,\pi$,$\infty$ \\ +\hline +\tt FAILED & \\ +\hline +\tt ECONST & Charge of the electron \\ +\tt DMASS & Dirac field mass \\ +\tt SMASS & Scalar field mass \\ +\hline +\tt GCONST & Gravitational constant \\ +\tt CCONST & Cosmological constants \\ +\hline +\tt LC0 LC1 LC2 LC3 & Parameters of the quadratic \\ +\tt LC4 LC5 LC6 & gravitational Lagrangian \\ +\tt MC1 MC2 MC3 & \\ +\hline +\tt AC0 & Nonminimal interaction constant \\ +\hline +\end{tabular} +\caption{Predefined constants}\label{predefconstants} +\end{center} +\end{table} + + +\subsection{Constants} +\index{Constants} + +Any constant must be declared by the command\cmdind{Constants} +\command{Constants \rpt{\parm{id}};} +The list of currently declared constants can be printed +by the command\cmdindx{Write}{Constants} +\command{Write Constants;} +There are also a number of built-in constants +which are listed in table \ref{predefconstants}. + +\subsection{Functions} + +Functions in \grg\ are the analogues of the \reduce\ \emph{operators} +but we prefer to use this traditional mathematical term. +The function must be declared by the command\cmdind{Functions} +\command{Functions \rpt{\parm{f}\opt{(\rpt{\parm{x}})}};} +Here \parm{f} is the function identifier. The optional list +of parameters \parm{x} defines function with \emph{implicit} +dependence. The \parm{x} must be either coordinate or constant. +The construction \comm{\parm{f}(*)} is a shortcut which +declares the function \parm{f} depending on \emph{all coordinates}. + +The following example declares three functions +\comm{fun1}, \comm{fun2} and \comm{fun3}. +The function \comm{fun1}, which was declared without implicit +coordinate list, must be always used in mathematical expressions +together with the explicit arguments like \comm{fun1(x+y)} etc. +The functions \comm{fun2} and \comm{fun3} can appear +in expressions in similar fashion but also as a single symbol +\comm{fun2} or \comm{fun3} +\begin{slisting} +<- Coordinates t, x, y, z; +<- Constant a; +<- Functions fun1, fun2(x,y), fun3(*); +<- Write functions; +Functions: + +fun1 fun2(x,y) fun3(t,x,y,z) + +<- d fun1(x+a); + +DF(fun1(a + x),x) d x + +<- d fun2; + +DF(fun2,x) d x + DF(fun2,y) d y + +<- d fun3; + +DF(fun3,t) d t + DF(fun3,x) d x + DF(fun3,y) d y + DF(fun3,z) d z +\end{slisting} + +The functions may have particular properties with respect +to their arguments permutation and sign. The corresponding +declarations are\cmdind{Symmetric}\cmdind{Antisymmetric}\cmdind{Odd}\cmdind{Even} +\command{Symmetric \rpt{\parm{f}};\\\tt +Antisymmetric \rpt{\parm{f}};\\\tt +Odd \rpt{\parm{f}};\\\tt +Even \rpt{\parm{f}};} +Notice that these commands are valid only after function \parm{f} +was declared by the command \comm{Function}. + +In addition to user-defined there is also large number of +functions predefined in \reduce. All these functions can be +used in \grg\ without declaration. The complete list of these +functions depends on \reduce\ versions. +Any function defined in the \reduce\ package (module) +is available too if the package is loaded before \grg\ was +started or during \grg\ session.\seethis{See \pref{packages} +to find out how to load the \reduce\ packages.} +For example the package \file{specfn} contains definitions +for various special functions. + +Finally there is also special declaration \cmdind{Generic Functions} +\command{Generic Functions \rpt{\parm{f}(\rpt{\parm{a}})};} +This command is valid iff the package \file{dfpart.red} is +installed on your \reduce\ system. Here unlike the usual +function declaration the list of parameters must be always +present and \parm{a} can be any identifier preferably +distinct from any other variable. +\seethis{See \pref{genfun} to find out about the generic functions.} +The role of \parm{a} is also completely different and is explained later. + +The list of declared functions can be printed by the command +\cmdindx{Write}{Functions} +\command{Write Functions;} +Generic functions in this output are marked by the label \comm{*}. + +\subsection{Affine Parameter} + +The variable which plays the role of affine parameter +in the geodesic equation must be declared by the command \label{affpar} +\command{Affine Parameter \parm{s};} +and can be printed by the command\cmdindx{Write}{Affine Parameter} +\command{Write Affine Parameter;} + +\vfill +\newpage + +\subsection{Case Sensitivity} +\label{case} + +Usually \reduce\ is case insensitive which means for example +that expression \comm{x-X} will be evaluated by \reduce\ as zero. +On the contrary all coordinates, constants and functions in \grg\ are +case sensitive, e.g. \comm{alpha}, \comm{Alpha} and \comm{ALPHA} +are all different. Notice that commands and switches in \grg\ +3.2 remain case insensitive. +\index{Internal \reduce\ case} + +Therefore all predefined by \grg\ constants and +all built-in objects must be used exactly as they +presented in this manual \comm{GCONST}, \comm{SMASS} etc. +The situation with the constants and functions which predefined +by \reduce\ is different. The point is that in spite of its default +case insensitivity internally \reduce\ converts everything +into some default case which may be upper or lower. +Therefore depending on the particular \reduce\ system they +must be typed either as +\begin{listing} + E I PI INFINITY SIN COS ATAN +\end{listing} +or in lower case +\begin{listing} + e i pi infinity sin cos atan +\end{listing} +For the sake of definiteness throughout this book we chose +the first upper case convention. + +When \grg\ starts it informs you about internal case of +your particular \reduce\ system by printing the message +\begin{slisting} +System variables are upper-cased: E I PI SIN ... +\end{slisting} +or +\begin{slisting} +System variables are lower-cased: e i pi sin ... +\end{slisting} +You can find out about the internal case +using the command\cmdind{Status}\cmdind{Show Status} +\command{\opt{Show} Status;} + +\vfill +\newpage + + +\subsection{Complex Conjugation} + +By default all variables and functions in \grg\ are considered to be +real excluding the imaginary unit constant \comm{I} (or \comm{i} as +explained above). But if two identifiers differ only by the trailing +character \comm{\cc} they are considered as a pair of +complex variables which are conjugated to each other. +In the following example coordinates +\comm{z} and \comm{z\cc} comprise such a pair: +\begin{slisting} +<- Coordinates u, v, z, z~; + +z & z~ - conjugated pair. + +<- Re(z); + + z + z~ +-------- + 2 + +<- Im(z~); + + I*(z - z~) +------------ + 2 +\end{slisting} + + + +\section{Objects} + +Objects play a fundamental role in \grg. They represent +mathematical quantities such as metric, connection, curvature +and any other spinor or tensor geometrical and physical fields +and equations. \grg\ has quite large number of built-in +objects and knows many formulas for their calculation. +But you are not obliged to use the built-in quantities +and can declare your own. The purpose of the declaration is +to tell \grg\ basic properties of a new quantity. + + +\subsection{Built-in Objects} + +\noindent +An object is characterized by the following properties and attributes: +\index{Built-in objects} +\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent\parsep=0mm} +\item Name +\item Identifier or symbol +\item Type of the component +\item List of indices +\item Symmetries with respect to index permutation +\item Density and pseudo-tensor property +\item Built-in ways of calculation +\item Value +\end{list} + +The object \emph{name} is a sequence of words which are +usually the common English name of corresponding quantity. +The name is case insensitive and is used to denote +a particular object in commands. +So called \emph{group names}\index{Group names} +refer to a collection of closely related objects. In particular +the name {\tt Curvature Spinors} (see page \pageref{curspincoll}) +refers to the irreducible components of the curvature tensor in +spinorial representation. +Actual content of the group may depend on the environment. +In particular the group {\tt Curvature Spinors} includes +three objects in the Riemann space (Weyl spinor, traceless +Ricci spinor and scalar curvature) while in the space with +torsion we have six irreducible curvature spinors. + +The object \emph{identifier} or \emph{symbol} is an identifier +which denotes the object in mathematical expressions. Object +symbols are case sensitive. + +The object \emph{type} is the type of its component: objects can be +scalar, vector or $p$-form valued. The \emph{density} and +\emph{pseudo-tensor} properties of the object characterizes its +behaviour under coordinate and frame transformations. + +Objects can have the following types of indices: +\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} +\item Upper and lower holonomic coordinate indices. +\item Upper and lower frame indices. +\item Upper and lower spinorial indices. +\item Upper and lower conjugated spinorial indices. +\item Enumerating indices. +\end{list} +The major part of \grg\ built-in objects has frame indices. +\seethis{See page \pageref{metric} about the frame in \grg.} +The frame in \grg\ can be arbitrary but you can easily specify +the frame to be holonomic or say orthogonal. Then built-in +object indices become holonomic or orthogonal respectively. + +\grg\ deals only with the SL(2,C) spinors which are restricted +to the 4-dimensional spaces of Lorentzian signature. +\seethis{See \pref{spinors} about the spinorial formalism in \grg.} +The corresponding SL(2,C) indices take values 0 and 1. +The conjugated indices are transformed with the help +of the complex conjugated SL(2,C) matrix. +If some spinor is totally symmetric in the group of $n$ spinorial +indices (irreducible spinor) then these indices can be +replaced by a single so called \emph{summed spinorial index} +of rank $n$ which take values from 0 to $n$. +The summed spinorial indices provide the most economic +way to store the irreducible spinor components. + +Enumerating indices just label a collection of +values and have no any covariant meaning. Accordingly there is +no difference between upper and lower enumerating indices. + +Notice that an index of any type in \grg\ always runs from +0 up to some maximal value which depend on the index type +and dimensionality: $d-1$ for frame and coordinate indices,\index{Dimension} +and $n$ the spinor indices of the rank $n$. + +\grg\ understands various types of index symmetries: +symmetry, antisymmetry, cyclic symmetry and Hermitian +symmetry. These symmetries can apply not only to single +indices but to any group of indices as well. +\index{Index symmetries}\index{Canonical order of indices} +\grg\ uses object symmetries to decrease the amount of memory +required to store the object components. It stores only components +with the indices in certain \emph{canonical} order +and any other component are automatically +restored if necessary by appropriate index permutation. +The canonical order of indices is defined as follows: +for symmetry, antisymmetry or Hermitian symmetry indices +are sorted in such a way that index values grows from +left to the right. For cyclic symmetry indices are shifted to +minimize the numerical value of the whole list of indices. + +Finally there are two special types of objects: equations +and connection 1-forms. +\index{Equations} +Equations have all the same properties as any +other object but in addition they have left and right hand side +and are printed in the form of equalities. +The connections are used by \grg\ to construct covariant derivatives. +\index{Connections}\seethis{See \pref{conn2} about the connections.} +There are only four types of connections: holonomic +connection 1-form, frame connection 1-form, spinor connection +1-form and conjugated spinor connection 1-form. + +Almost all built-in objects have associated built-in \emph{ways of +calculation} (one or several). +\index{Ways of calculation} +Each way is nothing but a formula which can be used +to obtain the object value. + +Every object can be in two states. Initially when \grg\ starts +all objects are in \emph{indefinite} state, i.e. nothing is known +about their value. \index{Object value} +Since \grg\ always works in some concrete frame and coordinate +system the object value is a table of the components. +As soon as the value of certain object +is obtained either by direct assignment or using some built-in +formula (way of calculation) \grg\ remember this value +and store it in some internal table. Later this value +can be printed, re-evaluated used in expression etc. +The object can be returned to its initial indefinite state +using the command \comm{Erase}.\cmdind{Erase} +\grg\ uses object symmetries to reduce total number of +components to store. + +The complete list of built-in \grg\ objects is given in +appendix C. The chapter 3 also describes built-in objects +but in the usual mathematical style. The equivalent commands +\cmdind{Show \parm{object}} +\command{Show \parm{object};\\\tt% +?~\parm{object};} +prints detailed information about the object \parm{object} +including object name, identifier, list of indices, +type of the component, current state (is the value of an +object known or not), symmetries and ways of calculation. +Here \parm{object} is either object name or its identifier. + +The command\cmdind{Show *} +\command{Show *;} +prints complete list of built-in object names. This list +is quite long and the command +\command{Show \parm{c}*;} +gives list of objects whose names begin with the character +\parm{c} (\comm{a}--\comm{z}). + +Finally the command \cmdind{Show All} +\command{Show All;} +prints list of objects whose values are currently known. + +Notice that some built-in objects has limited scope. +In particular some objects exists only in certain dimensionality, +the quantities which are specific to spaces with torsion +are defined iff switch \comm{TORSION} is turned on etc. + +Let us consider some examples. We begin with the +curvature tensor $R^a{}_{bcd}$ +\begin{slisting} +<- Show Riemann Tensor; + +Riemann tensor RIM'a.b.c.d is Scalar + Value: unknown + Symmetries: a(3,4) + Ways of calculation: + Standard way (D,OMEGA) +\end{slisting} +This object has name {\tt Riemann Tensor} and identifier +{\tt RIM}. The object is {\tt Scalar} (0-form) valued and +has four frame indices. Frame indices are denoted by the +lower-case characters and their upper or lower position +are denoted by \comm{'} or \comm{.} respectively. +The Riemann tensor is antisymmetric in two last indices +which is denoted by \comm{a(3,4)}. + +The curvature 2-form $\Omega^a{}_b$ +\begin{slisting} +<- ? OMEGA; + +Curvature OMEGA'e.f is 2-form + Value: unknown + Ways of calculation: + Standard way (omega) + From spinorial curvature (OMEGAU*,OMEGAD) +\end{slisting} +has name {\tt Curvature} and the identifier {\tt OMEGA} +and is 2-form valued. + +The traceless Ricci spinor (the quantity which is usually +denoted in the Newman-Penrose formalism as $\Phi_{AB\dot{C}\dot{D}}$) +\begin{slisting} +<- ? Traceless Ricci Spinor; + +Traceless ricci spinor RC.AB.CD~ is Scalar + Value: unknown + Symmetries: h(1,2) + Ways of calculation: + From spinor curvature (OMEGAU,SD,VOL) +\end{slisting} +Spinorial indices +are denoted by upper case characters with the trailing \comm{\cc} +for conjugated indices. Usual spinorial indices are denoted +by a \emph{single} upper case letter while summed indices +are denoted by several characters. Thus, the traceless Ricci +spinor has two summed spinorial indices +of rank 2 each taking the values from 0 to 2. The spinor +is hermitian \comm{h(1,2)}. + +The Einstein equation is an example of equation +\begin{slisting} +<- ? Einstein Equation; + +Einstein equation EEq.g.h is Scalar Equation + Value: unknown + Symmetries: s(1,2) + Ways of calculation: + Standard way (G,RIC,RR,TENMOM) +\end{slisting} +and 1-form $\Gamma^\alpha{}_\beta$ is an example of the connection \enlargethispage{2mm} +\begin{slisting} +<- Show Holonomic Connection; + +\reversemarginpar + +Holonomic connection GAMMA^x_y is 1-form Holonomic Connection + Value: unknown + Ways of calculation: + From frame connection (T,D,omega) +\end{slisting} +The coordinate indices are denoted by the lower-case +letters with labels \comm{\^} and \comm{\_} denoting +upper and lower index position respectively. +Notice that above the first ``{\tt Holonomic connection}'' is the +name of the object while second ``{\tt Holonomic Connection}'' +means that \grg\ recognizes it as the connection and will +use \comm{GAMMA} to construct covariant derivatives for quantities +having the coordinate indices. \seethis{See \pref{cder} about the covariant derivatives.} +You can define any number of other holonomic +connections and use them in the covariant derivatives +on the equal footing with the built-in object \comm{GAMMA}. + +\normalmarginpar + +The notation in which command \comm{Show} prints +information about a particular object is the same as in the +new object declaration and is explained in details below. + + +\subsection{Macro Objects} +\index{Macro Objects}\label{macro} + +There is also another class of built-in objects which are +called \emph{macro objects}. The main difference between the +usual and macro objects is that macro quantities has no +permanent storage to their components instead they are calculated +dynamically only when its component is required in some expression. +In addition +they do not have names and are denoted only by the identifier only. +Usually macro objects play auxiliary role. The complete +list of macro objects can be found in appendix B. + +The example of macro objects are the Christoffel symbols +of second and first kind $\{{}^\alpha_{\beta\gamma}\}$ +and $[{}_{\alpha,\beta\gamma}]$ having identifiers +\comm{CHR} and \comm{CHRF} respectively +\begin{slisting} +<- Show CHR; + +CHR^x_y_z is Scalar Macro Object + Symmetries: s(2,3) + +<- ? CHRF; + +CHRF_u_v_w is Scalar Macro Object + Symmetries: s(2,3) +\end{slisting} + + +\subsection{New Object Declaration} + +\grg\ has very large number of built-in quantities +but you are not obliged to use them in your calculations +instead you can define new quantities. The command\cmdind{New Object} +\command{New Object \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}};} +declares a new object. The words \comm{New} or \comm{Object} are +optional (but not both) so the above command are equivalent to +\command{Object \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}};\\\tt +New \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}}; } +Here \parm{ID} is an identifier of a new object. The identifier can +contain letters \comm{a}--\comm{z}, \comm{A}--\comm{Z} but neither +digits nor any other symbols. The identifier must be unique and cannot +coincide with the identifier of any other built-in or user-defined object. + +The \parm{ilist} is the list of indices having the form \label{indices} +\command{\rpt{\parm{ipos}\ \parm{itype}}} +where \parm{ipos} defines the index position and \parm{itype} +specifies its type. The coordinate holonomic and frame indices +are denoted by single lower-case letters with \parm{ipos} +\command{{\tt '}\rm\ \ upper frame index +\\{\tt .}\rm\ \ lower frame index +\\{\tt \^}\rm\ \ upper holonomic index +\\{\tt \_}\rm\ \ lower holonomic index} +The frame and holonomic indices in \grg\ take values from 0 to +$d-1$ where $d$ is the current space dimensionality.\index{Dimension} + +Spinorial indices are denoted by upper case letters +with trailing \comm{\cc} for conjugated spinorial indices: +\comm{A}, \comm{B\cc} etc. Summed spinorial index of rank $n$ is +denoted by $n$ upper-case letters. For example \comm{ABC} denotes +summed spinorial index of the rank 3 (runs from 0 to 3) +and \comm{AB\cc} denotes conjugated summed index of the rank 2 +(values 0, 1, 2). The upper position for spinorial indices +are denoted either by \comm{'} or \comm{\^} and lower one by +\comm{.} or \comm{\_}. + +Finally the enumerating indices are denoted by a single +lower-case letter followed either by digits or by \comm{dim}. +For example the index declared as \comm{i2} runs from 0 +to 2 while specification \comm{a13} denotes index whose +values runs from 0 to 13. +The specification \comm{idim} denotes enumerating index +which takes the values from 0 to $d-1$. +Upper of lower position for enumerating indices are identical, +thus in this case symbols \comm{' . \^ \_} are equivalent. + +The \parm{ctype} defines the type of new object component: +\command{Scalar \opt{Density \parm{dens}}\\\tt +\parm{p}-form \opt{Density \parm{dens}}\\\tt +Vector \opt{Density \parm{dens}}} +This part of the declaration can be omitted and then the object +is assumed to be scalar-valued. The \parm{dens} defines pseudo-scalar +and density properties of the object with respect to +coordinate and frame transformations: +\command{\opt{sgnL}\opt{*sgnD}\opt{*L\^\parm{n}}\opt{*D\^\parm{m}}} +where \comm{D} and \comm{L} is the coordinate transformation +determinant ${\rm det}(\partial x^{\alpha'}/\partial x^\beta)$ and +frame transformation determinant ${\rm det}(L^a{}_b)$ respectively. +If \comm{sgnL} or \comm{sgnD} is specified then under appropriate +transformation the object must be multiplied on the +sign of the corresponding determinant (pseudo tensor). +The specification \comm{L\^\parm{n}} or \comm{D\^\parm{m}} means +that the quantity must be multiplied on the appropriate +degree of the corresponding determinant (tensor density). +The parameters \parm{p}, \parm{n} and \parm{m} may be given +by expressions (must be enclosed in brackets) but value +of these expressions must be always integer and positive +in the case of \parm{p}. + +The symmetry specification \parm{slst} is a list +\command{\rpt{\parm{slst1}}} +where each element \parm{slst1} describes symmetries +for one group of indices and has the form +\command{\parm{sym}(\rpt{\parm{slst2}})} +The \parm{sym} determines type of the symmetry +\command{% +\tt s \ \rm symmetry \\ +\tt a \ \rm antisymmetry \\ +\tt c \ \rm cyclic symmetry \\ +\tt h \ \rm Hermitian symmetry} +and \parm{slst2} is either index number \parm{i} or list of +index numbers \comm{(\rpt{\parm{i}})} or another symmetry +specification of the form \parm{slst1}. Notice that $n$th +object index can be present only in one of the \parm{slst1}. + +Let us consider an object having four indices. +Then the following symmetry specifications are possible + +\begin{tabular}{ll} +\comm{s(1,2,3,4)} & total symmetry \\[1mm] +\comm{a(1,2),s(3,4)} & antisymmetry in first pair of indices and \\ + & symmetry in second pair \\[1mm] +\comm{s((1,2),(3,4))} & symmetry in pair permutation \\[1mm] +\comm{s(a(1,2),a(3,4))} & antisymmetry in first and second pair of indices \\ + & and symmetry in pair permutation +\end{tabular}\newline +The last example is the well known symmetry of Riemann curvature tensor. +The specification \comm{a(1,2),s(2,3)} is erroneous since +second index present in both parts of the specification +which is not allowed. + +Declaration for new equations is completely similar\cmdind{New Equation} +\command{\opt{New} Equation \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}};} + +\grg\ knows four types of connections:\cmdind{New Connection} \label{conn2} +\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} +\item Frame Connection 1-form $\omega^a{}_b$ having first upper and second lower frame indices +\item Holonomic Connection 1-form $\Gamma^\alpha{}_\beta$ having first upper and second lower coordinate indices +\item Spinor Connection 1-form $\omega_{AB}$ with lower spinor index of rank 2 +\item Conjugated Spinor Connection $\omega_{\dot{A}\dot{B}}$ 1-form with lower conjugated spinor index of rank 2 +\end{list} +Each of these connections are used to construct covariant derivatives +with respect to corresponding indices. In addition they are properly +transformed under the coordinate change and frame rotation. +There are complete set of built-in connections but you can declare +a new one by the command +\command{% +\opt{New} Connection \parm{ID}'a.b \opt{is 1-form};\\\tt +\opt{New} Connection \parm{ID}\^m\_n \opt{is 1-form};\\\tt +\opt{New} Connection \parm{ID}.AB\ \opt{is 1-form};\\\tt +\opt{New} Connection \parm{ID}.AB\cc\ \opt{is 1-form};} +Notice that any new connection must belong to one of the listed +above types and have indicated type and position of indices. This +representation of connection is chosen in \grg\ for the sake of +definiteness. + +There is one special case when new object can be declared +without explicit \comm{New Object} declaration. Let us +consider the following example: +\begin{slisting} +<- Coordinates t, x, y, z; +<- www=d x; +<- Show www; + +www is 1-form + Value: known +\end{slisting} +If we assign the value to some identifier \parm{id} +(\comm{www} in our example) +\seethis{See page \pageref{assig} about assignment command.} +and this identifier is not reserved yet by any other object then +\grg\ automatically declares a new object without indices +labeled by the identifier \parm{id} and having the type +of the expression in the right-hand side of the assignment +(1-form in our example). Notice that the \parm{id} must not include +digits since digits represent indices and any new object +with indices must be declared explicitly. + +The command +\command{Forget \parm{ID};} +completely removes the user-defined object with the +identifier \parm{ID}. + +Finally let us consider some examples: +\begin{slisting} +<- Coordinates t, x, y, z; +<- New RNEW'a.b_c_d is scalar density sgnD with a(3,4); +<- Show RNEW; + +RNEW'a.b_x_y is Scalar Density sgnD + Value: unknown + Symmetries: a(3,4) + +<- Null Metric; +<- Connection omnew.AA; +<- Show omnew; + +omnew.AB is 1-form Spinor Connection + Value: unknown +\end{slisting} +Here the first declaration defines a new scalar valued pseudo tensor +$\mbox{\comm{RNEW}}^a{}_{b\gamma\delta}$ which is antisymmetric +in the last pair of indices. Second declaration introduce new spinor +connection \comm{omnew}. Notice that new connection is automatically +declared 1-form and the type of connection is derived by the +type of new object indices (lower spinorial index of rank 2 in our +example). + + +\section{Assignment Command} +\index{Assignment (command)}\label{assig} + +The assignment command sets the value to the particular +components of the object. In general it has the form +\command{\opt{\parm{Name}} \rpt{\parm{comp} = \parm{expr}};} +or for equations +\command{\opt{\parm{Name}} \rpt{\parm{comp} = \parm{lhs}=\parm{rhs}};} +Here \parm{Name} is the optional object name. If the object +has no indices then \parm{comp} is the object identifier. +If the object has indices then \parm{comm} consist of identifier +with additional digits denoting indices. +For example the following command assigns standard spherical flat +value to the frame $\theta^a$ +\begin{listing} + Frame + T0 = d t, + T1 = d r, + T2 = r*d theta, + T3 = r*SIN(theta)*d phi; +\end{listing} +and the command +\begin{listing} + RIM0123 = 100; +\end{listing} +assigns the value to the $R^0{}_{123}$ component of the Riemann tensor. +Notice that in this notation each digit is considered as one index, +thus it does not work if the value of some index is greater than 9 +(e.g. if dimensionality is 10 or greater). In this case another +notation can be used in which indices are added to the object +identifier as a list of digits enclosed in brackets +\command{\opt{\parm{Name}} \parm{ID}(\rpt{\parm{n}})~= \parm{expr};} +In particular the command +\begin{listing} + RIM(0,1,2,3) = 100; +\end{listing} +is equivalent to the example above. + +The assignment set value only to the certain components of an object +leaving other components unchanged. But if before assignment +the object was in indefinite state (no value is known) then assignment +turns it to the definite state and all other components of the object +are assumed to be zero. + +The digits standing for object indices in the left-hand side +of an assignment can be replaced by identifiers +\index{Assignment (command)!tensorial} +\command{\opt{\parm{Name}} \parm{ID}(\rpt{\parm{id}})~= \parm{expr};} +Such assignment is called \emph{tensorial} one. +For example the following tensorial assignment set the value to the +curvature 2-form $\Omega^a{}_b$ +\begin{listing} + OMEGA(a,b) = d omega(a,b) + omega(a,m){\w}omega(m,b); +\end{listing} +This command is equivalent to $d\times d$ of assignments where \comm{a} +and \comm{b} take values from 0 to $d-1$ ($d$ is the space dimensionality).\index{Dimension} +Notice that identifiers in the left-hand side of tensorial assignment +must not coincide with any predefined or declared by the user +constant or coordinate. It is possible to mix digits and identifiers: +\begin{listing} + FT(0,a) = 0; +\end{listing} +Here \comm{FT} is identifier of the built-in object +{\tt EM Tensor} which is the electromagnetic strength tensor $F_{ab}$ +and this command sets the electric part of the tensor to zero. + +The assignment command takes into account symmetries of the +objects. For example {\tt EM Tensor} is antisymmetric +and in order to assign value say to the components $F_{01}=-F_{10}$ +it suffices to do this just for one of them +\begin{slisting} +<- Coordinates t, x, y, z; +<- EM Tensor FT01=111, FT(3,2)=222; +<- Write FT; +EM tensor: + +FT = 111 + t x + +FT = -222 + y z +\end{slisting} +We can see that \grg\ automatically transforms indices to the +\emph{canonical} order. This rule works in the case or +tensorial assignment as well +\begin{slisting} +<- Coordinates t, x, y, z; +<- Function ff; +<- EM Tensor FT(a,b)=ff(a,b); +<- Write FT; +EM tensor: + +FT = ff(0,1) + t x + +FT = ff(0,2) + t y + +FT = ff(0,3) + t z + +FT = ff(1,2) + x y + +FT = ff(1,3) + x z + +FT = ff(2,3) + y z + +<- FT(2,1); + + - ff(1,2) +\end{slisting} +In this case both parameters \comm{a} and \comm{b} runs from 0 to 3 +but \grg\ assigns the value only to the components +having indices in the canonical order \comm{a}$<$\comm{b}. +\grg\ follows this rule also if in the left-hand +side of tensorial assignment digits are mixed with +parameters which may sometimes produce unexpected result: +\begin{slisting} +<- Coordinates t, x, y, z; +<- Function ee; +<- FT(0,a)=ee(a); +<- Write FT; +EM tensor: + +FT = ee(1) + t x + +FT = ee(2) + t y + +FT = ee(3) + t z + +<- Erase FT; +<- FT(3,a)=ee(a); +<- Write FT; +EM tensor: + +0 +\end{slisting} +Observe the difference between these two assignments (the command +\comm{Erase FT;} destroys the previously assigned value). +In fact second assignment assigns no values since +\comm{3} and \comm{a} are not in the canonical order +\comm{3}$\geq$\comm{a} for \comm{a} running from 0 to 3. +Notice the difference from the case when all indices in +the left-hand side are given by the explicit numerical values. +In this case \grg\ automatically transforms the indices to their +canonical order and \comm{FT(3,2)=222;} is equivalent +to \comm{FT(2,3)=-222;}. + + +Finally there is one more form of the tensorial assignment +which can be applied to the summed spinorial indices. +\index{Assignment (command)!summed spinor indices} +Let us consider the spinorial analogue of electromagnetic strength +tensor $\Phi_{AB}$. This spinor is irreducible (i.e. symmetric in $\scriptstyle AB$). +The corresponding \grg\ built-in object {\tt Undotted EM Spinor} +(identifier \comm{FIU}) has one summed spinorial index of rank 2. +Let us consider two different assignment commands +\begin{slisting} +<- Coordinates u, v, z, z~; + +z & z~ - conjugated pair. + +<- Null Metric; +<- Function ee; +<- FIU(a)=ee(a); +<- Write FIU; +Undotted EM spinor: + + + +FIU = ee(0) + 0 + +FIU = ee(1) + 1 + +FIU = ee(2) + 2 + +<- Erase FIU; +<- FIU(a+b)=ee(a,b); +<- Write FIU; +Undotted EM spinor: + +FIU = ee(0,0) + 0 + +FIU = ee(0,1) + 1 + +FIU = ee(1,1) + 2 +\end{slisting} +In the first case \comm{a} is treated as a summed index +and runs from 0 to 2 but in the second case \comm{a} and \comm{b} +are considered as usual single SL(2,C) spinorial indices +each having values 0 and 1. + +The notation for the object components in the left-hand +side of assignment do not distinguishes upper and lower +indices. Actually the indices are always assumed to be in +the default position. +You can always check the default index types and positions +using the command \comm{Show \parm{object};}.\cmdind{Show \parm{object}} +For example the {\tt Riemann Tensor} has first upper and +three lower frame indices and the command \comm{RIM0123=100;} +and \comm{RIM(0,1,2,3)=100;} both assign value to the +$R^0{}_{123}$ component of the tensor where indices are +represented with respect to the current frame. + + +\section{Geometry} + +The number of built-in objects in \grg\ is rather large. +They all described in chapter 3 and appendices B and C. +In this section we consider only the most important ones. + +\subsection{Metric, Frame and Line-Element} +\index{Metric}\index{Frame} +\label{metric} + +The line-element in \grg\ is defined by the +following equation +\begin{equation} +ds^2 = g_{ab}\,\theta^a\!\otimes\theta^b +\end{equation} +where $\theta^a=h^a_\mu dx^\mu$ is the frame 1-form and $g_{ab}$ is the +frame metric. The corresponding built-in objects are +\comm{Frame} (identifier \comm{T}) and \comm{Metric} +(identifier \comm{G}). There are also the ``inverse'' +counterparts $\partial_a=h_a^\mu\partial_\mu$ ({\tt Vector Frame}, +identifier \comm{D}) and $g^{ab}$ ({\tt Inverse Metric}, identifier +\comm{GI}). To determine the metric properties of the space +you can assign some values to both the metric and the frame. +There are two well known special cases. First is the usual +coordinate formalism in which frame is holonomic $\theta^a=dx^\alpha$. +In this case there is no difference between frame and coordinate +indices. Another representation is known as the tetrad (in dimension 4) +formalism. In this case frame metric equals to some constant +matrix $g_{ab}=\eta_{ab}$ and significant information about +line-element ``is encoded'' in the frame. + +In general both metric and frame can be nontrivial but not +necessarily. If no any value is given by user to the frame +when \grg\ automatically assumes that frame is \emph{holonomic} +\index{Frame!default value} +\begin{equation} +\theta^a=dx^\alpha +\end{equation} +Thus if we assign the value to metric only we automatically +get standard coordinate formalism. On the contrary if +no value is assigned to the metric then \grg\ automatically +assumes\index{Signature} \label{defaultmetric} +\index{Metric!default value} +\begin{equation} +g_{ab} = {\rm diag}(+1,-1,\dots) +\end{equation} +where $+1$ and $-1$ on the diagonal of the matrix +correspond to the current signature specification. + +Notice that current signature is printed among other +information by the command\cmdind{Show Status}\cmdind{Status} +\command{\opt{Show} Status;} +and current line-element is printed by the command +\cmdind{ds2} +\command{ds2;} +or equivalently\cmdind{Line-Element} +\command{Line-Element;} + +Finally if neither frame nor metric are specified by user +then both these quantities acquire default value and we +automatically obtain flat space of the default signature: +\begin{slisting} +<- Dimension 4 with Signature(-,+,+,+); +<- Coordinates t, x, y, z; +<- ds2; +Assuming Default Metric. +Metric calculated By default. 0.05 sec +Assuming Default Holonomic Frame. +Frame calculated By default. 0.05 sec + + 2 2 2 2 2 + ds = - d t + d x + d y + d z + +\end{slisting} + + +\subsection{Spinors} +\label{spinors} + +Spinorial representations exist in spaces of various dimensions +and signatures but in \grg\ spinors are restricted +to the 4-dimensional spaces of Lorentzian signature ${\scriptstyle(-,+,+,+)}$ +or ${\scriptstyle(+,-,-,-)}$ only. Another restriction is that in the +spinorial formalism the metric must be the \index{Metric!Standard Null} +\emph{standard null metric}: +\index{Standard null metric}\index{Spinors}\index{Spinors!Standard null metric} +\begin{equation} +g_{ab}=g^{ab}=\pm\left(\begin{array}{rrrr} +0 & -1 & 0 & 0 \\ +-1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 \\ +0 & 0 & 1 & 0 +\end{array}\right) +\end{equation} +where upper sign correspond to the signature ${\scriptstyle(-,+,+,+)}$ and +lower sign to the signature ${\scriptstyle(+,-,-,-)}$. +There is special command\cmdind{Null Metric} +\command{Null Metric;} +which assigns this standard value to the metric. + +Thus spinorial frame (tetrad) in \grg\ must be null +\begin{equation} +ds^2 = \pm(-\theta^0\!\otimes\theta^1 +-\theta^1\!\otimes\theta^0 ++\theta^2\!\otimes\theta^3 ++\theta^3\!\otimes\theta^2) +\end{equation} +and conjugation rules for this tetrad must be +\begin{equation} +\overline{\theta^0}=\theta^0,\quad +\overline{\theta^1}=\theta^1,\quad +\overline{\theta^2}=\theta^3,\quad +\overline{\theta^3}=\theta^2 +\end{equation} + +For the sake of efficiency the sigma-matrices $\sigma^a\!{}_{A\dot{B}}$ +for such a tetrad are chosen in the simplest form. The only +nonzero components of the matrices are\index{Sigma matrices} +\begin{eqnarray} +&&\sigma_0{}^{1\dot{1}}= +\sigma_1{}^{0\dot{0}}= +\sigma_2{}^{1\dot{0}}= +\sigma_3{}^{0\dot{1}}=1 \\[1mm] && +\sigma^0{}_{1\dot{1}}= +\sigma^1{}_{0\dot{0}}= +\sigma^2{}_{1\dot{0}}= +\sigma^3{}_{0\dot{1}}=\mp1 +\end{eqnarray} + + +\subsection{Connection, Torsion and Nonmetricity} +\label{conn} + +As was explained above \grg\ recognizes four types of connections: +holonomic $\Gamma^\alpha{}_\beta$, frame $\omega^a{}_b$, +spinorial $\omega_{AB}$ and conjugated spinorial +$\omega_{\dot{A}\dot{B}}$. Accordingly there are four +built-in objects: {\tt Holonomic Connection} (id. \comm{GAMMA}), +{\tt Frame Connection} (id. \comm{omega}), {\tt Undotted Connection} +(id. \comm{omegau}), {\tt Dotted Connection} (id. \comm{omegad}). +Connections are used in \grg\ in covariant derivatives. In addition +they are properly transformed under frame and coordinate +transformations. + +By default the connection in \grg\ are assumed to be Riemannian. +In particular in this case holonomic connection is nothing but +Christoffel symbols $\Gamma^\alpha{}_\beta= +\{{}^\alpha_{\beta\pi}\}dx^\pi$. +If it is necessary to work with torsion and/or nonmetricity +\swind{TORSION}\swind{NONMETR} +then the switches \comm{TORSION} and/or \comm{NONMETR} +must be turned on. \seethis{See \pref{conn2} about the built-in connections.} +In this case the Riemannian analogues +or the aforementioned four connections are available as well. + + +\section{Expressions} + +Expressions in \grg\ can be algebraic (scalar), vector or +p-form valued. \grg\ knows all the usual mathematical operations +on algebraic expressions, exterior forms and vectors. + +\subsection{Operations and Operators} + +The operations known to \grg\ are presented in the form of the table. +Operations are subdivided into six groups separated by horizontal +lines. Operations in each group have equal level of precedence and +the precedence level decreases from the top to the bottom of the table. +As in usual mathematical notation we can use brackets \verb"( )" +to change operation precedence. + +Other constructions which can be used in expression are +described below. + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|c|} +\hline +{\bf Operation} & {\bf Description} & {\bf Grouping} \\ +\hline +{\tt [$v_1$,$v_2$]} & Vector bracket & \\ +\hline +{\tt @} $x$ & Holonomic vector $\partial_x$ & \\ +\cline{1-2} +{\tt d} $a$ & Exterior differential & \\ +{\tt d} $\omega$ & & + {\tt d} \cc$a$ $\Leftrightarrow$ {\tt (d(}\cc$a${\tt))} \\ +\cline{1-2} +{\tt \dd} $a$ & Dualization & \\ +{\tt \dd} $\omega$ & & \\ +\cline{1-2} +{\tt \cc} $e$ & Complex conjugation & \\ +\hline +$a_1${\tt **}$a_2$ & Exponention & \\ +$a_1${\tt\^} $a_2$ & & \\ +\hline +$e$\ {\tt /}\ $a$ & Division & + $e${\tt /}$a_1${\tt /}$a_2$ $\Leftrightarrow$ +{\tt (}$e${\tt /}$a_1${\tt )/}$a_2$ \\ +\hline +$a$\ {\tt *}\ $e$ & Multiplication & \\ +\cline{1-2} +$v$\ {\tt |}\ $a$ & Vector acting on scalar & +$v$\ii$\omega_1$\w$\omega_2${\tt *}$a$ \\ +\cline{1-2} +$v$\ \ip\ $\omega$ & Interior product & $\Updownarrow$ \\ +\cline{1-2} +$v_1$\ {\tt.}\ $v_2$& Scalar product & +$v$\ii{\tt (}$\omega_1$\w{\tt(}$\omega_2${\tt *}$a${\tt ))} \\ +$v$\ {\tt.}\ $o$ & & \\ +$o_1$\ {\tt.}\ $o_2$& & \\ +\cline{1-2} +$\omega_1$\ \w\ $\omega_2$ & Exterior product & \\ +\hline +{\tt +}\ $e$ & Prefix plus & \\ +\cline{1-2} +{\tt -}\ $e$ & Prefix minus & \\ +\cline{1-2} +$e_1$\ {\tt +}\ $e_2$ & Addition & \\ +\cline{1-2} +$e_1$\ {\tt -}\ $e_2$ & Subtraction & \\ +\hline +\end{tabular} +\end{center} +\label{operators} +\caption{Operation and operators. Here: +$e$ is any expression, +$a$ is any scalar valued (algebraic) expressions, +$v$ is any vector valued expression, +$x$ is a coordinate, +$o$ is any 1-form valued expression, +$\omega$ is any form valued expression.} +\end{table} + + + +\subsection{Variables and Functions} + +Operator listed in the table 2.2 act on +the following types of the operands: +\begin{itemize} +\item[(i)] integer numbers (e.g. {\tt 0}, {\tt 123}), +\item[(ii)] symbols or identifiers (e.g. {\tt I}, {\tt phi}, {\tt RIM0103}), +\item[(iii)] functional expressions (e.g. {\tt SIN(x)}, {\tt G(0,1)} etc). +\end{itemize} + +Valid identifier must belong to one of the following types: +\begin{itemize} +\item Coordinate. +\item User-defined or built-in constant. +\item Function declared with the implicit dependence list. +\item Component of an object. +\end{itemize} + +Any valid functional expression must belong to one of the following types: +\itemsep=0.5mm +\begin{itemize} +\item User-defined function. +\item Function defined in \reduce\ (operator). +\item Component of built-in or user-defined object in functional notation. +\item Some special functional expressions listed below. +\end{itemize} + +\subsection{Derivatives} + +The derivatives in \grg\ and \reduce\ are written as +\command{DF(\parm{a},\rpt{\parm{x}\opt{,\parm{n}}})} +where \parm{a} is the differentiated expression, \parm{x} is +the differentiation variable and integer number \parm{n} is +the repetition of the differentiation. For example +\[ +\mbox{\tt DF(f(x,y),x,2,y)}=\frac{\partial^3f(x,y)}{\partial^2x\partial y} +\] + +There are also another type of derivatives +\command{DFP(\parm{a},\rpt{\parm{x}\opt{,\parm{n}}})} +\seethis{See section \ref{genfun} about the generic functions.} +They are valid only after {\tt Generic Function} +declaration if the package \file{dfpart} +is installed on your system. + +\subsection{Complex Conjugation} + +Symbol \comm{\cc\cc} in the sum of terms is an abbreviation: +\command{% +\tt $e$ + \cc\cc\ $=$\ $e$ + \cc$e$ \\ +\tt $e$ - \cc\cc\ $=$\ $e$ - \cc$e$ } + +Functions \comm{Re} and \comm{Im} gives real and imaginary +parts of an expression: +\command{% +\tt Re($e$)\ $=$\ ($e$+\cc$e$)/2 \\ +\tt Im($e$)\ $=$\ I*(-$e$+\cc$e$)/2} +\subsection{Sums and Products} +The following expressions represent sum and product +\command{Sum(\rpt{\parm{iter}},\parm{e})\\\tt +Prod(\rpt{\parm{iter}},\parm{e})} +where \parm{e} is the summed expression and \parm{iter} +defines summation variables. +The range of summation can be \label{iter} +specified by two methods. First ``long'' notation is +\command{\parm{id} = \parm{low}..\parm{up}} +and the identifier \parm{id} runs from \parm{low} up to +\parm{up}. Both \parm{low} and \parm{up} can be given +by arbitrary expressions but value of these expressions +must be integer. The \parm{low} can be omitted +\command{\parm{id} = \parm{up}} +and in this case \parm{id} runs from 0 to \parm{up}. +The identifier \parm{id} should not coincide with any +built-in or user-defined variable. + + +In ``short'' notation \parm{iter} is just identifier \label{siter} +\parm{id} and its range is determined using +the following rules +\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} +\item Mixed letter-digit \parm{id} runs from 0 to $d-1$ + where $d$ is the space dimensionality. +\begin{verbatim} + Aid j2s +\end{verbatim} +\item The \parm{id} consisting of lower-case letters runs from + $0$ to $d-1$ +\begin{verbatim} + j a abc kkk +\end{verbatim} +\item The \parm{id} consisting of upper-case letters runs from + $0$ to the number of letters in \parm{id}, e.g. the following + identifiers run from 0 to 1 and from 0 to 3 respectively +\begin{verbatim} + B ABC +\end{verbatim} +\item Letters with one trailing digit run from 0 to the value + of this digit. Both \parm{id} below runs from 0 to 3: +\begin{verbatim} + j3 A3 +\end{verbatim} +\item Letters with two digits run from the value of the + first digit to the value of the second digit. The \parm{id} below + run from 2 to 3: +\begin{verbatim} + j23 A23 +\end{verbatim} +\item Letters with 3 or more digits are incorrect +\begin{verbatim} + j123 +\end{verbatim} +\end{list} + +Two or more summation parameters are separated either +by commas or by one of the relational operators +\begin{listing} + < > <= => +\end{listing} +This means that only the terms satisfying these relations +will be included in the sum. For example +\[ +\mbox{\tt Sum(i24<=ABC,k=1..d-1,f(i24,ABC,k))} = +\sum_{i=2}^{4} \sum_{\scriptstyle a=0\atop\scriptstyle i\leq a}^{3} \sum^{d-1}_{k=1} f(i,a,k) +\] + +\enlargethispage{5mm} + +\grg's \comm{Sum} and \comm{Prod} +\seethis{Use \comm{SUM}, \comm{PROD} or \comm{sum}, \comm{prod} +depending on \reduce\ internal case as explained on page +\pageref{case}.} +should not be confused with \reduce's \comm{SUM} and \comm{PROD} +which are also available in \grg. \grg's \comm{Sum} apply +to any scalar, vector or form-valued expressions and always +expanded by \grg\ into the appropriate explicit sum of terms. On the +contrary \comm{SUM} defined in \reduce\ can be applied to the +algebraic expressions only. \grg\ leaves such expression unchanged +and passes +it to the \reduce\ algebraic evaluator. Unlike \comm{Sum} the +summation limits in \comm{SUM} can be given by algebraic +expressions. If value of these expressions is integer then +result of the \comm{SUM} will be the same as for \comm{Sum} +but if summation limits are symbolic sometimes \reduce\ is capable +to find a closed expression for such a sum but not always. +See the following example +\begin{slisting} +<- Coordinates t, x, y, z; +<- Function f; +<- Constants n, m; +<- Sum(k=1..3,f(k)); + +f(3) + f(2) + f(1) + +<- SUM(f(n),n,1,3); + +f(3) + f(2) + f(1) + +<- SUM(n,n,1,m); + + m*(m + 1) +----------- + 2 + +<- SUM(f(n),n,1,m); + +SUM(f(n),n,1,m) +\end{slisting} + +\newpage + +\subsection{Einstein Summation Rule} + +According to the Einstein summation rule if \grg\ encounters +some unknown repeated identifier \parm{id} then summation over this +\parm{id} is performed. The range of the summation variable +is determined according to the ``short'' notation explained in +the previous section. + + +\subsection{Object Components and Index Manipulation} + +The components of built-in or user-defined object can be +denoted in expressions by two methods which are +similar to the notation used in the left-hand side of the +assignment command. The first method uses the object identifier +with additional digits denoting the indices {\tt T0}, {\tt RIM0213}. +The second method uses the functional +notation {\tt T(0)}, {\tt RIM(0,2,1,3)}, {\tt OMEGA(j,k)}. + +In functional notation the default index type and position +\index{Index manipulations} +can be changed using the markers: {\tt '} upper frame, +{\tt .} lower frame, {\tt \^} upper holonomic, {\tt \_} lower +holonomic. For example expression {\tt RIM(a,b,m,n)} +gives components of Riemann tensor with the default indices +$R^a{}_{bmn}$ (first upper frame and three lower frame indices) +while expression {\tt RIM('a,'b,\_m,\_n)} gives +$R^{ab}{}_{\mu\nu}$ with two upper frame and two lower coordinate +indices. For enumerating indices position markers are ignored +and only {\tt '} and {\tt .} works for spinorial indices. + +In the spinorial formalism +\seethis{See \pref{spinors} about spinorial formalism.} +each frame index can be replaced by a pair if spinorial indices +according to the formulas: +\[ +A^a\sigma_a{}^{B\dot{D}}=A^{B\dot{D}},\qquad +B_a\sigma^a\!{}_{B\dot{D}}=B_{B\dot{D}} +\] +Accordingly any frame index can be replaced by a pair of +spinorial indices. +\label{sumspin} +Similarly one summed spinorial index or rank $n$ can be +replaced by $n$ single spinor indices. +There is only one restriction. If an object has several +frame and/or summed spinorial indices then \emph{all} +must be represented in such expanded form. +In the following example the null frame $\theta^a$ +is printed in the usual and spinorial $\theta^{B\dot C}$ +representations. The relationship +$\theta^a\sigma_a{}^{B\dot C}-\theta^{B\dot C}=0$ is +verifies as well +\begin{slisting} +<- Coordinates u, v, z, z~; + +z & z~ - conjugated pair. + +<- Null Metric; +<- Frame T(a)=d x(a); +<- ds2; +\newpage + 2 + ds = (-2) d u d v + 2 d z d z~ + +<- T(a); + +a=0 : d u + +a=1 : d v + +a=2 : d z + +a=3 : d z~ + +<- T(B,C); + +B=0 C=0 : d v + +B=0 C=1 : d z~ + +B=1 C=0 : d z + +B=1 C=1 : d u + +<- T(a)*sigmai(a,B,C)-T(B,C); + +0 +\end{slisting} + + +\subsection{Parts of Equations and Solutions} +\index{Equations!in expressions} + +The functional expressions +\command{LHS(\parm{eqcomp})\\\tt +RHS(\parm{eqcomp})} +give access to the left-hand and right-hand side of an +equation respectively. Here \parm{eqcomp} is the +component of the equation as explained in the +previous section. + +The \comm{LHS}, \comm{RHS} also provide access to the \parm{n}'th +\seethis{See page \pageref{solutions} about solutions.} +solution if \parm{eqcomp} is \comm{Sol(\parm{n})}. + + +\subsection{Lie Derivatives} +\index{Lie derivatives} + +The Lie derivative is given by the expression +\command{Lie(\parm{v},\parm{objcomp})} +where \parm{objcomp} is the component of an object in +functional notation. For example the following +expression is the Lie derivative of the metric $\pounds_vg_{ab}$ +\begin{listing} + Lie(vec,G(a,b)); +\end{listing} +The index manipulations in the Lie derivatives are permitted. +In particular the expression +\begin{listing} + Lie(vec,G(^m,b)); +\end{listing} +is the Lie derivative of the frame $\pounds_vg^\mu{}_{b} +\equiv \pounds_vh^\mu_a$ +and must vanish. + + + + +\subsection{Covariant Derivatives and Differentials} +\index{Covariant derivatives}\index{Covariant differentials} +\label{cder} + +The covariant differential +\command{Dc(\parm{objcomp}\opt{{\upshape\tt ,}\rpt{\parm{conn}}})} +and covariant derivative +\command{Dfc(\parm{v},\parm{objcomp}\opt{{\upshape\tt ,}\rpt{\parm{conn}}})} +Here \parm{objcomp} is an object component in functional notation +and \parm{v} is a vector-valued expression. +The optional parameters \parm{conn} are the identifiers of +connections. +\seethis{See page \pageref{conn} about the built-in connections.} +If \parm{conn} is omitted then \grg\ uses default +connection for each type of indices: frame, coordinate, +spinor and conjugated spinor. If \parm{conn} is indicated +then \grg\ uses this connection instead of default one +for appropriate type of indices. For example expression +\begin{listing} + Dc(OMEGA(a,b)) +\end{listing} +is the covariant differential of the curvature 2-form $D\Omega^a{}_b$. +This expression should vanish in Riemann space and should be +proportional to the torsion in Riemann-Cartan space. +Here \grg\ will use default object {\tt Frame connection} +(id. \comm{omega}). The expression +\begin{listing} + Dc(OMEGA(a,b),romega) +\end{listing} +is similar but it uses another built-in connection +{\tt Riemann frame connection } (id. \comm{romega}) which +are different if torsion or nonmetricity are nonzero. +The index manipulations are allowed in the covariant derivatives. +For example the expression +\begin{listing} + Dfc(v,RIC(\^m,\_n)) +\end{listing} +gives the covariant derivative of the curvature of the +Ricci tensor with first coordinate upper and second coordinate lower +indices $\nabla_vR^\mu{}_\nu$. + +\subsection{Symmetrization} + +The functional expressions works iff the switch \swind{EXPANDSYM} +\comm{EXPANDSYM} is on +\command{% +Asy(\rpt{\parm{i}},\parm{e})\\\tt +Sy(\rpt{\parm{i}},\parm{e})\\\tt +Cy(\rpt{\parm{i}},\parm{e})} +They produce antisymmetrization, symmetrization and cyclic symmetrization +of the expression \parm{e} with respect to \parm{i} without +corresponding $1/n$ or $1/n!$. + + +\subsection{Substitutions} +\index{Substitutions}\label{subs} + +The expression +\command{SUB(\rpt{\parm{sub}},\parm{e})} +is similar to the analogous expression in \reduce\ with two +generalizations: (i) it applies not only to algebraic +but to form and vector valued expression \parm{e} as well, +\seethis{See page \pageref{solutions} about solutions.} +(ii) as in {\tt Let} command \parm{sub} can be either +the relation {\tt \parm{l}\,=\,\parm{r}} or solution +{\tt Sub(\parm{n})}. + + +\subsection{Conditional Expressions} +\index{Conditional expressions}\index{Boolean expressions} + +The conditional expression +\command{If(\parm{cond},\parm{e1},\parm{e2})} +chooses \parm{e1} or \parm{e2} depending on the value of the +boolean expression \parm{cond}. + +Boolean expression appears in (i) the conditional expression +\label{bool} +{\tt If}, (ii) in {\tt For all Such That} substitutions. +Any nonzero expression is considered as {\bf true} and +vanishing expression as {\bf false}. Boolean expressions +may contain the following usual relations and logical +operations: {\tt < > <= >= = |= not and or}. They also may +contain the following predicates \vspace*{2mm} + +\begin{tabular}{|l|l|} +\hline +\tt OBJECT(\parm{obj}) & Is \parm{obj} an object identifier or not \\ +\hline +\tt ON(\parm{switch}) & Test position of the \parm{switch} \\ +\tt OFF(\parm{switch}) & \\ +\hline +\tt ZERO(\parm{object}) & Is the value of the \parm{object} zero or not \\ +\hline +\tt HASVALUE(\parm{object}) & Whether the \parm{object} has any value or not \\ +\hline +\tt NULLM(\parm{object}) & Is the \parm{object} the standard null metric \\ +\hline +\end{tabular} \vspace*{2mm} \newline +Here \parm{object} is an object identifier. + +The expression \comm{ERROR("\parm{message}")} causes an error +with the \comm{"\parm{message}"}. It can be used +to test any required conditions during the batch file execution. + + +\subsection{Functions in Expressions} + +Any function which appear in expression must be +either declared by the \comm{Function} declaration or +be defined in \reduce\ (in \reduce\ functions are called +operators). In general arguments of functions in \grg\ must be +algebraic expression with one exception. If one (and only one) +argument of some function $f$ is form-valued $\omega=a d x + b d y$ then +\grg\ applies $f$ to the algebraic +multipliers of the form $f(\omega) = f(a) d x+ f(b) d y$. +The same rule works for vector-valued arguments. +Let us consider the example in the \reduce\ +operator \comm{LIMIT} is applied to the +form-valued expression +\begin{slisting} +<- Coordinates t, x, y, z; +<- www=(x+y)\^2/(x\^2-1)*d x+(x+y)/(x-z)*d y; +<- www; + + 2 2 + x + 2*x*y + y x + y +(-----------------) d x + (-------) d y + 2 x - z + x - 1 + +<- LIMIT(www,x,INFINITY); + + d x + d y +\end{slisting} + +I would like to remind also that depending on the +particular \reduce\ system \reduce\ operators must be +used in \grg\ in upper \comm{LIMIT} or lower case \comm{limit}. +See page \pageref{case} for more details. + +Any function or operator defined in the \reduce\ package +can be used in \grg\ as well. Some examples are +considered in section \ref{packages}. + + +\subsection{Expression Evaluation} +\index{Expression evaluation} + +\grg\ evaluates expressions in several steps: + +(1) All \grg-specific constructions such as +\comm{Sum}, \comm{Prod}, \comm{Re}, \comm{Im} etc are +explicitly expanded. + +(2) If expression contains components of some built-in +or user defined object they are replaced by the appropriate value. +If the object is in indefinite state +\seethis{See page \pageref{find} about the \comm{Find} command.} +(no value of the object is known) then \grg\ tries to +calculate its value by the method used by the \comm{Find} command. +The automatic object calculation can be prevented by +\swind{AUTO} +turning the switch \comm{AUTO} off. +If due to some reason the object cannot be calculated then +expression evaluation is terminated with the error message. + +(3) After all object components are replaced by their +values \grg\ performs all ``geometrical'' operations: exterior +and interior products, scalar products etc. If expression is +form-valued when it is reduced to the form +$a\,dx^0\wedge dx^1\dots+b\,d x^1\wedge+\dots$ where $a$ and $b$ +are algebraic expressions (similarly for the vector-valued expressions). + +(4) The \reduce\ algebraic simplification routine +is applied to the algebraic expressions $a$, $b$. +\seethis{In the anholonomic mode the basis $b^i\wedge b^j\dots$ +is used instead. See section \ref{amode}.} +Final expression consist of exterior products of basis +coordinate differentials $dx^i\wedge dx^j\dots$ (or basis +vectors $\partial_{x^i}$) multiplied by the algebraic expressions. +The algebraic expressions contain only the coordinates, +constants and functions. + +\subsection{Controlling Expression Evaluation} + +There are many \reduce\ switches which control +algebraic expression evaluation. The number of these switches +and details of their work depend on the \reduce\ version. +Here we consider some of these switches. All examples below +are made with the \reduce\ 3.5. On other \reduce\ versions +result may be a bit different. + +Switches {\tt EXP} and {\tt MCD} control expansion and +reduction of rational expressions to a common denominator +respectively. +\begin{slisting} +<- (x+y)\^2; + + 2 2 +x + 2*x*y + y + +<- Off EXP; +<- (x+y)\^2; + + 2 +(x + y) + +<- On EXP; +<- 1/x+1/y; + + x + y +------- + x*y + +<- Off MCD; +<- 1/x+1/y; + + -1 -1 +x + y +\end{slisting} +These switches are normally on. + +Switches {\tt PRECISE} and {\tt REDUCED} control evaluation of +square roots:\label{PRECISE}\label{REDUCED} +\begin{slisting} +<- SQRT(-8*x\^2*y); + +2*SQRT( - 2*y)*x + +<- On REDUCED; +<- SQRT(-8*x\^2*y); + +2*SQRT(y)*SQRT(2)*I*x + +<- Off REDUCED; +<- On PRECISE; +<- SQRT(-8*x\^2*y); + +2*SQRT(y)*SQRT(2)*I*x + +<- On REDUCED, PRECISE; +<- SQRT(-8*x\^2*y); + +2*SQRT(y)*SQRT(2)*ABS(x) +\end{slisting} + + +Combining rational expressions the system by default +calculates the least common multiple of denominators but +turning the switch {\tt LCM} off prevents this calculation. + +Switch {\tt GCD} (normally off) makes the system +search and cancel the greatest common divisor of the +numerator and denominator of rational expressions. +Turning {\tt GCD} on may significantly slow down the +calculations. There is also another switch {\tt EZGCD} +which uses other algorithm for g.c.d. calculation. + + +Switches {\tt COMBINELOGS} and {\tt EXPANDLOGS} control +the evaluation of logarithms +\begin{slisting} +<- On EXPANDLOGS; +<- LOG(x*y); + +LOG(x) + LOG(y) + +<- LOG(x/y); + +LOG(x) - LOG(y) + +<- Off EXPANDLOGS; +<- On COMBINELOGS; +<- LOG(x)+LOG(y); + +LOG(x*y) +\end{slisting} + +By default all polynomials are considered by \reduce\ as +the polynomials with integer coefficients. The switches +{\tt RATIONAL} and {\tt COMPLEX} allow rational and +complex coefficients in polynomials respectively: +\begin{slisting} +<- (x\^2+y\^2+x*y/3)/(x-1/2); + + 2 2 + 2*(3*x + x*y + 3*y ) +----------------------- + 3*(2*x - 1) + +<- On RATIONAL; +<- (x\^2+y\^2+x*y/3)/(x-1/2); + + 2 1 2 + x + ---*x*y + y + 3 +------------------- + 1 + x - --- + 2 + +<- Off RATIONAL; +<- 1/I; + + 1 +--- + I + +<- (x\^2+y\^2)/(x+I*y); + + 2 2 + x + y +--------- + I*y + x + +<- On COMPLEX; +<- 1/I; + + - I + +<- (x\^2+y\^2)/(x+I*y); + +x - I*y +\end{slisting} +Switch {\tt RATIONALIZE} removes complex numbers from the +denominators of the expressions but it works even if +{\tt COMPLEX} is off. + +Turning off switch {\tt EXP} and on {\tt GCD} one can +make the system to factor expressions +\begin{slisting} +<- Off EXP; +<- On GCD; +<- x\^2+y\^2+2*x*y; + + 2 +(x + y) +\end{slisting} +Similar effect can be achieved by turning on switch {\tt FACTOR}. +Unfortunately this works only when \grg\ prints expressions and +internally expressions remain in the expanded form. +To make \grg\ to work with factored expressions internally one +must turn on {\tt FACTOR} and {\tt AEVAL}. +\swind{AEVAL} +The \grg\ switch {\tt AEVAL} make \grg\ to use an alternative +\reduce\ routine for algebraic expression evaluation and simplification. +This routine works well with {\tt FACTOR} on. +\seethis{See section \ref{tuning} about configuration files.} +Possibly it +is good idea to turn switch {\tt AEVAL} on by default. +This can be done using \grg\ configuration files. + +\subsection{Substitutions} +\index{Substitutions} + +The substitution commands in \grg\ are the same as the +corresponding \reduce\ instructions +\cmdind{Let}\cmdind{Match}\cmdind{For All Let} +\command{\opt{For All \rpt{\parm{x}}\,\opt{Such That \parm{cond}}} Let \rpt{\parm{sub}};\\\tt +\opt{For All \rpt{\parm{x}}\,\opt{Such That \parm{cond}}} Match \rpt{\parm{sub}};} +\seethis{See page \pageref{solutions} about solutions.} +where \parm{sub} is either relation {\tt \parm{l}\,=\,\parm{r}} +or the solution in the form \comm{Sol(\parm{n})}. +After the substitution is activated every appearance of \parm{l} will be +replaced by \parm{r}. The {\tt For All} substitutions have additional list +of parameters \parm{x} and will work for any value +of \parm{x}. The optional condition \parm{cond} imposes restrictions +on the value of the parameters \parm{x}. The \parm{cond} is +the boolean expression (see page \pageref{bool}). + +The substitution can be deactivated by the command +\cmdind{Clear} +\command{\opt{For All \rpt{\parm{x}}\,\opt{Such That \parm{cond}}} Clear \rpt{\parm{sub}};} +Notice that the variables \parm{x} must be exactly the same +as in the corresponding {\tt For All Let} command. + +The difference between \comm{Match} and \comm{Let} +is that the former matches the degrees of the +expressions exactly while \comm{Let} matches all powers which +are greater than one indicated in the substitution: +\begin{slisting} +<- Const a; +<- (a+1)\^8; + + 8 7 6 5 4 3 2 +a + 8*a + 28*a + 56*a + 70*a + 56*a + 28*a + 8*a + 1 + +<- Let a\^3=1; +<- (a+1)\^8; + + 2 +85*a + 86*a + 85 + +<- Clear a\^3; +<- Match a\^3=1; +<- (a+1)\^8; + + 8 7 6 5 4 2 +a + 8*a + 28*a + 56*a + 70*a + 28*a + 8*a + 57 +\end{slisting} + +Substitutions can be used for various purposes, for example: +(i) to define additional mathematical relations such as +trigonometric ones; +(ii) to ``assign'' value to the user-defined and built-in constants; +(iii) to define differentiation rules for functions. + +After some substitution is activated it applies to every +evaluated expression but value of the objects calculated +\emph{before} remain unchanged. +The command \comm{Evaluate} re-simplifies the value of the object +\cmdind{Evaluate} +\command{Evaluate \parm{object};} +here \parm{object} is the object name, or identifier, or the +group object name. +Let us consider a simple \grg\ task which +calculates the volume 4-form of some metric +\begin{slisting} +<- Coordinates t, x, y, z; +<- Constant a; +<- Tetrad T0=d t, T1=d x, T2=SIN(a)*d y+COS(a)*d z, + T3=-COS(a)*d y+SIN(a)* d z; +<- Find and Write Volume; +Volume : + + 2 2 +VOL = (SIN(a) + COS(a) ) d t \w\ d x \w\ d y \w\ d z +\end{slisting} +We see that \reduce\ do not know the +appropriate trigonometric rule. +Thus we are going to apply substitution +\begin{slisting} +<- For all x let SIN(x)\^2 = 1-COS(x)\^2; +<- Write Volume; +Volume : + +VOL = d t \w\ d x \w\ d y \w\ d z +\end{slisting} +The situation has been improved. +But actually, the \emph{internal} representation +of {\tt VOL} remains unchanged. {\tt Write} by default +re-simplifies expressions before printing. +\swinda{WRS} +By turning switch {\tt WRS} off we can prevent this +re-simplification: +\begin{slisting} +<- Off WRS; +<- Write Volume; +Volume : + 2 2 +VOL = (SIN(a) + COS(a) ) d t \w\ d x \w\ d y \w\ d z +\end{slisting} +Now we can apply \comm{Evaluate}: +\begin{slisting} +<- Evaluate Volume; +<- Write Volume; +Volume : + +VOL = d t \w\ d x \w\ d y \w\ d z +\end{slisting} +We see that the internal value of {\tt VOL} now has been +replaced by re-simplified expression. + +Notice that the command +\command{Evaluate All;} +applies \comm{Evaluate} to all objects whose value is +currently known. + +\subsection{Generic Functions} +\index{Generic Functions}\label{genfun} + +Unfortunately \reduce\ lacks the notion of partial derivative of a function. +The expression \comm{DF(f(x,y),x)} is treated by \reduce\ as the +``derivative of the expression \comm{f(x,y)} with respect to +the variable \comm{x}'' rather than the ``derivative of the function +\comm{f} with respect to its first argument''. +Due to this \reduce\ cannot handle +chain differentiation rule etc. This problem is fixed by the +package \file{dfpart} written by H.~Melenk. +This package introduces notion of generic function and +partial derivative \comm{DFP}. If \file{dfpart} is installed +on your \reduce\ system \grg\ provides the interface +to these facilities. + + + +Let us consider an example. First we declare +one usual and two generic functions +\begin{slisting} +<- Coordinates t, x, y, z; +<- Function f; +<- Generic Function g(a,b), h(b); +<- Write Functions; +Functions: + +g*(a,b) h*(b) f +\end{slisting} +Generic functions must be always declared with +the list of parameters (\comm{a} and \comm{b} in our example). +These parameters play the role of labels which denotes +arguments of the generic function and the partial +derivatives with respect to these arguments +are defined. Due to this generic functions allow the +chain differentiation rule +\begin{slisting} +<- DF(f(SIN(x),y),x); + +DF(f(SIN(x),y),x) + +<- DF(g(SIN(x),y),x); + +COS(x)*g (SIN(x),y) + a +\end{slisting} +Here subscript \comm{a} denotes +the derivative of the function \comm{g} with respect to the +first argument. \enlargethispage{5mm} +The operator \comm{DFP} is introduced to denotes such +derivatives in expressions: +\begin{slisting} +<- DF(g(x,y)*h(y),b); + +0 + +<- DFP(g(x,y)*h(y),b); + +g (x,y)*h(y) + h (y)*g(x,y) + b b +\end{slisting} + +\newpage + +If switch \swind{DFPCOMMUTE} +\comm{DFPCOMMUTE} is turned on then \comm{DFP} +derivatives commute. + + +\section{Using Built-in Formulas In Calculations} + +\grg\ has large number of built-in objects and almost +each object has built-in formulas or so called +\emph{ways of calculation} which can be used to find +the value of the object. This section explains how +these formulas (ways) can be used. + +\subsection{\comm{Find} Command} +\index{Ways of calculation}\cmdind{Find}\label{find} + +Almost each \grg\ built-in object has associated +\emph{ways of calculation}. Each way is nothing but +a formula or equation which allows to compute +the value of the object. All these formulas +are described in the usual mathematical style in +chapter 3. +The command\cmdind{Show \parm{object}} +\command{Show \parm{object};} +or equivalently +\command{?~\parm{object};} +prints information about object's ways of calculation. + +The command \comm{Find} applies built-in formulas to +calculate the object value +\command{Find \parm{object} \opt{\parm{way}};} +where \parm{object} is the object name, or identifier, or +group object name. +The optional specification \parm{way} indicates the +particular way if the \parm{object} has several built-in ways +of calculation. + +\enlargethispage{3mm} + +Consider the curvature 2-form $\Omega^a{}_b$ +(object \comm{Curvature}, id. \comm{OMEGA}): +\begin{slisting} +<- Show Curvature; + +Curvature OMEGA'a.b is 2-form + Value: unknown + Ways of calculation: + Standard way (omega) + From spinorial curvature (OMEGAU*,OMEGAD) +\end{slisting} + +\noindent +We can see that this object has two built in ways of +calculation. First way named {\tt Standard way} is the +usual equation +$\Omega^a{}_b=d\omega^a{}_b+\omega^a{}_m\wedge\omega^m{}_b$. +Second way under the name {\tt From spinorial curvature} +uses spinor $\tsst$ tensor relationship to compute the curvature 2-form +using its spinor analogues $\Omega_{AB}$ and +$\Omega_{\dot{A}\dot{B}}$ as the source data. +The ways of calculation are printed by the command {\tt Show} +in the form +\command{\parm{wayname} (\rpt{\parm{SI}})} +where \parm{wayname} is the way name and \seethis{See Eq. (\ref{omes}) on \pref{omes}.} +the \parm{SI} are the identifiers of the \emph{source} objects which are +present in the right-hand side of the equation. The value of +these objects must be known before the formula can be applied. + +%\enlargethispage{5mm} + +The \parm{way} in the \comm{Find} command allows one to +choose the particular way which can be done by two methods. +In the first form \parm{way} is just the name exactly as +it printed by the \comm{Show} command +\command{wayname} +or {\tt Using standard way} or {\tt By standard way} if the way +name is {\tt Standard way}. Another method to specify +the way is to indicate the appropriate source object +\command{From \parm{object}\\\tt% +Using \parm{object}} +where \parm{object} is the name or the identifier of the source object. +For example second (spinorial) way of calculation for the curvature +2-form can be chosen by the following equivalent commands \vspace{-1mm} +\begin{listing} + Find curvature from spinorial curvature; + Find curvature using OMEGAU; +\end{listing} +while first way is activated by the commands \vspace*{-1mm} +\begin{listing} + Find curvature by standard way; + Find curvature using omega; +\end{listing} +Recall that object identifiers are case sensitive +and \comm{omega} is the identifier +of the frame connection 1-form $\omega^a{}_b$ and should not be +confused with \comm{OMEGA}. + + +The \parm{way} specification in the \comm{Find} +can be omitted and in this case +\grg\ uses the following algorithm to choose +a particular way of calculation. Observe that the identifier +of the undotted curvature 2-form $\Omega_{AB}$ is marked +by the symbol $*$. This label marks so called \emph{main} +objects. If no way of calculation is specified when +\grg\ tries to choose the way, browsing the way list +form top to the bottom, for which the value of the \emph{main} +object is already known. If no switch way exists then +\grg\ just picks up the first way in the list. +Therefore in our example the command +\begin{listing} + Find curvature; +\end{listing} +will use the second way if the value of the object $\Omega_{AB}$ +(id. \comm{OMEGAU}) is known and second way otherwise. + +As soon as some way of calculation is chosen \grg\ tries to +calculate the values of the source objects which are present +in the right-hand side of corresponding equations. +\grg\ tries to do this by applying the \comm{Find} command without way +specification to these objects. Thus a single \comm{Find} +can cause quite long chain of calculations. +This recursive work is reflected by the appropriate +tracing messages. The tracing can be eliminated by turning off +switch \comm{TRACE}.\swind{TRACE} + +Here we present the sample \grg\ session which computes +curvature 2-form for the flat gravitational waves +\begin{slisting} + +<- Cord u, v, z, z~; + +z & z~ - conjugated pair. + +<- Null Metric; +<- Function H(u,z,z~); +<- Frame T0=d u, T1=d v+H*d u, T2=d z, T3=d z~; +<- ds2; + + 2 2 + ds = ( - 2*H) d u + (-2) d u d v + 2 d z d z~ + +<- Find Curvature; +Sqrt det of metric calculated. 0.16 sec +Volume calculated. 0.16 sec +Vector frame calculated From frame. 0.16 sec +Inverse metric calculated From metric. 0.16 sec +Frame connection calculated. 0.22 sec +Curvature calculated. 0.22 sec +<- Write Curvature; +Curvature: + + 1 +OMEGA = ( - DF(H,z,2)) d u \w d z + ( - DF(H,z,z~)) d u \w d z~ + 2 + + 1 +OMEGA = ( - DF(H,z,z~)) d u \w d z + ( - DF(H,z~,2)) d u \w d z~ + 3 + + 2 +OMEGA = ( - DF(H,z,z~)) d u \w d z + ( - DF(H,z~,2)) d u \w d z~ + 0 +\newpage + 3 +OMEGA = ( - DF(H,z,2)) d u \w d z + ( - DF(H,z,z~)) d u \w d z~ + 0 +\end{slisting} + + +Finally we want to emphasize that ways associated +with some object may depend on the concrete environment. +In particular the {\tt Standard way} for +the curvature 2-form is always available but second +way which is essentially related to spinors works +\seethis{See \pref{spinors} about the spinorial formalism.} +only in the 4-dimensional spaces of Lorentzian signature +and iff the metric is null. +If some way is not valid in the current environment +it simply disappears from the way list printed by the \comm{Show}. + +It should be noted also that the \comm{Find \parm{object};} +command works only if the \parm{object} is in the indefinite state +and is rejected if the value of the \parm{object} is already known. +If you want to re-calculate the object then previous value must be +cleared by the \comm{Erase} command. + +\subsection{\comm{Erase} command} +\cmdind{Erase} + +The command +\command{Erase \parm{object};} +destroys the \parm{object} value and returns it to initial +indefinite state. It can be used also to free the +memory. + +\subsection{\comm{Zero} command} +\cmdind{Zero} + +Command +\command{Zero \parm{object};} +assigns zero values to all \parm{object} components. + +\subsection{\comm{Normalize} command} +\cmdind{Normalize} + +Command +\command{Normalize \parm{object};} +applies to equations. It replaces equalities +of the form $l=r$ by the equalities $l-r=0$ +and re-simplifies the result. + +\subsection{\comm{Evaluate} command} +\cmdind{Evaluate} + +The command +\command{Evaluate \parm{object};} +re-simplifies existing value of the \parm{object}. +This command is useful if we want to apply new substitutions +\seethis{See page \pageref{subs} about substitutions.} +to the object whose value is already known. +The command +\command{Evaluate All;} +re-simplifies all objects whose value is currently known. + + +\section{Printing Result of Calculations} + +\subsection{\comm{Write} Command} +\cmdind{Write} + +The command +\command{Write \parm{object};} +prints value of the \parm{object}. Here \parm{object} +id the object name or identifier.\index{Group name} +Group names denoting a collection of several objects +\seethis{See page \pageref{macro} about macro objects.} +and macro object identifiers can be used in the \comm{Write} +command as well. In addition word \comm{All} +can be used to print all currently known objects. + + +The command \comm{Write} can print declarations as well if +\parm{object} is {\tt functions}, {\tt constants}, or +{\tt affine parameter}. + + +The command +\command{Write \rpt{\parm{object}}~to~"\parm{file}";} +or equivalently +\command{Write \rpt{\parm{object}}~>~"\parm{file}";} +writes result into the \comm{"\parm{file}"}. Notice +that \comm{Write} always destroys previous contents of the +file. Therefore we have another command +\command{Write to "\parm{file}";\\\tt% +Write > "\parm{file}";} +which redirects all output into the file. The standard output +can be restored by the commands\cmdind{End of Write}\cmdind{EndW} +\command{EndW;\\\tt% +End of Write;} + +\enlargethispage{3mm} + +By default \comm{Write} re-simplifies the expressions +before printing them. \swind{WRS} +\seethis{See page \pageref{subs} about substitutions.} +This is convenient when substitutions are activated +but slows down the printing especially for very large +expressions. The re-simplification can be abolished +by turning off switch \comm{WRS}. +If switch \comm{WMATR} is turned on then +\swind{WMATR} +\grg\ prints all 2-index scalar-valued objects in +the matrix form +\begin{slisting} +<- Coordinates t, x, y, z; +<- On wmatr; +<- Find and Write metric; +Assuming Default Metric. +Metric calculated By default. 0.06 sec +Metric: + +[-1 0 0 0] +[ ] +[0 1 0 0] +[ ] +[0 0 1 0] +[ ] +[0 0 0 1] +\end{slisting} + + +\comm{Write} prints frame, spinor and enumerating indices as +numerical subscripts while holonomic indices are printed as +the coordinate identifiers. If frame is holonomic +and there is no difference between frame and coordinate indices then +by default all frame indices are also labelled by the +appropriate identifiers. But is switch \comm{HOLONOMIC} \swinda{HOLONOMIC} +is turned off they are still printed as numbers. + +\subsection{\comm{Print} Command} +\cmdind{Print} + +The \comm{Write} command described in the previous section +prints value of an object. This value must be +calculated beforehand by the \comm{Find} command +or established by the assignment. +The command \comm{Print} evaluates expression and +immediately prints its value. It has several forms +\command{% +\opt{Print} \parm{expr} \opt{For \parm{iter}};\\\tt +For \parm{iter} Print \parm{expr};} +Here \parm{expr} is expression to be evaluated and +\parm{iter} indicates that expression must be +evaluated for several value of some variable. +The specification \parm{iter} is completely the same as +is the \comm{Sum} expression and is described in details +in section \ref{iter} on page \pageref{iter}. +It consists of the list of parameters +separated by commas \comm{,} or relational operators +{\tt < > => =<}. For example the command +\begin{listing} + G(a,b) for a "\parm{file}";\\\tt +Unload \parm{object} To "\parm{file}";} +writes \parm{object} value into \comm{"\parm{file}"} in some +special format. +Here \parm{object} is name or identifier of an object. + +The data can be later restored with help of the command\cmdind{Load} +\command{Load "\parm{file}";} + +The command {\tt Unload} always overwrites previous \comm{"\parm{file}"} +contents. To save several objects in one file one must use +the following sequence of commands\cmdind{EndU}\cmdind{End of Unload} +\begin{listing} + Unload > "\parm{file}"; + Unload \parm{object}; + Unload \parm{object}; + ... + Unload \parm{object}; + End Of Unload; +\end{listing} +Here command \comm{Unload > "\parm{file}";} opens +\comm{"\parm{file}"} and {\tt End Of Unload;} closes it. +The last command has the short form +\command{EndU;} +In fact presented above sequence of commands can be +abbreviated as +\command{Unload \rpt{\parm{object}}~>~"\parm{file}";} + +One needs to stress that only the commands {\tt Unload \dots;} +can be used between {\tt Unload > \dots} and +{\tt End Of Unload;}. If this rule does not hold then {\tt Load} +may fail to restore the file. +The only additional command which can be used among these +{\tt Unload \parm{object};} commands is the comment +{\tt \% \parm{text};}. This command insertes +the comment \parm{text} into the \comm{"\parm{file}"}. +Later when \comm{"\parm{file}"} will be restored by the +{\tt Load} the \parm{text} message will be printed. +This allows one to attach comments to unreadable files +produced by {\tt Unload} command. + +As in other commands \parm{object} in \comm{Unload} command +is either the name or identifier of an object. Names {\tt Coordinates}, +{\tt Constants} and {\tt Functions} can also be used to +save declarations. And finally, the command +\command{Unload All > "\parm{file}";} +saves all objects whose value is currently known +\seethis{See section \ref{amode} about anholonomic basis.} +and all declarations. Moreover, in the anholonomic basis mode this +command saves full information about an anholonomic basis. + +When data or coordinates declarations are restored from a file +they replace current values. Function and constant declarations +are added to current declarations. + +One should realize that serious troubles may appear when different +coordinates are used in the current session and in the restored file. +Even the order of coordinates is extremely important. +We strongly recommend saving all declarations (especially coordinates) +in addition to other objects. It ensures at least that will \grg\ print a +warning message if some contradictions are detected between +current declarations and declarations stored into a file. +The best way to avoid these troubles is to use the command +\command{Unload All > "\parm{file}";} +Loading the file saved by this command at the very beginning of +a new \grg\ task completely restores the previous \grg\ state +with all data and declarations. + +Sometimes one needs to prevent the {\tt Load}/{\tt Unload} operations +with coordinates.\swind{UNLCORD} +If switch {\tt UNLCORD} is turned off (normally on) +then all {\tt Load} and {\tt Unload} operations +with coordinates are blocked. + +Since {\tt Unload} writes data in human-unreadable form there +is the command\cmdind{Show File}\cmdind{File}\cmdind{Show {"\parm{file}"}} +\command{Show \opt{File} "\parm{file}";} +or equivalently +\command{?~\opt{File}~"\parm{file}";\\\tt +File "\parm{file}";} +which prints short information about objects and declarations +contained in the \comm{"\parm{file}"}. +It also prints comments contained in the file. + + +\subsection{Coordinate Transformations} +\index{Coordinate transformations} + +Command\cmdind{New Coordinates} +\command{New Coordinates \rpt{\parm{new}} with \rpt{\parm{old}=\parm{expr}};} +introduces new coordinates \parm{new} and +defines how old coordinates \parm{old} are expressed in terms +of new ones. If the specified transformation is nonsingular +\grg\ converts all existing objects to the new coordinate system. + + +The {\tt New Coordinates} command properly transforms all +objects having coordinate indices. The transformation +of frame indices depend on the switch \comm{HOLONOMIC}. \swind{HOLONOMIC} +In general case when frame is not holonomic then objects +having frame indices remain unchanged and only their components +are transformed into the new coordinate system. But if frame +is holonomic then by default all frame indices are transformed +similarly to the coordinate ones. Notice that in such situation +the frame after transformation once again will be holonomic +in the new coordinate system. +But if switch \comm{HOLONOMIC} is turned off the system +distinguishes frame and coordinate indices in spite of the current +frame type. In such situation the holonomic frame +ceases to be holonomic after coordinate transformation. + +\subsection{Frame Transformations} +\index{Frame transformations} + +Spinorial rotations are performed by +the command\cmdind{Make Spinorial Rotation}\cmdind{Spinorial Rotation} +\command{\opt{Make} Spinorial Rotation \opt{ +((\parm{expr}${}_{00}$,\parm{expr}${}_{01}$), +(\parm{expr}${}_{10}$,\parm{expr}${}_{11}$))};} +where expressions $\mbox{\parm{expr}}_{AB}$ comprise the SL(2,C) +transformation matrix +\[ +\phi'_A=L_A{}^B\phi_B,\ \ +\mbox{\parm{expr}}_{AB}=L_A{}^B +\] + +If the specified matrix is really a SL(2,C) one then \grg\ +performs appropriate transformation on all objects whose +value is currently known. + +Matrix specification in the command can be omitted +\command{\opt{Make} Spinorial Rotation;} +In this case the SL(2,C) matrix $L_A{}^B$ must be specified as +the value of a special object {\tt Spinorial Transformation LS.A'B} +(identifier {\tt LS}). + +Command for frame rotation is analogously\cmdind{Make Rotation}\cmdind{Rotation} +\command{\opt{Make} Rotation \opt{ +((\parm{expr}${}_{00}$,\parm{expr}${}_{01}$,...), +(\parm{expr}${}_{10}$,\parm{expr}${}_{11}$,...),...)};} +with the nonsingular $d\times d$ rotation matrix +\[ +A'^a=L^a{}_bA^b,\ \ \mbox{\parm{expr}}_{ab}=L^a{}_b +\] +\grg\ verifies that this matrix is a valid \emph{rotation} +by checking that frame metric $g_{ab}$ \emph{remains unchanged} +under this transformation +\[ +g'_{ab} = L^m{}_a L^n{}_b g_{mn} = g_{ab} +\] + +Once again the matrix specification +can be omitted and transformation $L^a{}_b$ can be specified as the value +of the object {\tt Frame Transformation L'a.b} (identifier {\tt L}) +\command{\opt{Make} Rotation;} + +Frame rotation commands correctly transform frame and +spinor connection 1-forms. + + +Finally, there is a special form of the frame +transformation command\cmdind{Change Metric} +\command{Change Metric \opt{ +((\parm{expr}${}_{00}$,\parm{expr}${}_{01}$,...), +(\parm{expr}${}_{10}$,\parm{expr}${}_{11}$,...),...)};} +The only difference between this command and {\tt Make Rotation} +is that {\tt Change Metric} does not impose +any restriction on the transformation matrix and +transformed metric does not necessary coincides +with the original one. + +Sometimes it is convenient to keep some object unchanged +under the frame transformation. The command\cmdind{Hold} +\command{Hold \parm{object};} +makes the system to keep the \parm{object} unchanged +during frame and spinor transformations. The command\cmdind{Release} +\command{Release \parm{object};} +discards the action of the \comm{Hold} command. + + +\subsection{Algebraic Classification} +\index{Algebraic classification} + +The command\cmdind{Classify} +\command{Classify \parm{object};} +performs algebraic classification of the \parm{object} +specified by its name or identifier. +Currently \grg\ knows algorithms for classifying +the following irreducible spinors + +\begin{tabular}{ll} +$X_{ABCD}$ & Weyl spinor type \\ +$X_{AB\dot{C}\dot{D}}$ & Traceless Ricci spinor type \\ +$X_{AB}$ & Electromagnetic stress spinor type \\ +$X_{A\dot{B}}$ & Vector in the spinorial representation +\end{tabular} \newline + +\reversemarginpar + +The {\tt Classify} command can be applied to any built-in or +user-defined object having one of the listed above +\seethis{See page \pageref{sumspin} about summed spinor indices.} +types of indices. Notice that all spinors must be irreducible +(totally symmetric in dotted and undotted indices) +and $X_{AB\dot{C}\dot{D}}$, $X_{A\dot{B}}$ must be Hermitian. +Groups of the irreducible indices must be represented +as a single summed index. + +\normalmarginpar + +\grg\ uses the algorithm by F.~W.~Letniowski and R.~G.~McLenaghan +[Gen. Rel. Grav. 20 (1988) 463-483] for Petrov-Penrose +classification of Weyl spinor $X_{ABCD}$. The obvious +simplification of this algorithm is applied to +the spinor analog of electromagnetic strength tensor $X_{AB}$. +The spinor $X_{AB\dot{C}\dot{D}}$ is classified by the algorithm +by G.~C.~Joly, M.~A.~H.~McCallum and W.~Seixas +[Class. Quantum Grav. 7 (1990) 541-556, +Class. Quantum Grav. 8 (1991) 1577-1585]. + +The classification process is accompanied by the +tracing messages which can be eliminated by turning \swinda{TRACE} +off the switch \comm{TRACE}. +On the contrary if one turns on \swind{SHOWEXPR} +the switch \comm{SHOWEXPR} then \grg\ prints +all expressions which appear during the classification +to let you check whether the decision about +nonvanishing of these expressions is really correct or not. +This facility is important also in classifying +$X_{AB\dot{C}\dot{D}}$ and $X_{A\dot{B}}$ +since algebraic type for this objects may depend on +the \emph{sign} of some expressions which +cannot be determined by \grg\ correctly. + + +\subsection{\reduce\ Packages and Functions in \grg} +\index{Using \reduce\ packages} +\label{packages} + +Any procedure or function defined +in \reduce\ package can be used in \grg. +The package must be loaded either before +\grg\ is started or during \grg\ session by one of the +equivalent commands +\cmdind{Package}\cmdind{Use Package}\cmdind{Load} +\command{\opt{Use} Package \parm{package};\\\tt +Load \parm{package};} +where \parm{package} is the package name. Notice that an +identifier must be used for the package name unlike +the \comm{Load "\parm{file}";} command described in \enlargethispage{5mm} +section \ref{UnloadLoad}. Let us consider some examples. +The \reduce\ package \file{specfn} contains +definitions of various special functions and +below we demonstrate 11th Legendre polynomial +\begin{slisting} +<- Coordinates t, x, y, z; +<- package specfn; +<- LEGENDREP(11,x); + + 10 8 6 4 2 + x*(88179*x - 230945*x + 218790*x - 90090*x + 15015*x - 693) +------------------------------------------------------------------- + 256 +\end{slisting} + +\newpage + +Another example demonstrates the \file{taylor} package +\begin{slisting} +<- Coordinates t, x, y, z; +<- www=d(E^(x+y)*SIN(x)); +<- www; + + x + y x + y +(E *(COS(x) + SIN(x))) d x + (E *SIN(x)) d y + +<- load taylor; +<- TAYLOR(www,x,0,5); + + y y + y y y 2 E 4 E 5 6 y y 2 +(E + 2*E *x + E *x - ----*x - ----*x + O(x )) d x + (E *x + E *x + 6 15 + + y y + E 3 E 5 6 + + ----*x - ----*x + O(x )) d y + 3 30 +\end{slisting} + +You can also define your own operators and procedures +in \reduce\ and later use them in \grg. +In the following example file \file{lasym.red} contains +a definition of little \reduce\ procedure +which computes a leading term of asymptotic expansion +of the rational function at large values of some +variable. This file is inputted in \reduce\ before +\grg\ is started +\begin{slisting} + +1: in "lasym.red"; + +procedure leadingterm(w,x); + lterm(num(w),x)/lterm(den(w),x); + +leadingterm + +end; + +2: load grg; + +This is GRG 3.2 release 2 (Feb 9, 1997) ... + +System directory: c:{\bs}red35{\bs}grg32{\bs} +System variables are upper-cased: E I PI SIN ... +Dimension is 4 with Signature (-,+,+,+) + +<- Coordinates t, r, theta, phi; +<- OMEGA01=(123*r^3+2*r+t)/(r+t)^5*d theta{\w}d phi; +<- OMEGA01; + + 3 + 123*r + 2*r + t +(-------------------------------------------------) d theta \w d phi + 5 4 3 2 2 3 4 5 + r + 5*r *t + 10*r *t + 10*r *t + 5*r*t + t + +<- LEADINGTERM(OMEGA01,r); + + 123 +(-----) d theta \w d phi + 2 + r +\end{slisting} + + +\subsection{Anholonomic Basis Mode} +\index{Anholonomic basis mode}\index{Basis}\label{amode} + +\grg\ may work in both holonomic and anholonomic basis modes. +In the first default case, values of all expressions are +represented in a natural holonomic (coordinate) basis: +$d x^\mu,~d x^\mu\wedge x^\nu\dots$ for exterior +forms and $\partial_\mu=\partial/\partial x^\mu$ +for vectors. In the second case an +arbitrary basis $b^i=b^i_\mu d x^\mu$ is used for +forms and inverse vector basis $e_i=e_i^\mu\partial_\mu$ for vectors +($b^i_\mu e^\mu_j=\delta^i_j$). You can specify this basis +assigning a value to built-in object +{\tt Basis} (identifier {\tt b}). If {\tt Basis} is not +specified by user then \grg\ assumes that it coincides +with the frame $b^i=\theta^i$. + +Frame should not be confused with basis. Frame $\theta^a$ is used +only for ``external'' purposes to represent tensor indices +while basis $b^i$ and vector basis $e_i$ is used for ``internal'' +purposes to represent form and vector valued object components. + +The command\cmdind{Anholonomic} +\command{Anholonomic;} +switches the system to the anholonomic basis mode and +the command\cmdind{Holonomic} +\command{Holonomic;} +switches it back to the standard holonomic mode. + +Working in anholonomic mode \grg\ creates some internal tables +for efficient calculation of exterior differentiation and +complex conjugation. In anholonomic mode the command +\cmdind{Unload} +\begin{listing} + Unload All > "\parm{file}"; +\end{listing} +automatically saves these tables into the \comm{"\parm{file}"}. +Subsequent\cmdind{Load} +\begin{listing} + Load "\parm{file}"; +\end{listing} +restores the tables and automatically switches the current mode to +anholonomic one. Note that automatic anholonomic mode +saving/restoring works only if {\tt All} is used in +{\tt Unload} command. + +One can find out the current mode with the help of the command +\cmdind{Show Status}\cmdind{Status} +\command{\opt{Show} Status;} + + +\subsection{Synonymy} +\index{Synonymy} + +Sometimes \grg\ commands may be rather long. For +instance, in order to find the curvature 2-form $\Omega_{ab}$ +from the spinorial curvature $\Omega_{AB}$ and $\Omega_{\dot{A}\dot{B}}$ +the following command should be used +\begin{listing} + Find Curvature From Spinorial Curvature; +\end{listing} +Certainly, this command is clear but typing of such long +phrases may be very dull. \grg\ has synonymy mechanism +which allows one to make input much shorter. + +The synonymous words in commands and object names +are considered to be equivalent. The complete list +of predefined \grg\ synonymy is given in appendix D. +Here we present just the most important ones +\begin{verbatim} + Connection Con + Constants Const Constant + Coordinates Cord + Curvature Cur + Dotted Do + Equation Equations Eq + Find F Calculate Calc + Functions Fun Function + Next N + Show ? + Spinor Spin Spinorial Sp + Switch Sw + Symmetries Sym Symmetric + Undotted Un + Write W +\end{verbatim} +Words in each line are considered as equivalent +in all commands. Thus the above command can be abbreviated as +\begin{listing} + F cur from sp cur; +\end{listing} + +Section \ref{tuning} explains how to change built-in synonymy +and how to define a new one. + + +\subsection{Compound Commands} +\index{Compound commands} + +Sometime one may need to perform several consecutive actions +with one object. In this case we can use so called +\emph{compound commands} to shorten the input. +Internally \grg\ replaces each compound command by several usual +ones. For example the compound command +\begin{listing} + Find and Write Einstein Equation; +\end{listing} +to a pair of usual ones +\begin{listing} + Find Einstein Equation; + Write Einstein Equation; +\end{listing} +Actions (commands) can be attached to the end of the +compound command as well: +\begin{listing} + Find, Write Curvature and Erase It; +\qquad\qquad \udr + Find \& Write \& Erase Curvature; +\qquad\qquad \udr + Find Curvature; + Write Curvature; + Erase Curvature; +\end{listing} +Note that we have used {\tt ,} and {\tt \&} instead of {\tt and} +in this example. All these separators are equivalent in compound +commands. + +Now let us consider the case when one needs to perform a single action +with several objects. The command +\begin{listing} + Write Frame, Vector Frame and Metric; +\end{listing} +is equivalent to +\begin{listing} + Write Frame; + Write Vector Frame; + Write Metric; +\end{listing} +Way specification can be attached to the {\tt Find} command: +\begin{listing} + Find QT, QP From Torsion using spinors; +\qquad\qquad \udr + Find QT From Torsion using spinors; + Find QP From Torsion using spinors; +\end{listing} +One can combine several actions and several objects. +For example, the command +\begin{listing} + Find omega, Curvature by Standard Way and Write and Erase Them; +\end{listing} +is equivalent to the sequence of +$(2{\rm\ objects})\times(3{\rm\ commands}) =6$ +commands +\begin{listing} + Find omega by Standard Way; + Find Curvature by Standard Way; + Write omega; + Write Curvature; + Erase omega; + Erase Curvature; +\end{listing} +Note that the way specification is attached only to ``left'' +commands ({\tt Find} in our case). + +The compound commands mechanism works only with +{\tt Find}, {\tt Erase}, {\tt Write} and {\tt Evaluate} commands. + +And finally, \grg\ always replaces {\tt Re-\parm{command};} by +{\tt Erase and \parm{command};}. For example +\begin{listing} + Re-Calculate Maxwell Equations; +\qquad\qquad \udr + Erase and Calculate Maxwell Equations; +\end{listing} + +You can see how \grg\ expand compound commands into the +\swind{SHOWCOMMANDS} +usual ones by turning switch \comm{SHOWCOMMANDS} on. + + +\section{Tuning \grg} +\label{tuning} + +\grg\ can be tuned according to your needs and preferences. +The configuration files allow one to change some default settings +and the environment variable \comm{grg} defines the system +directory which can be used as the depository for +frequently used files. + +\subsection{Configuration Files} +\label{configsect} + +The configuration files allows one to establish +\begin{list}{$\bullet$}{\labelwidth=8mm\leftmargin=10mm} +\item Default dimension and signature. +\item Initial position of switches. +\item \reduce\ packages which must be preloaded. +\item Synonymy. +\item Default \grg\ start up method. +\end{list} + +There are two configuration files. First \emph{global} +configuration file \file{grgcfg.sl} defines the settings +\index{Global configuration file} +during system installation when \grg\ is compiled. +These global settings become permanent and can be changed only +if \grg\ is recompiled. The \emph{local} +configuration file \file{grg.cfg} allows one to override +global settings locally. +\index{Local configuration file} +When \grg\ starts it search the file \file{grg.cfg} +in current directory (folder) and if it is present +uses the corresponding settings. + +Below we are going to explain how to change settings in +both global and local configuration files but before +doing this we must emphasize that this need some care. +First, the configuration files use LISP command format +which differs from usual \grg\ commands. +Second, is something is wrong with configuration file +then no clear diagnostic is provided. +Finally, if global configuration is damaged you will +not be able to compile \grg. The best strategy is to +make a back-up copy of the configuration files before start +editing them. +Notice that lines preceded by the percent sign +\comm{\%} are ignored by the system (comments). + +Both local \file{grg.cfg} and global \file{grgcfg.sl} +configuration files have similar structure and can include +the following commands. + +Command\index{Signature!default}\index{Dimension!default} +\begin{listing} + (signature!> - + + + +) +\end{listing} +establishes default dimension 5 with the signature +$\scriptstyle(-,+,+,+,+)$. Do not forget \comm{!} and spaces between +\comm{+} and \comm{-}. This command \emph{must be present} +in the global configuration file \file{grgcfg.sl} +otherwise \grg\ cannot be compiled. + +The commands +\begin{listing} + (on!> page) + (off!> allfac) +\end{listing} +change default switch position. In this example we +turn on the switch \comm{PAGE} (this switch is defined +in DOS \reduce\ only and allows one to scroll back and forth +through input and output) and turn off switch +\comm{ALLFAC}. + +The command +\begin{listing} + (package!> taylor) +\end{listing} +makes the system to load \reduce\ package \file{taylor} +during \grg\ start. + +The command of the form\index{Synonymy} +\begin{listing} + (synonymous!> + ( affine aff ) + ( antisymmetric asy ) + ( components comp ) + ( unload save ) + ) +\end{listing} +defines synonymous words. The words in each line will be +equivalent in all \grg\ commands. + +Finally the command +\begin{listing} + (setq ![autostart!] nil) +\end{listing} +alters default \grg\ start up method. It makes sense only +in the global configuration file \file{grgcfg.sl}. +By default \grg\ is launched by single command +\begin{listing} + load grg; +\end{listing} +which firstly load the program into memory and then +automatically starts it. Unfortunately on some systems +this short method does not work properly: \grg\ shows wrong +timing during computations, the \comm{quit;} command returns +the control to \reduce\ session instead of terminating the +whole program. If the aforementioned option is activated then +\grg\ must be launched by two commands +\begin{listing} + load grg; + grg; +\end{listing} +which fixes the problems. Here first command just loads the program +into memory and second one starts it manually. Notice that +one can always use commands +\begin{listing} + load grg32; + grg; +\end{listing} +to start \grg\ manually. Command \comm{load grg32;} always +loads \grg\ into memory without starting it independently +on the option under consideration. + + +\subsection{System Directory} +\index{System directory} + +The environment variable \comm{grg} or \comm{GRG} +defines so called \grg\ system directory (folder). +The way of setting this variable is operating system +dependent. For example the following commands +can be used to set \comm{grg} variable in DOS, UNIX and +VAX/VMS respectively: +\begin{listing} + set grg=d:{\bs}xxx{\bs}yyy{\bs} {\rm DOS} + setenv grg /xxx/yyy/ {\rm UNIX} + define grg SYS$USER:[xxx.yyy] {\rm VAX/VMS} +\end{listing} +The value of the variable \comm{grg} must point +out to some directory. +In DOS and UNIX the directory +name must include trailing \comm{\bs} or \comm{/} +respectively. The command\cmdind{Show Status}\cmdind{Status} +\command{\opt{Show} Status;} +prints current system directory. + +When \grg\ tries to input some batch file containing +\grg\ commands it first searches it in the current working +directory and if the file is absent then it tries +to find it in the system directory. Therefore if you have +some frequently used files you can define the system directory +and move these files there. In this case it is not necessary +to keep them in each working directory. Notice \grg\ uses +the same strategy when opening local configuration file +\file{grg.cfg}. Thus if system directory is defined and it +contains the file \file{grg.cfg} the settings contained in +this file effectively overrides global settings without +recompiling \grg. + + +\section{Examples} + +In this section we want to demonstrate how \grg\ can be applied +to solve simple but realistic problem. +We want to calculate the Ricci tensor for the Robertson-Walker +metric by three different methods. + +First \grg\ task (program) +\begin{listing} + Coordinates t,r,theta,phi; + Function a(t); + Frame T0=d t, T1=a*d r, T2=a*r*d theta, T3=a*r*SIN(theta)*d phi; + ds2; + Find and Write Ricci Tensor; + RIC(\_j,\_k); +\end{listing} +defines the Robertson-Walker metric using the tetrad +formalism with the orthonormal Lorentzian tetrad $\theta^a$. +Using built-in formulas for the Ricci tensor the only one command +is required to accomplish out goal +{\tt Find and Write Ricci Tensor;}. The command {\tt ds2;} +just shows the metric we are dealing with. Notice that +command {\tt Find ...} gives the \emph{tetrad} components of the Ricci +tensor $R_{ab}$. Thus, in addition we print coordinate +components of the tensor $R_{\mu\nu}$ by the command +{\tt RIC(\_j,\_k);}. The hard-copy of the corresponding +\grg\ session is presented below \enlargethispage{4mm} +\begin{slisting} +<- Coordinates t, r, theta, phi; +<- Function a(t); +<- Frame T0=d t, T1=a*d r, T2=a*r*d theta, T3=a*r*SIN(theta)*d phi; +<- ds2; +Assuming Default Metric. +Metric calculated By default. 0.16 sec + + 2 2 2 2 2 2 2 2 2 2 2 + ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi + +<- Find and Write Ricci Tensor; +Sqrt det of metric calculated. 0.21 sec +Volume calculated. 0.21 sec +Vector frame calculated From frame. 0.21 sec +Inverse metric calculated From metric. 0.21 sec +Frame connection calculated. 0.38 sec +Curvature calculated. 0.49 sec +Ricci tensor calculated From curvature. 0.54 sec +Ricci tensor: + + - 3*DF(a,t,2) +RIC = ---------------- + 00 a +\newpage + 2 + DF(a,t,2)*a + 2*DF(a,t) +RIC = -------------------------- + 11 2 + a + + 2 + DF(a,t,2)*a + 2*DF(a,t) +RIC = -------------------------- + 22 2 + a + + 2 + DF(a,t,2)*a + 2*DF(a,t) +RIC = -------------------------- + 33 2 + a + +<- RIC(_j,_k); + + - 3*DF(a,t,2) +j=0 k=0 : ---------------- + a + + 2 +j=1 k=1 : DF(a,t,2)*a + 2*DF(a,t) + + 2 2 +j=2 k=2 : r *(DF(a,t,2)*a + 2*DF(a,t) ) + + 2 2 2 +j=3 k=3 : SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) +\end{slisting} +Tracing messages demonstrate that \grg\ automatically +applied several built-in equations to obtain required value of +$R_{ab}$. The metric is automatically assumed to be +Lorentzian $g_{ab}={\rm diag}(-1,1,1,1)$. +First \grg\ computed the frame connection 1-form $\omega^a{}_b$. +Next the curvature 2-form $\Omega^a{}_b$ was computed using +standard equation (\ref{omes}) on page \pageref{omes}. +Finally the Ricci tensor was obtained using +relation (\ref{rics}) on page \pageref{rics}. + +Second \grg\ task is similar to the first one: +\begin{listing} + Coordinates t,r,theta,phi; + Function a(t); + Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; + ds2; + Find and Write Ricci Tensor; +\end{listing} +The only difference is that now we work in the coordinate +formalism by assigning value to the metric rather than +frame. The frame is assumed to be holonomic automatically. +\begin{slisting} +<- Coordinates t, r, theta, phi; +<- Function a(t); +<- Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; +<- ds2; +Assuming Default Holonomic Frame. +Frame calculated By default. 0.11 sec + + 2 2 2 2 2 2 2 2 2 2 2 + ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi + +<- Find and Write Ricci Tensor; +Sqrt det of metric calculated. 0.22 sec +Volume calculated. 0.22 sec +Vector frame calculated From frame. 0.22 sec +Inverse metric calculated From metric. 0.27 sec +Frame connection calculated. 0.33 sec +Curvature calculated. 0.60 sec +Ricci tensor calculated From curvature. 0.60 sec +Ricci tensor: + + - 3*DF(a,t,2) +RIC = ---------------- + t t a + + 2 +RIC = DF(a,t,2)*a + 2*DF(a,t) + r r + + 2 2 +RIC = r *(DF(a,t,2)*a + 2*DF(a,t) ) + theta theta + + 2 2 2 +RIC = SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) + phi phi +\end{slisting} +Once again \grg\ uses the same built-in formulas to compute +the Ricci tensor but now all quantities have holonomic +indices instead of tetrad ones. + +Finally the third task demonstrate how \grg\ can be used +without built-in equations. Once again we use coordinate +formalism and declare two new objects the Christoffel symbols +\comm{Chr} and Ricci tensor \comm{Ric} +(since \grg\ is case sensitive they are different from the built-in +objects \comm{CHR} and \comm{RIC}). Next we use +well-known equations to compute these quantities +\begin{listing} + Coordinates t,r,theta,phi; + Function a(t); + Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; + ds2; + New Chr^a_b_c with s(2,3); + Chr(j,k,l)= 1/2*GI(j,m)*(@x(k)|G(l,m)+@x(l)|G(k,m)-@x(m)|G(k,l)); + New Ric_a_b with s(1,2); + Ric(j,k) = @x(n)|Chr(n,j,k) - @x(k)|Chr(n,j,n) + + Chr(n,m,n)*Chr(m,j,k) - Chr(n,m,k)*Chr(m,n,j); + Write Ric; +\end{listing} +The hard-copy of the corresponding session is +\begin{slisting} +<- Coordinates t, r, theta, phi; +<- Function a(t); +<- Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; +<- ds2; +Assuming Default Holonomic Frame. +Frame calculated By default. 0.16 sec + + 2 2 2 2 2 2 2 2 2 2 2 + ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi + +<- New Chr^a_b_c with s(2,3); +<- Chr(j,k,l)=1/2*GI(j,m)*(@x(k)|G(l,m)+@x(l)|G(k,m)-@x(m)|G(k,l)); +Inverse metric calculated From metric. 0.27 sec +<- New Ric_a_b with s(1,2); +<- Ric(j,k)=@x(n)|Chr(n,j,k)-@x(k)|Chr(n,j,n)+Chr(n,m,n)*Chr(m,j,k) + -Chr(n,m,k)*Chr(m,n,j); +<- Write Ric; +The Ric: + + - 3*DF(a,t,2) +Ric = ---------------- + t t a + + 2 +Ric = DF(a,t,2)*a + 2*DF(a,t) + r r +\newpage + 2 2 +Ric = r *(DF(a,t,2)*a + 2*DF(a,t) ) + theta theta + + 2 2 2 +Ric = SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) + phi phi +\end{slisting} + + + +\chapter{Formulas} +\parindent=0pt +\arraycolsep=1pt +\parskip=1.6mm plus 1mm minus 1mm + +This chapter describes in usual mathematical manner all \grg\ +built-in objects and formulas. The description is extremely short +since it is intended for reference only. +If not stated explicitly we use lower case greek letters +${\scriptstyle \alpha,\beta,\dots}$ for +holonomic (coordinate) indices; ${\scriptstyle a,b,c,d,m,n}$ for +anholonomic frame indices and ${\scriptstyle i,j,k,l}$ +for enumerating indices. + +To establish the relationship between \grg\ built-in object6s +and mathematical quantities we use the following notation +\[ +\mbox{\tt Frame Connection omega'a.b} = \omega^a{}_b +\] +This equality means that there is built-in object named +{\tt Frame Connection} having identifier {\tt omega} +which represent the frame connection 1-form $\omega^a{}_b$. +If the name is omitted then we deal with \emph{macro} object +(see page \pageref{macro}). The notation for indices +in the left-hand side of such equalities is the same +as in the {\tt New object} declaration and +is explained on page \pageref{indices}. + +This chapter contains not only definitions of all built-in +objects but all formulas which \grg\ knows and can apply +to find their value. If an object has +several formulas for its computation when each formula +is given together with the corresponding name which is printed +in the typewriter font. +In the case then an object has only one associated +formula the way name is usually omitted. + + +\section{Dimension and Signature} + +Let us denote the space-time dimensionality by $d$ +and $n$'th element of the signature specification +${\rm diag}{\scriptstyle(+1,-1,\dots)}$ by ${\rm diag}_n$ +($n$ runs from 0 to $d-1$). + +There are several macro objects which gives access to +the dimension and signature +\object{dim}{d} +\object{sdiag.idim}{{\rm diag}_i} +\object{sgnt \mbox{=} sign}{s=\prod^{d-1}_{i=0}{\rm diag}_i} +\object{mpsgn}{{\rm diag}_0} +\object{pmsgn}{-{\rm diag}_0} + +The macros (two equivalent ones) which give access to +coordinates +\object{X\^m \mbox{=} x\^m}{x^\mu} + + +\section{Metric, Frame and Basis} + +Frame $\theta^a$ and metric $g_{ab}$ plays the +fundamental role in \grg. Together they determine the +space-time line element +\begin{equation} +ds^2 = g_{ab}\,\theta^a\!\otimes\theta^b = + g_{\mu\nu}\,dx^\mu\!\otimes dx^\nu +\end{equation} + +The corresponding objects are +\object{Frame T'a}{\theta^a=h^a_\mu dx^\mu} +\object{Metric G.a.b}{g_{ab}} +and ``inverse'' objects are +\object{Vector Frame D.a}{\partial_a=h^\mu_a\partial_\mu} +\object{Inverse Metric GI'a'b}{g^{ab}} + +The frame can be computed by two ways. First, {\tt By default} +frame is assumed to be holonomic +\begin{equation} +\theta^a = dx^\alpha +\end{equation} +and {\tt From vector frame} +\begin{equation} +\theta^a= |h_a^\mu|^{-1} d x^\mu +\end{equation} + +The vector frame can be obtained {\tt From frame} +\begin{equation} +\partial_a= |h^a_\mu|^{-1} \partial_\mu +\end{equation} + +The metric can be computed {\tt By default} \index{Metric!default value} +\begin{equation} +g_{ab} = {\rm if}\ a=b\ {\rm then}\ {\rm diag}_a\ {\rm else}\ 0 +\end{equation} +or {\tt From inverse metric} +\begin{equation} +g_{ab} = |g^{ab}|^{-1} +\end{equation} + +The inverse metric can be computed {\tt From metric} +\begin{equation} +g^{ab} = |g_{ab}|^{-1} +\end{equation} + +The holonomic metric $g_{\mu\nu}$ and frame $h^a_\mu$ +are given by the macro objects: +\object{g\_m\_n}{g_{\mu\nu}} +\object{gi\^m\^n}{g^{\mu\nu}} +\object{h'a\_m}{h^a_\mu} +\object{hi.a\^m}{h_a^\mu} + +The metric determinants and related densities +\object{Det of Metric detG}{g={\rm det}|g_{ab}|} +\object{Det of Holonomic Metric detg}{{\rm det}|g_{\mu\nu}|} +\object{Sqrt Det of Metric sdetG}{\sqrt{sg}} + +The volume $d$-form +\object{Volume VOL}{\upsilon = \sqrt{sg}\,\theta^0\wedge\dots\wedge\,\theta^{d-1} +=\frac{1}{d!}{\cal E}_{a_0\dots a_{d-1}}\,\theta^{a_0}\wedge\dots\wedge\,\theta^{a_{d-1}}} + +The so called s-forms play the role of basis in the space of the +2-forms +\object{S-forms S'a'b}{S^{ab}=\theta^a\wedge\theta^b} + +The basis and corresponding inverse vector basis are used +when \grg\ works in the anholonomic mode +\seethis{See page \pageref{amode}.} +\object{Basis b'idim }{b^i=b^i_\mu dx^\mu} +\object{Vector Basis e.idim }{e_i=b_i^\mu\partial_\mu} +The basis can be computed {\tt From frame} +\begin{equation} +b^i=\theta^i +\end{equation} +or {\tt From vector basis} +\begin{equation} +b^i = |b_i^\mu|^{-1}dx^\mu +\end{equation} +The vector basis can be computed {\tt From basis} +\begin{equation} +e_i = |b^i_\mu|^{-1}\partial_\mu +\end{equation} + + +\section{Delta and Epsilon Symbols} + +Macro objects for Kronecker delta symbols +\object{del\^m\_n}{\delta^\mu_\nu} +\object{delh'a.b}{\delta^a_b} +and totally antisymmetric tensors +\object{eps.a.b.c.d}{{\cal E}_{abcd},\quad{\cal E}_{0123}=\sqrt{sg}} +\object{epsi'a'b'c'd}{{\cal E}^{abcd},\quad{\cal E}_{0123}=\frac{s}{\sqrt{sg}}} +\object{epsh\_m\_n\_k\_l}{{\cal E}_{\mu\nu\kappa\lambda},\quad{\cal E}_{0123}=\sqrt{s\,{\rm det}|g_{\mu\nu}|}} +\object{epsih\^m\^n\^k\^l}{{\cal E}^{\mu\nu\kappa\lambda},\quad{\cal E}_{0123}=\frac{s}{\sqrt{s\,{\rm det}|g_{\mu\nu}|}}} +The definition for epsilon-tensors is given for dimension 4. +The generalization to other dimensions is obvious. + + +\section{Dualization} + +We use the following definition for the dualization +operation. For any $p$-form +\begin{equation} +\omega_p=\frac{1}{p!}\omega_{\alpha_1\dots\alpha_p}dx^{\alpha_1}\wedge +\dots\wedge dx^{\alpha_p} +\end{equation} +the dual $(d-p)$-form is +\begin{equation} +*\omega_p=\frac{1}{p!(d-p)!}{\cal E}_{\alpha_1\dots\alpha_{d-p}} +{}^{\beta_1\dots\beta_p}\,\omega_{\beta_1\dots\beta_p}\, +dx^{\alpha_1}\wedge\dots\wedge dx^{\alpha_{d-p}} +\end{equation} + +The equivalent relation which also uniquely defines the $*$ +operation is +\begin{equation} +*(\theta^{a_1}\wedge\dots\wedge \theta^{a_p}) = +(-1)^{p(d-p)} \partial_{a_p}\ipr\dots\partial_{a_1}\ipr\,\upsilon +\end{equation} + +With such convention we have the following identities +\begin{eqnarray} +**\omega_p &=& s(-1)^{p(d-p)}\,\omega_p \\[0.5mm] +*\upsilon &=& s \\[0.5mm] +*1 &=& \upsilon +\end{eqnarray} + + +\section{Spinors} +\label{spinors1} + +The notion of spinors in \grg\ is restricted to + 4-dimensional spaces of Lorentzian signature ${\scriptstyle(-,+,+,+)}$ +or ${\scriptstyle(+,-,-,-)}$ only. In this section the upper sign relates to the +signature ${\scriptstyle(-,+,+,+)}$ and lower one to +${\scriptstyle(+,-,-,-)}$. + +In addition to work with spinors the metric must have the following +form which we call the \emph{standard null metric} \index{Metric!Standard Null} +\index{Standard null metric}\index{Spinors}\index{Spinors!Standard null metric} +\begin{equation} +g_{ab}=g^{ab}=\pm\left(\begin{array}{rrrr} +0 & -1 & 0 & 0 \\ +-1 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 \\ +0 & 0 & 1 & 0 +\end{array}\right) +\end{equation} +Such value of the metric can be established by the command +\cmdind{Null Metric} +{\tt Null metric;}. + +Therefore the line-element for spinorial formalism has the form +\begin{equation} +ds^2 = \pm(-\theta^0\!\otimes\theta^1 +-\theta^1\!\otimes\theta^0 ++\theta^2\!\otimes\theta^3 ++\theta^3\!\otimes\theta^2) +\end{equation} + +We require also the conjugation rules for this null tetrad (frame) be +\begin{equation} +\overline{\theta^0}=\theta^0,\quad +\overline{\theta^1}=\theta^1,\quad +\overline{\theta^2}=\theta^3,\quad +\overline{\theta^3}=\theta^2 +\end{equation} + +For such a metric and frame we fix sigma-matrices in the +following form \index{Sigma matrices} +\begin{eqnarray} \label{sigma} +&&\sigma_0{}^{1\dot{1}}= +\sigma_1{}^{0\dot{0}}= +\sigma_2{}^{1\dot{0}}= +\sigma_3{}^{0\dot{1}}=1 \\[1mm] && +\sigma^0{}_{1\dot{1}}= +\sigma^1{}_{0\dot{0}}= +\sigma^2{}_{1\dot{0}}= +\sigma^3{}_{0\dot{1}}=\mp1 +\end{eqnarray} + +The sigma-matrices obey the rules +\begin{eqnarray} +g_{mn}\sigma^m\!{}_{A\dot B}\sigma^n\!{}_{C\dot D} &=& +\mp \epsilon_{AC}\epsilon_{\dot B\dot D} \\[1mm] +\sigma^{aM\dot N}\sigma^b\!{}_{M\dot N} &=& \mp g^{ab} +\end{eqnarray} + +The antisymmetric SL(2,C) spinor metric +\begin{equation} +\epsilon_{AB}=\epsilon^{AB} +=\epsilon_{\dot A\dot B} +=\epsilon^{\dot A\dot B}= +\left(\begin{array}{rr} +0 & 1 \\ +-1 & 0 +\end{array}\right) +\end{equation} +can be used to raise and lower spinor indices +\begin{equation} +\varphi^A=\varphi_B\,\epsilon^{BA},\qquad +\varphi_A=\epsilon_{AB}\,\varphi^B +\end{equation} + +The following macro objects represent standard +spinorial quantities +\object{DEL'A.B}{\delta^A_B} +\object{EPS.A.B}{\epsilon_{AB}} +\object{EPSI'A'B}{\epsilon^{AB}} +\object{sigma'a.A.B\cc}{\sigma^a\!{}_{A\dot B}} +\object{sigmai.a'A'B\cc}{\sigma_a{}^{A\dot B}} + +The relationship between tensors and spinors +is established by the sigma-matrices +\begin{eqnarray} +X^a &\tsst& X^{A\dot A}=A^a\sigma_a{}^{A\dot A} \\ +X_a &\tsst& X_{A\dot A}=A_a\sigma^a\!{}_{A\dot A} +\end{eqnarray} +where sigma-matrices are given by Eq. (\ref{sigma}) +We shall denote similar equations by the sign $\tsst$ +conserving alphabetical relationship between tensor indices in the +left-hand side and spinorial one in the right-hand side: +$\scriptstyle a\tsst A\dot A$, $\scriptstyle b\tsst B\dot B$. + +There is one quite important special case. Any real +antisymmetric tensor $X_{ab}$ are equivalent to the +pair of conjugated irreducible (symmetric) spinors +\begin{eqnarray} +&& X_{ab}=X_{[ab]} \tsst X_{A\dot AB\dot B}= +\epsilon_{AB} X_{\dot A\dot B} + \epsilon_{\dot A\dot B}X_{AB} +\nonumber\\[1mm] +&& X_{AB}=\frac{1}{2}X_{A\dot AB\dot B}\epsilon^{\dot A\dot B},\ + X_{\dot A\dot B}=\frac{1}{2}X_{A\dot AB\dot B}\epsilon^{AB} +\end{eqnarray} +The explicit form of these relations for the sigma-matrices +(\ref{sigma}) is +\begin{equation} +\begin{array}{rclrcl} +X_0 &=& X_{13} & X_{\dot0} &=& X_{12} \\[1mm] +X_1 &=&-\frac{1}{2}(X_{01}-X_{23})\qquad & X_{\dot1} &=& +-\frac{1}{2}(X_{01}+X_{23}) \\[1mm] +X_2 &=& -X_{02} & X_{\dot2} &=& -X_{03} +\end{array}\label{asys} +\end{equation} +and the ``inverse'' relation +\begin{equation} +\begin{array}{rclrcl} +X_{01} &=& -X_1-X_{\dot1},\qquad & X_{23} &=& X_1-X_{\dot1}, \\[1mm] +X_{02} &=& -X_2, & X_{12} &=& X_{\dot0}, \\[1mm] +X_{03} &=& -X_{\dot 2}, & X_{13} &=& X_0 +\end{array}\label{asyt} +\end{equation} + +We shall apply the relations (\ref{asys}) and (\ref{asyt}) to various +antisymmetric quantities. In particular the {\tt Spinorial S-forms} +\object{Undotted S-forms SU.AB}{S_{AB}} +\object{Dotted S-forms SD.AB\cc}{S_{\dot A\dot B}} +The {\tt Standard way} to compute these quantities uses +relations (\ref{asys}) +\begin{equation} + S_{ab}=\theta_a\wedge\theta_b \tsst +\epsilon_{AB} S_{\dot A\dot B} + \epsilon_{\dot A\dot B}S_{AB} +\end{equation} +Spinorial S-forms are self dual +\begin{equation} +*S_{AB}=iS_{AB},\qquad +*S_{\dot A\dot B}=-iS_{\dot A\dot B} +\end{equation} +and exteriorly orthogonal +\begin{equation} +S_{AB}\wedge S_{CD}=-\frac{i}2\upsilon(\epsilon_{AC}\epsilon_{BD}+ +\epsilon_{AD}\epsilon_{BC}),\quad S_{AB}\wedge S_{\dot C\dot D}=0 +\end{equation} + +There is one subtle pint concerning tensor quantities in the +spinorial formalism. Since spinorial null tetrad is complex +with the conjugation rule $\overline{\theta^2}=\theta^3$ +all tensor quantities represented in this frame also becomes +complex with similar conjugation rules for any tensor index. +There is special macro object {\tt cci} which performs such +``index conjugation'': {\tt cci{0}=0}, {\tt cci(1)=1}, +{\tt cci{2}=3}, {\tt cci(3)=2}. Therefore the correct expression +for the $\overline{\theta^a}$ is {\tt \cc T(cci(a))} but not +{\tt \cc T(a)}. + + + +\section{Connection, Torsion and Nonmetricity} +\label{conn1} + +Covariant derivatives and differentials for +quantities having frame and coordinate indices are +\begin{eqnarray} +DX^a{}_b &=& dX^a{}_b ++ \omega^a{}_m\wedge X^m{}_b - \omega^m{}_b\wedge X^a{}_m \\[1mm] +DX^\mu{}_\nu &=& dX^\mu{}_\nu ++ \Gamma^\mu{}_\pi\wedge X^\pi{}_\nu - \Gamma^\pi{}_\nu\wedge X^\mu{}_\pi +\end{eqnarray} + +The corresponding built-in connection 1-forms are +\object{Frame Connection omega'a.b}{\omega^a{}_b=\omega^a{}_{b\mu}dx^\mu} +\object{Holonomic Connection GAMMA\^m\_n} +{\Gamma^\mu{}_\nu=\Gamma^\mu{}_{\nu\pi}dx^\pi} + +Frame connection can be computed {\tt From holonomic connection} +\begin{equation} +\omega^a{}_b = \Gamma^a{}_b + dh^\mu_b\,h^a_\mu +\end{equation} +and inversely holonomic connection can be obtained +{\tt From frame connection} +\begin{equation} +\Gamma^\mu{}_\nu=\omega^\mu{}_\nu + dh^b_\nu\,h^\mu_b +\end{equation} + +By default these connections are Riemannian (i.e. symmetric and +metric compatible). To work with nonsymmetric +connection with torsion the switch \comm{TORSION}\swinda{TORSION} +must be turned on. Then the torsion 2-form is +\object{Torsion THETA'a}{\Theta^a=\frac12Q^a{}_{pq}S^{pq},\quad +Q^a{}_{bc}=\Gamma^a{}_{bc}-\Gamma^a_{cb}} +Finally to work with non metric-compatible +spaces with nonmetricity the switch \comm{NONMETR}\swinda{NONMETR} +must be turned on. The nonmetricity 1-form is +\object{Nonmetricity N.a.b}{N_{ab}=N_{ab\mu}dx^\mu, +\quad N_{ab\mu}=-\nabla_\mu g_{ab}} +In general any torsion or nonmetricity related object is +defined iff the corresponding switch is on. + +If either \comm{TORSION} or \comm{NONMETR} is on then Riemannian +versions of the connection 1-forms are available as well +\object{Riemann Frame Connection romega'a.b} +{\rim{\omega}{}^a{}_b} +\object{Riemann Holonomic Connection RGAMMA\^m\_n} +{\rim{\Gamma}{}^\mu{}_\nu} + +The Riemann holonomic connection can be obtained +{\tt From Riemann frame connection} +\begin{equation} +\rim{\Gamma}{}^\mu{}_\nu=\rim{\omega}{}^\mu{}_\nu + dh^b_\nu\,h^\mu_b +\end{equation} + + + +If torsion is nonzero but nonmetricity vanishes +(\comm{TORSION} is on, \comm{NONMETR} is off) then +the difference between the connection and Riemann connection +is called the contorsion 1-form +\object{Contorsion KQ'a.b}{\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b= +\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_{b\mu}dx^\mu= +\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} + +If nonmetricity is nonzero but torsion vanishes +(\comm{TORSION} is off, \comm{NONMETR} is on) then +the difference between the connection and Riemann connection +is called the nonmetricity defect +\object{Nonmetricity Defect KN'a.b} +{\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b= +\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_{b\mu}dx^\mu= +\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} + +Finally if both torsion and nonmetricity are nonzero +(\comm{TORSION} and \comm{NONMETR} are on) then we +\object{Connection Defect K'a.b} +{K^a{}_b=K^a{}_{b\mu}dx^\mu= +\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} +\begin{equation} +K^a{}_b = \stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b ++ \stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b +\end{equation} + + +For the sake of convenience we introduce also macro objects +which compute the usual Christoffel symbols +\object{CHR\^m\_n\_p }{ \{{}^\mu_{\nu\pi}\} = +\frac{1}{2}g^{\mu\tau}(\partial_\pi g_{\nu\tau} ++\partial_\nu g_{\pi\tau} +-\partial_\tau g_{\nu\pi})} +\object{CHRF\_m\_n\_p }{ [{}_{\mu},_{\nu\pi}] = +\frac{1}{2}(\partial_\pi g_{\nu\mu} ++\partial_\nu g_{\pi\mu} +-\partial_\mu g_{\nu\pi})} +\object{CHRT\_m }{ \{{}^\pi_{\pi\mu}\} = +\frac{1}{2{\rm det}|g_{\alpha\beta}|}\partial_\mu\left( +{\rm det}|g_{\alpha\beta}|\right)} + +The connection, frame, metric, torsion and nonmetricity are +related to each other by the so called structural equations +which in the most general case read +\begin{eqnarray} +&& D\theta^a + \Theta^a = 0 \nonumber\\[2mm] +&& Dg_{ab} + N_{ab} = 0 \label{str0} +\end{eqnarray} +or in the equivalent ``explicit'' form +\begin{equation} +\begin{array}{ll} +\omega^a{}_b\wedge\theta^b = -t^a,\qquad & t^a=d\theta^a+\Theta^a,\\[2mm] +\omega_{ab}+\omega_{ba} = n_{ab},\qquad & n_{ab}=dg_{ab}+N_{ab} \label{str} +\end{array} +\end{equation} + +The solution to equations (\ref{str}) are given by the relation +\begin{equation} +\omega^a{}_b = +\frac{1}{2}\left[ -\partial^a\ipr t_b + \partial_b\ipr t^a + n^a{}_b ++\big(\partial^a\ipr(\partial_b\ipr t_c-n_{bc}) ++\partial_b\ipr n^a{}_c\big)\theta^c\right] \label{solstr} +\end{equation} + +For various specific values of $n_{ab}$ and $t^a$ equations +(\ref{str}) and (\ref{solstr}) can be used for different purposes. + +In the most general case (\ref{solstr}) is the {\tt Standard way} to +compute connection 1-form $\omega^a{}_b$. +The torsion and nonmetricity are included in +these equations depending on the switches \comm{TORSION} and +\comm{NONMETR}. + +The same equation (\ref{solstr}) with $n_{ab}=dg_{ab}$ and +$t^a=d\theta^a$ is the {\tt Standard way} to find Riemann +frame connection $\rim{\omega}{}^a{}_b$. + +If torsion is nonzero then $\omega^a{}_b$ can be computed +{\tt From contorsion} +\begin{equation} +\omega^a{}_b = \rim{\omega}{}^a{}_b ++ \stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b \label{a1} +\end{equation} +where $\rim{\omega}{}^a{}_b$ is given by Eq. (\ref{solstr}). + +Similarly if nonmetricity is nonzero then $\omega^a{}_b$ can be computed +{\tt From nonmetricity defect} +\begin{equation} +\omega^a{}_b = \rim{\omega}{}^a{}_b ++ \stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b \label{a2} +\end{equation} +where $\rim{\omega}{}^a{}_b$ is given by Eq. (\ref{solstr}). + +Finally if both torsion and nonmetricity are +nonzero then $\omega^a{}_b$ can be computed +{\tt From connection defect} +\begin{equation} +\omega^a{}_b = \rim{\omega}{}^a{}_b + K^a{}_b \label{a3} +\end{equation} +where $\rim{\omega}{}^a{}_b$ is given by Eq. (\ref{solstr}). + +The Riemannian part of connection in Eqs. (\ref{a1}), +(\ref{a2}), (\ref{a3}) are directly computed by Eq. (\ref{solstr}) +(not via the object \comm{romega}). + +The contorsion $\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b$ +is obtained {\tt From torsion} by (\ref{solstr}) +with $t^a=\Theta^a$, $n_{ab}=0$. + +The nonmetricity defect $\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b$ +is obtained {\tt From nonmetricity} by (\ref{solstr}) +with $t^a=0$, $n_{ab}=N_{ab}$. + +Analogously (\ref{solstr}) with $t^a=\Theta^a$, $n_{ab}=N_{ab}$ +is the {\tt Standard way} to compute the connection defect $K^a{}_b$. + +The torsion $\Theta^a$ can be calculated {\tt From contorsion} +\begin{equation} +\Theta^a = -\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b\wedge\theta^b +\end{equation} +or {\tt From connection defect} +\begin{equation} +\Theta^a = -K^a{}_b\wedge\theta^b +\end{equation} + +The nonmetricity $N_{ab}$ can be computed {\tt From nonmetricity defect} +\begin{equation} +N_{ab} = \stackrel{\scriptscriptstyle N}{K}_{ab}+ +\stackrel{\scriptscriptstyle N}{K}_{ba} +\end{equation} +or {\tt From connection defect} +\begin{equation} +N_{ab} = K_{ab}+K_{ba} +\end{equation} + + +\section{Spinorial Connection and Torsion} + +Spinorial connection is defined in \grg\ iff nonmetricity +is zero and switch \comm{NONMETR} is turned off. +The upper sign in this section correspond to the signature +${\scriptstyle(-,+,+,+)}$ while lower one to the signature +${\scriptstyle(+,-,-,-)}$. + +Spinorial connection is defined by the equation +\begin{equation} +DX^A_{\dot B} = dX^A{}_{\dot B} +\mp\omega^A{}_M\,X^M{}_{\dot B} +\pm\omega^{\dot M}{}_{\dot B}\,X^A{}_{\dot M} +\end{equation} +where due to antisymmetry of the frame connection +$\omega_{ab}=\omega_{[ab]}$ we have {\tt Spinorial connection} +1-forms +\begin{equation} +\omega_{ab} \tsst +\epsilon_{AB} \omega_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \omega_{AB} +\end{equation} +\object{Undotted Connection omegau.AB}{\omega_{AB}} +\object{Dotted Connection omegad.AB\cc}{\omega_{\dot A\dot B}} + +The spinorial connection 1-forms +$\omega_{AB}$ and $\omega_{\dot A\dot B}$ +can be calculated {\tt From frame connection} by the +standard spinor $\tsst$ tensor relation (\ref{asys}). + +Inversely the frame connection $\omega_{ab}$ can be +found {\tt From spinorial connection} by relation (\ref{asyt}). + +Since $\omega_{ab}$ is real the spinorial equivalents +$\omega_{AB}$ and $\omega_{\dot A\dot B}$ can be computed from +each other {\tt By conjugation} +\begin{equation} +\omega_{\dot A\dot B}=\overline{\omega_{AB}},\qquad +\omega_{AB}=\overline{\omega_{\dot A\dot B}} +\end{equation} + +If torsion is nonzero (\comm{TORSION} is on) when we have +in addition the {\tt Riemann spinorial connection} +\object{Riemann Undotted Connection romegau.AB}{\rim{\omega}_{AB}} +\object{Riemann Dotted Connection romegad.AB\cc}{\rim{\omega}_{\dot A\dot B}} + +The Riemann spinorial connection $\rim{\omega}_{AB}$ +can be calculated by {\tt Standard way} +\begin{equation} +\stackrel{{\scriptscriptstyle\{\}}}{\omega}_{AB}= \label{ssolver} +\pm i*[ d S_{AB}\wedge\theta_{C\dot C} + -\epsilon_{C(A} d S_{B)M}\wedge \theta^M_{\ \ \dot C}]\theta^{C\dot C} +\end{equation} +The conjugated relation is used for $\rim{\omega}_{\dot A\dot B}$. + +The {\tt Spinorial contorsion} 1-forms +\object{Undotted Contorsion KU.AB}{\stackrel{\scriptscriptstyle Q}{K}\!{}_{AB}} +\object{Dotted Contorsion KD.AB\cc}{\stackrel{\scriptscriptstyle Q}{K}\!{}_{\dot A\dot B}} +are the spinorial analogues of the contorsion 1-form +\begin{equation} +\stackrel{\scriptscriptstyle Q}{K}_{ab} \tsst +\epsilon_{AB} \stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \stackrel{\scriptscriptstyle Q}{K}_{AB} +\end{equation} + +The spinorial contorsion 1-forms +$\stackrel{\scriptscriptstyle Q}{K}_{AB}$ and $\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}$ +can be calculated {\tt From contorsion} by the +standard spinor $\tsst$ tensor relation (\ref{asys}). + +Inversely the contorsion $\stackrel{\scriptscriptstyle Q}{K}_{ab}$ can be +found {\tt From spinorial contorsion} by relation (\ref{asyt}). + +The spinorial equivalents +$\stackrel{\scriptscriptstyle Q}{K}_{AB}$ and $\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}$ +can be computed from +each other {\tt By conjugation} +\begin{equation} +\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}=\overline{\stackrel{\scriptscriptstyle Q}{K}_{AB}},\qquad +\stackrel{\scriptscriptstyle Q}{K}_{AB}=\overline{\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}} +\end{equation} + +The {\tt Standard way} to find $\omega_{AB}$ is +\begin{equation} +\omega_{AB} = \rim{\omega}_{AB}+\stackrel{\scriptscriptstyle Q}{K}_{AB} +\end{equation} +where $\rim{\omega}_{AB}$ is given directly by Eq. (\ref{ssolver}). +The conjugated Eq. is used for $\omega_{\dot A\dot B}$. + + +\section{Curvature} + +The curvature 2-form +\object{Curvature OMEGA'a.b}{\Omega^a{}_b= +\frac{1}{2}R^a_{bcd}\,S^{cd}} +can be computed {\tt By standard way} +\begin{equation} +\Omega^a{}_b = d\omega^a{}_b + \omega^a{}_n \wedge \omega^n{}_b \label{omes} +\end{equation} + +The Riemann curvature tensor is given by the relation +\object{Riemann Tensor RIM'a.b.c.d}{R^a{}_{bcd}= +\partial_d\ipr\partial_c\ipr\Omega^a{}_b} + +The Ricci tensor +\object{Ricci Tensor RIC.a.b}{R_{ab}} +can be computed {\tt From Curvature} +\begin{equation} +R_{ab} = \partial_b\ipr\partial_m\ipr\Omega^m{}_a \label{rics} +\end{equation} +or {\tt From Riemann tensor} +\begin{equation} +R_{ab} = R^m{}_{amb} +\end{equation} + +The +\object{Scalar Curvature RR}{R} +can be computed {\tt From Ricci Tensor} +\begin{equation} +R = R_{mn}\,g^{mn} +\end{equation} + +The Einstein tensor is given by the relation +\object{Einstein Tensor GT.a.b}{G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R} + +If nonmetricity is nonzero (\comm{NONMETR} is on) then we have +\object{Homothetic Curvature OMEGAH}{\OO{h}} +\object{A-Ricci Tensor RICA.a.b}{\RR{A}_{ab}} +\object{S-Ricci Tensor RICS.a.b}{\RR{S}_{ab}} + +They can be calculated {\tt From curvature} by the +relations +\begin{equation} +\OO{h}=\Omega^n{}_n +\end{equation} +\begin{equation} +\RR{A}_{ab}= \partial_b\ipr\partial^m\ipr\Omega_{[ma]} +\end{equation} +\begin{equation} +\RR{S}_{ab}= \partial_b\ipr\partial^m\ipr\Omega_{(ma)} +\end{equation} +and the scalar curvature can be computed {\tt From A-Ricci tensor} +\begin{equation} +R = \RR{A}_{mn}g^{mn} +\end{equation} + + +\section{Spinorial Curvature} + +Spinorial curvature is defined in \grg\ iff nonmetricity +is zero and switch \comm{NONMETR} is turned off. +The upper sign in this section correspond to the signature +${\scriptstyle(-,+,+,+)}$ while lower one to the signature +${\scriptstyle(+,-,-,-)}$. + +The {\tt Spinorial curvature} 2-forms +\object{Undotted Curvature OMEGAU.AB}{\Omega_{AB}} +\object{Dotted Curvature OMEGAD.AB\cc}{\Omega_{\dot A\dot B}} +is related to the curvature 2-form $\Omega_{ab}$ by the standard +relation +\begin{equation} +\Omega_{ab} \tsst +\epsilon_{AB} \Omega_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \Omega_{AB} +\end{equation} + +The spinorial curvature 1-forms +$\Omega_{AB}$ and $\Omega_{\dot A\dot B}$ +can be calculated {\tt From curvature} by the +relation (\ref{asys}). + +The frame curvature $\Omega_{ab}$ can be +found {\tt From spinorial curvature} by relation (\ref{asyt}). + +The $\Omega_{AB}$ and $\Omega_{\dot A\dot B}$ can be +computed from each other {\tt By conjugation} +\begin{equation} +\Omega_{\dot A\dot B}=\overline{\Omega_{AB}},\qquad +\Omega_{AB}=\overline{\Omega_{\dot A\dot B}} +\end{equation} + +The {\tt Standard way} to calculate $\Omega_{AB}$ is +\begin{equation} +\Omega_{AB} = d\omega_{AB} \pm \omega_A{}^M\wedge\omega_{MB} +\end{equation} +The conjugated relation is used for $\Omega_{\dot A\dot B}$. + + +\section{Curvature Decomposition} + +In general curvature consists of 11 irreducible pieces. +If nonmetricity is nonzero then one can +perform decomposition +\begin{equation} +R_{abcd}=\RR{A}_{abcd}+\RR{S}_{abcd},\qquad +\RR{A}_{abcd}=R_{[ab]cd},\qquad +\RR{S}_{abcd}=R_{(ab)cd} +\end{equation} +Here the S-part of the curvature vanishes identically if +nonmetricity is zero and we consider further decomposition +of A and S parts independently. + +First we consider the A-part of the curvature. It can be +decomposed into 6 pieces +\begin{equation} +\RR{A}_{abcd} = +\RR{w}_{abcd}+ +\RR{c}_{abcd}+ +\RR{r}_{abcd}+ +\RR{a}_{abcd}+ +\RR{b}_{abcd}+ +\RR{d}_{abcd} +\end{equation} +Here first three terms are the well-known irreducible pieces +of the Riemannian curvature while last three terms vanish if +torsion is zero. The corresponding 2-forms are +\object{Weyl 2-form OMW.a.b } +{\OO{w}_{ab} = \frac12 \RR{w}_{abcd}\,S^{cd}} +\object{Traceless Ricci 2-form OMC.a.b } +{\OO{c}_{ab} = \frac12 \RR{c}_{abcd}\,S^{cd}} +\object{Scalar Curvature 2-form OMR.a.b } +{\OO{r}_{ab} = \frac12 \RR{r}_{abcd}\,S^{cd}} +\object{Ricanti 2-form OMA.a.b } +{\OO{a}_{ab} = \frac12 \RR{a}_{abcd}\,S^{cd}} +\object{Traceless Deviation 2-form OMB.a.b } +{\OO{b}_{ab} = \frac12 \RR{b}_{abcd}\,S^{cd}} +\object{Antisymmetric Curvature 2-form OMD.a.b } +{\OO{d}_{ab} = \frac12 \RR{d}_{abcd}\,S^{cd}} + +The {\tt Standard way} to find these quantities is given +by the following formulas. +\begin{equation} +\OO{r}_{ab} = \frac{1}{d(d-1)}R\,S_{ab} +\end{equation} +\begin{equation} +\OO{c}_{ab} = \frac{1}{(d-2)}\left[ +C_{am}\,\theta^m\!\wedge\theta_b +-C_{bm}\,\theta^m\!\wedge\theta_a\right],\quad +C_{ab}=\RR{A}_{(ab)}-\frac{1}{d}g_{ab}R +\end{equation} +\begin{equation} +\OO{a}_{ab} = \frac{1}{(d-2)}\left[ +A_{am}\,\theta^m\!\wedge\theta_b +-A_{bm}\,\theta^m\!\wedge\theta_a\right],\quad +A_{ab}=\RR{A}_{[ab]} +\end{equation} +\begin{equation} +\OO{d}_{ab} = \frac{1}{12}\partial_b\ipr\partial_a\ipr +(\OO{A}_{mn}\wedge\theta^m\!\wedge\theta^n) +\end{equation} +\begin{equation} +\OO{b}_{ab} =\frac{1}{2}\left[ +\partial_b\ipr(\theta^m\!\wedge\OO{A0}_{am}) +-\partial_a\ipr(\theta^m\!\wedge\OO{A0}_{bm}) +\right] +\end{equation} +where +\[ +\OO{A0}_{ab} = +\OO{A}_{ab} +-\OO{c}_{ab} +-\OO{r}_{ab} +-\OO{a}_{ab} +-\OO{d}_{ab} +\] +And finally +\begin{equation} +\OO{w}_{ab} = +\OO{A}_{ab} +-\OO{c}_{ab} +-\OO{r}_{ab} +-\OO{a}_{ab} +-\OO{b}_{ab} +-\OO{d}_{ab} +\end{equation} + +If $d=2$ then $\OO{A}_{ab}$ turns out to be irreducible and +coincides with the scalar curvature irreducible piece +\begin{equation} +\OO{A}_{ab} = \OO{r}_{ab} +\end{equation} + +Now we consider the decomposition of the S curvature part which +is nonzero iff nonmetricity is nonzero. First we consider +the case $d\geq3$. In this case we have 5 irreducible components +\begin{equation} +\RR{S}_{abcd} = +\RR{h}_{abcd}+ +\RR{sc}_{abcd}+ +\RR{sa}_{abcd}+ +\RR{v}_{abcd}+ +\RR{u}_{abcd} +\end{equation} + +The corresponding 2-forms are +\object{Homothetic Curvature 2-form OSH.a.b } +{\OO{h}_{ab} = \frac12 \RR{h}_{abcd}\,S^{cd}} +\object{Antisymmetric S-Ricci 2-form OSA.a.b } +{\OO{sa}_{ab} = \frac12 \RR{sa}_{abcd}\,S^{cd}} +\object{Traceless S-Ricci 2-form OSC.a.b } +{\OO{sc}_{ab} = \frac12 \RR{sc}_{abcd}\,S^{cd}} +\object{Antisymmetric S-Curvature 2-form OSV.a.b } +{\OO{v}_{ab} = \frac12 \RR{v}_{abcd}\,S^{cd}} +\object{Symmetric S-Curvature 2-form OSU.a.b } +{\OO{u}_{ab} = \frac12 \RR{u}_{abcd}\,S^{cd}} + + +The {\tt Standard way} to compute the decomposition is +\begin{equation} +\OO{h}_{ab}=-\frac{1}{(d^2-4)}\left[ +\theta_a\wedge\partial_b\ipr\OO{h}{} ++\theta_b\wedge\partial_a\ipr\OO{h}{} +-g_{ab}\OO{h}{}d\right] +\end{equation} +\begin{equation} +\OO{sa}_{ab} =\frac{d}{(d^2-4)}\left[ +\theta_a\wedge(\RR{S}_{[bm]}\wedge\theta^m) ++\theta_b\wedge(\RR{S}_{[am]}\wedge\theta^m) +-\frac{2}{d}g_{ab}\,\RR{S}_{cd}S^{cd}\right] +\end{equation} +\begin{equation} +\OO{sc}_{ab} =\frac{1}{d}\left[ +\theta_a\wedge(\RR{S}_{(bm)}\wedge\theta^m) ++\theta_b\wedge(\RR{S}_{(am)}\wedge\theta^m)\right] \label{ccc} +\end{equation} +\begin{equation} +\OO{v}_{ab} = \frac{1}{4}\left[ +\partial_a\ipr(\OO{S0}_{bm}\wedge\theta^m) ++\partial_b\ipr(\OO{S0}_{am}\wedge\theta^m)\right] +\end{equation} +where +\[ +\OO{S0}_{ab} = +\OO{S}_{ab} +-\OO{h}_{ab} +-\OO{sa}_{ab} +-\OO{sc}_{ab} +\] +And finally +\begin{equation} +\OO{u}_{ab} = +\OO{S}_{ab} +-\OO{h}_{ab} +-\OO{sa}_{ab} +-\OO{sc}_{ab} +-\OO{v}_{ab} +\end{equation} + +If $d=2$ then only the h- and sc-components are nonzero. +The $\OO{sc}_{ab}$ are given by (\ref{ccc}) and +\begin{equation} +\OO{h}_{ab} = \OO{S}_{ab}-\OO{sc}_{ab} +\end{equation} + +\begin{center} +\begin{tabular}{|c|c|c|} +\hline object & exists if & and has $n$ components \\ +\hline +\vv$R_{abcd}$ & & $\frac{d^3(d-1)}{2}$ \\[1mm] +\hline\vv$\rim{R}{}_{abcd}$ & & $\frac{d^2(d^2-1)}{12}$ \\[1mm] +\hline\vv$\RR{A}_{abcd}$ & & $\frac{d^2(d-1)^2}{4}$ \\[1mm] +\hline\vv$\RR{S}_{abcd}$ & & $\frac{d^2(d^2-1)}{4}$ \\[1mm] +\hline\vv$\RR{w}_{abcd}$ & $d\geq4$ & $\frac{d(d+1)(d+2)(d-3)}{12}$ \\ +\vv$\RR{c}_{abcd}$ & $d\geq3$ & $\frac{(d+2)(d-1)}{2}$ \\ +\vv$\RR{r}_{abcd}$ & & $1$ \\[1mm] +\hline\vv$\RR{a}_{abcd}$ & $d\geq3$ & $\frac{d(d-1)}{2}$ \\ +\vv$\RR{b}_{abcd}$ & $d\geq4$ & $\frac{d(d-1)(d+2)(d-3)}{8}$ \\ +\vv$\RR{d}_{abcd}$ & $d\geq4$ & $\frac{d(d-1)(d-2)(d-3)}{24}$ \\[1mm] +\hline\vv$\RR{h}_{abcd}$ & & $\frac{d(d-1)}{2}$ \\ +\vv$\RR{sa}_{abcd}$ & $d\geq3$ & $\frac{d(d-1)}{2}$ \\ +\vv$\RR{sc}_{abcd}$ & & $\frac{(d+2)(d-1)}{2}$ \\ +\vv$\RR{v}_{abcd}$ & $d\geq4$ & $\frac{d(d+2)(d-1)(d-3)}{8}$ \\ +\vv$\RR{u}_{abcd}$ & $d\geq3$ & $\frac{(d-2)(d+4)(d^2-1)}{8}$ \\[1mm] +\hline +\end{tabular} +\end{center} + + + +\section{Spinorial Curvature Decomposition} + +Spinorial curvature is defined in \grg\ iff nonmetricity +is zero and switch \comm{NONMETR} is turned off. +The upper sign in this section correspond to the signature +${\scriptstyle(-,+,+,+)}$ while lower one to the signature +${\scriptstyle(+,-,-,-)}$. + +Let us introduce the spinorial analog of the curvature tensor +\begin{eqnarray} +R_{abcd}&\tsst& +\ \ R_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}} ++R_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD} \nonumber\\[1mm] +&&+R_{AB\dot{C}\dot{D}}\epsilon_{\dot{A}\dot{B}}\epsilon_{CD} ++R_{\dot{A}\dot{B} CD}\epsilon_{AB}\epsilon_{\dot{C}\dot{D}}, \\[1.5mm] +R_{ABCD}&=&-i*(\Omega_{AB}\wedge S_{CD}),\ \ +R_{AB\dot{C}\dot{D}}\ =\ i*(\Omega_{AB}\wedge S_{\dot{C}\dot{D}})\\[1.5mm] +R_{\dot{A}\dot{B}\dot{C}\dot{D}}&=&\overline{R_{ABCD}},\ \ +R_{\dot{A}\dot{B} CD}\ =\ \overline{R_{AB\dot{C}\dot{D}}} +\end{eqnarray} + +The quantities $R_{ABCD}$ and $R_{AB\dot C\dot D}$ can be used to compute +the {\tt Curvature spinors} ($\equiv$ {\tt Curvature components}) +\object{Weyl Spinor RW.ABCD}{C_{ABCD}} +\object{Traceless Ricci Spinor RC.AB.CD\cc}{C_{AB\dot C\dot D}} +\object{Scalar Curvature RR}{R} +\object{Ricanti Spinor RA.AB}{A_{AB}} +\object{Traceless Deviation Spinor RB.AB.CD\cc}{B_{AB\dot C\dot D}} +\object{Scalar Deviation RD}{D} +All these spinors are irreducible (totally symmetric). + +Weyl spinor $C_{ABCD}$ {\tt From spinor curvature} is +\begin{eqnarray} +C_{abcd}&\tsst& C_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}} + +C_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD} \\[1mm] +C_{ABCD}&=&R_{(ABCD)} \label{RW} +\end{eqnarray} + +Traceless Ricci spinor $C_{AB\dot{A}\dot{B}}$ {\tt From spinor curvature} is +\begin{eqnarray} +C_{ab}&\tsst&C_{AB\dot{A}\dot{B}}\\[2mm] +C_{AB\dot{C}\dot{D}}&=&\pm(R_{AB\dot{C}\dot{D}}+R_{\dot{C}\dot{D} AB}) +\end{eqnarray} + +Scalar curvature {\tt From spinor curvature} is +\begin{equation} R=2(R^{MN}_{\ \ \ \ MN}+R^{\dot{M}\dot{N}}_{\ \ \ \ \dot{M}\dot{N}}) +\end{equation} + +Antisymmetric Ricci spinor $A_{AB}$ {\tt From spinor curvature} is +\begin{eqnarray} +A_{ab}&\tsst& A_{AB}\epsilon_{\dot{A}\dot{B}}+A_{\dot{A}\dot{B}}\epsilon_{AB}\\[1mm] +A_{AB}&=&\mp R^{\ \ \ \,M}_{(A|\ \ M|B)} +\end{eqnarray} + +Traceless deviation spinor $B_{AB\dot{A}\dot{B}}$ {\tt From spinor curvature} is +\begin{eqnarray} +B_{ab}&\tsst&B_{AB\dot{A}\dot{B}}\\[1mm] +B_{AB\dot{C}\dot{D}}&=&\pm i(R_{AB\dot{C}\dot{D}}-R_{\dot{C}\dot{D} AB}) +\end{eqnarray} + +Deviation trace {\tt From spinor curvature} is +\begin{equation} +D=-2i(R^{MN}_{\ \ \ \ MN}-R^{\dot{M}\dot{N}}_{\ \ \ \ \dot{M}\dot{N}}) +\end{equation} + +Note that spinors $C_{AB\dot{A}\dot{B}},B_{AB\dot{A}\dot{B}}$ are Hermitian +\begin{equation} +C_{AB\dot{C}\dot{D}}=\overline{C_{CD\dot{A}\dot{B}}},\ \ +B_{AB\dot{C}\dot{D}}=\overline{B_{CD\dot{A}\dot{B}}} +\end{equation} + +Finally we introduce the decomposition for the spinorial +curvature 2-form +\begin{equation} +\Omega_{AB}= +\OO{w}_{AB}+\OO{c}_{AB}+\OO{r}_{AB} ++\OO{a}_{AB}+\OO{b}_{AB}+\OO{c}_{AB} +\end{equation} +where the {\tt Undotted curvature 2-forms} +\object{Undotted Weyl 2-form OMWU.AB }{\OO{w}_{AB}} +\object{Undotted Traceless Ricci 2-form OMCU.AB }{\OO{c}_{AB}} +\object{Undotted Scalar Curvature 2-form OMRU.AB }{\OO{r}_{AB}} +\object{Undotted Ricanti 2-form OMAU.AB }{\OO{a}_{AB}} +\object{Undotted Traceless Deviation 2-form OMBU.AB }{\OO{b}_{AB}} +\object{Undotted Scalar Deviation 2-form OMDU.AB }{\OO{d}_{AB}} +are given by +\begin{eqnarray} +\OO{w}_{AB}&=&C_{ABCD}S^{CD} \\[1mm] +\OO{c}_{AB}&=&\pm\frac12 C_{AB\dot{C}\dot{D}}S^{\dot{C}\dot{D}} \\[1mm] +\OO{r}_{AB}&=&\frac1{12}S_{AB}R \\[1mm] +\OO{a}_{AB}&=&\pm A_{(A}^{\ \ \ M}S_{M|B)} \\[1mm] +\OO{b}_{AB}&=&\mp\frac{i}2 B_{AB\dot{C}\dot{D}}S^{\dot{C}\dot{D}} \\[1mm] +\OO{d}_{AB}&=&\frac{i}{12}S_{AB}D +\end{eqnarray} + + + + + + + +\section{Torsion Decomposition} + +The torsion tensor +\begin{equation} +Q_{abc}=Q_{a[bc]},\qquad +\Theta^a=\frac{1}{2}Q^a{}_{bc}\,S^{bc} +\end{equation} +consists of three irreducible pieces +\begin{equation} +Q_{abc} = +\stackrel{\rm c}{Q}_{abc} ++\stackrel{\rm t}{Q}_{abc} ++\stackrel{\rm a}{Q}_{abc} +\end{equation} + +\begin{center} +\begin{tabular}{|c|c|c|} +\hline object & exists if & and has $n$ components \\ +\hline +\vv$Q_{abc}$ & & $\frac{d^2(d-1)}{2}$ \\[1mm] +\hline\vv$\stackrel{\rm c}{Q}_{abc}$ & $d\geq3$ & $\frac{d(d^2-4)}{3}$ \\ +\vv$\stackrel{\rm t}{Q}_{abc}$ & & $d$ \\ +\vv$\stackrel{\rm a}{Q}_{abc}$ & $d\geq3$ & $\frac{d(d-1)(d-2)}{6}$ \\[1mm] +\hline +\end{tabular} +\end{center} + +The corresponding union of three objects {\tt Torsion 2-forms} is +\object{Traceless Torsion 2-form THQC'a} +{\stackrel{\rm c}{\Theta}\!{}^a=\frac{1}{2} + \stackrel{\rm c}{Q}\!{}^a{}_{bc}\,S^{bc}} +\object{Torsion Trace 2-form THQT'a} +{\stackrel{\rm t}{\Theta}\!{}^a=\frac{1}{2} + \stackrel{\rm t}{Q}\!{}^a{}_{bc}\,S^{bc}} +\object{Antisymmetric Torsion 2-form THQA'a} +{\stackrel{\rm a}{\Theta}\!{}^a=\frac{1}{2} + \stackrel{\rm a}{Q}\!{}^a{}_{bc}\,S^{bc}} + +And the auxiliary quantities +\object{Torsion Trace QT'a}{Q^a} +\object{Torsion Trace 1-form QQ}{Q=-\partial_a\ipr\Theta^a} +\object{Antisymmetric Torsion 3-form QQA}{\stackrel{\rm a}{Q}=\theta_a\wedge\Theta^a} + +The torsion trace $Q^a=Q^m{}_{am}$ can be obtained {\tt From torsion +trace 1-form} +\begin{equation} +Q^a = \partial^a\ipr Q +\end{equation} + +The {\tt Standard way} for the irreducible torsion 2-forms is +\begin{equation} +\stackrel{\rm t}{\Theta}\!{}^a = -\frac{1}{(d-1)}\theta^a\wedge Q +\end{equation} +\begin{equation} +\stackrel{\rm t}{\Theta}\!{}^a = \frac{1}{3}\partial^a\ipr\stackrel{\rm a}{Q} +\end{equation} +\begin{equation} +\stackrel{\rm c}{\Theta}\!{}^a = \Theta^a +-\stackrel{\rm t}{\Theta}\!{}^a +-\stackrel{\rm a}{\Theta}\!{}^a +\end{equation} + +The rest of this section is valid in dimension 4 only. + +In this case one can introduce the torsion pseudo trace +\object{Torsion Pseudo Trace QP'a}{ +P^a = \stackrel{*}{Q}\!{}^{ma}{}_{m}, +\ \stackrel{*}{Q}\!{}^a{}_{bc} = \frac{1}{2}{\cal E}_{bc}{}^{pq} +Q^a{}_{pq}} +which can be computed {\tt From antisymmetric torsion 3-form} +\begin{equation} +P^a = \partial^a\ipr\,*\!\stackrel{\rm a}{Q} +\end{equation} + +Finally let us consider the spinorial representation of the +torsion. +Below the upper sign corresponds to the +\seethis{See \pref{spinors}\ or \ref{spinors1}.} +signature ${\scriptstyle(-,+,+,+)}$ and lower one to the +signature ${\scriptstyle(+,-,-,-)}$. + +First we introduce the spinorial analog of the torsion tensor +\begin{equation} +Q_{abc}\tsst Q_{A\dot{A} BC}\epsilon_{\dot{B}\dot{C}} ++Q_{A\dot{A}\dot{B}\dot{C}}\epsilon_{BC} +\end{equation} +where +\begin{equation} +Q_{A\dot{A} BC}=-i*(\Theta_{A\dot{A}}\wedge S_{BC}),\qquad +Q_{A\dot{A}\dot{B}\dot{C}}=i*(\Theta_{A\dot{A}}\wedge S_{\dot{B}\dot{C}}) +\end{equation} +These spinors are reducible but the +\object{Traceless Torsion Spinor QC.ABC.D\cc}{C_{ABC\dot D}} +\[ +\stackrel{\rm c}{Q}_{abc}\tsst C_{ABC\dot A}\epsilon_{\dot{B}\dot{C}} ++Q_{\dot{A}\dot{B}\dot{C}A}\epsilon_{BC},\quad +C_{\dot{A}\dot{B}\dot{C} A}=\overline{C_{ABC\dot{A}}} +\] +is irreducible (symmetric in $\scriptstyle ABC$). And it can be +computed {\tt From torsion} by the relation +\begin{equation} +C_{ABC\dot A} = Q_{(A|\dot{A}|BC)} +\end{equation} + +The torsion trace can be calculated {\tt From torsion using spinors} +\begin{equation} +Q^a\tsst Q^{A\dot{A}},\quad +Q_{A\dot{B}}=\mp(Q^M{}_{\dot{B}MA}+Q_A{}^{\dot M}{}_{\dot M\dot{B}}) +\end{equation} + +And similarly the torsion pseudo-trace can be found +{\tt From torsion using spinors} +\begin{equation} +P^a\tsst P^{A\dot{A}},\quad +P_{A\dot{B}}=\mp i(Q^M{}_{\dot{B}MA}-Q_A{}^{\dot M}{}_{\dot M\dot{B}}) +\end{equation} + +Finally we introduce the {\tt Undotted trace 2-forms} +which are selfdual parts of the irreducible torsion 2-forms +\object{Undotted Traceless Torsion 2-form THQCU'a} +{\stackrel{\rm c}{\vartheta}\!{}^a} +\object{Undotted Torsion Trace 2-form THQTU'a} +{\stackrel{\rm t}{\vartheta}\!{}^a} +\object{Undotted Antisymmetric Torsion 2-form THQAU'a} +{\stackrel{\rm a}{\vartheta}\!{}^a} \seethis{See \pref{thetau}.} +These quantities will be used in the gravitational equations. + +This complex 2-forms can be obtained by the equations +({\tt Standard way}): +\begin{eqnarray} +\stackrel{\rm c}{\vartheta}\!{}^a &\tsst& \stackrel{\rm c}{\vartheta}\!{}^{A\dot A} +=C^A_{\ \ BC}{}^{\dot{A}}S^{BC}\\[1mm] +\stackrel{\rm t}{\vartheta}\!{}^a &\tsst& \stackrel{\rm t}{\vartheta}\!{}^{A\dot A} +=\mp\frac13 Q_{M}^{\ \ \ \dot{A}}S^{AM}\\[1mm] +\stackrel{\rm a}{\vartheta}\!{}^a &\tsst& \stackrel{\rm a}{\vartheta}\!{}^{A\dot A} +=\pm\frac{i}3 P_{M}^{\ \ \ \dot{A}}S^{AM} +\end{eqnarray} + + + +\section{Nonmetricity Decomposition} + +In general the nonmetricity tensor +\begin{equation} +N_{abc}=N_{(ab)c},\qquad N_{ab}=N_{abc}\theta^c +\end{equation} +consist of 4 irreducible pieces +\begin{equation} +N_{abcd} = +\stackrel{\rm c}{N}_{abc} ++\stackrel{\rm a}{N}_{abc} ++\stackrel{\rm t}{N}_{abc} ++\stackrel{\rm w}{N}_{abc} +\end{equation} + +\begin{center} +\begin{tabular}{|c|c|c|} +\hline object & exists if & and has $n$ components \\ +\hline +\vv$N_{abc}$ & & $\frac{d^2(d+1)}{2}$ \\[1mm] +\hline\vv$\stackrel{\rm c}{N}_{abc}$ & & $\frac{d(d-1)(d+4)}{6}$ \\ +\vv$\stackrel{\rm a}{N}_{abc}$ & $d\geq3$ & $\frac{d(d^2-4)}{3}$ \\ +\vv$\stackrel{\rm t}{N}_{abc}$ & & $d$ \\ +\vv$\stackrel{\rm w}{N}_{abc}$ & & $d$ \\[1mm] +\hline +\end{tabular} +\end{center} + +The corresponding union of objects {\tt Nonmetricity 1-forms} +consist of +\object{Symmetric Nonmetricity 1-form NC.a.b} +{\stackrel{\rm c}{N}_{ab}=\stackrel{\rm c}{N}_{abc}\theta^c} +\object{Antisymmetric Nonmetricity 1-form NA.a.b} +{\stackrel{\rm a}{N}_{ab}=\stackrel{\rm a}{N}_{abc}\theta^c} +\object{Nonmetricity Trace 1-form NT.a.b} +{\stackrel{\rm t}{N}_{ab}=\stackrel{\rm t}{N}_{abc}\theta^c} +\object{Weyl Nonmetricity 1-form NW.a.b} +{\stackrel{\rm w}{N}_{ab}=\stackrel{\rm w}{N}_{abc}\theta^c} + +We have also two auxiliary 1-forms +\object{Weyl Vector NNW}{\stackrel{\rm w}{N}} +\object{Nonmetricity Trace NNT}{\stackrel{\rm t}{N}} + +They are computed according to the following formulas +\begin{equation} +\stackrel{\rm w}{N} = N^a{}_a +\end{equation} +\begin{equation} +\stackrel{\rm t}{N} = \theta^a\,\partial^b\ipr N_{ab} +- \frac{1}{d} \stackrel{\rm w}{N} +\end{equation} +\begin{equation} +\stackrel{\rm w}{N}_{ab} = \frac{1}{d}g_{ab}\stackrel{\rm w}{N} +\end{equation} +\begin{equation} +\stackrel{\rm t}{N}_{ab}=\frac{d}{(d-1)(d+2)}\left[ +\theta_b\partial_a\ipr\stackrel{\rm t}{N} ++\theta_a\partial_b\ipr\stackrel{\rm t}{N} +-\frac{2}{d} g_{ab} \stackrel{\rm t}{N}\right] +\end{equation} +\begin{equation} +\stackrel{\rm a}{N}_{ab}=\frac{1}{3}\left[ +\partial_a\ipr(\theta^m\wedge\stackrel{0}{N}_{bm}) ++\partial_b\ipr(\theta^m\wedge\stackrel{0}{N}_{am})\right] +\end{equation} +where +\[ +\stackrel{\rm 0}{N}_{ab}= +N_{abc} +-\stackrel{\rm t}{N}_{abc} +-\stackrel{\rm w}{N}_{abc} +\] +And finally +\begin{equation} +\stackrel{\rm c}{N}_{ab}= +N_{abc} +-\stackrel{\rm a}{N}_{abc} +-\stackrel{\rm t}{N}_{abc} +-\stackrel{\rm w}{N}_{abc} +\end{equation} + +\section{Newman-Penrose Formalism} + +The method of spinorial differential forms described in the +previous sections are essentially equivalent to the well +known Newman-Penrose formalism but for the sake of convenience +\grg\ has complete set of macro objects which allows to +write the Newman-Penrose equations in +traditional notation. All these objects refer (up to some sign +and 1/2 factors) to other \grg\ built-in objects. + +In this section upper sign corresponds to the +signature ${\scriptstyle(-,+,+,+)}$ and lower one to the +signature ${\scriptstyle(+,-,-,-)}$. +\seethis{See \pref{spinors}.} +The frame must be null as explained in section \ref{spinors}. + +For the Newman-Penrose formalism we use notation and conventions +of the book \emph{Exact Solutions of the Einstein Field Equations} +by D. Kramer, H. Stephani, M. MacCallum and E. Herlt, ed. +E. Schmutzer (Berlin, 1980). We denote this book as ESEFE. + +We chose the relationships between NP null tetrad and \grg\ null +frame as follows +\begin{equation} +l^\mu=h^\mu_0,\quad +k^\mu=h^\mu_1,\quad +\overline{m}\!{}^\mu=h^\mu_2,\quad +m^\mu=h^\mu_3 +\end{equation} + +The NP vector operators are just the components of the +vector frame $\partial_a$ +\begin{eqnarray} +\mbox{\tt DD}&=& D =\partial_1 \\ +\mbox{\tt DT}&=& \Delta=\partial_0 \\ +\mbox{\tt du}&=& \delta=\partial_3 \\ +\mbox{\tt dd}&=& \overline\delta=\partial_2 +\end{eqnarray} + +The spin coefficient are the components of the connection +1-form +\object{SPCOEF.AB.c}{ \omega_{AB\,c}=\partial_c\ipr\omega_{AB}} +or in the NP notation +\begin{eqnarray} +\mbox{\tt alphanp }&=& \alpha =\pm\omega_{(1)2} \\ +\mbox{\tt betanp }&=& \beta =\pm\omega_{(1)3} \\ +\mbox{\tt gammanp }&=& \gamma =\pm\omega_{(1)0} \\ +\mbox{\tt epsilonnp }&=& \epsilon =\pm\omega_{(1)1} \\ +\mbox{\tt kappanp }&=& \kappa =\pm\omega_{(0)1} \\ +\mbox{\tt rhonp }&=& \rho =\pm\omega_{(0)2} \\ +\mbox{\tt sigmanp }&=& \sigma =\pm\omega_{(0)3} \\ +\mbox{\tt taunp }&=& \tau =\pm\omega_{(0)0} \\ +\mbox{\tt munp }&=& \mu =\pm\omega_{(2)3} \\ +\mbox{\tt nunp }&=& \nu =\pm\omega_{(2)0} \\ +\mbox{\tt lambdanp }&=& \lambda =\pm\omega_{(2)2} \\ +\mbox{\tt pinp }&=& \pi =\pm\omega_{(2)1} \\ +\end{eqnarray} +where the first index of the +quantity $\omega_{(AB)c}$ is included inn parentheses to remind +that it is summed spinorial index. + +Finally for the curvature we have +\object{PHINP.AB.CD\cc }{ +\Phi_{AB\dot{C}\dot{D}} = \pm\frac{1}{2}C_{AB\dot C\dot D} } +\object{PSINP.ABCD }{\Psi_{ABCD}=C_{ABCD}} +the conventions for the scalar curvature $R$ in ESEFE and +in \grg\ are the same. + +For the signature ${\scriptstyle(-,+,+,+)}$ the Newman-Penrose equations for +the quantities introduced above can be found in section 7.1 of ESEFE. +For other signature ${\scriptstyle(+,-,-,-)}$ one must alter the sign of +$\Psi_{ABCD}$, $\Phi_{AB\dot{C}\dot{D}}$ and $R$ in Eqs. (7.28)--(7.45). + +\section{Electromagnetic Field} + +Formulas in this section are valid only in spaces +with the signature ${\scriptstyle(-,+,\dots,+)}$ and +${\scriptstyle(+,-,\dots,-)}$. +The sign factor $\sigma$ in the expressions below is +$\sigma=-{\rm diag}_0$ ($+1$ for the first signature and $-1$ +for the second). + +Let us introduce the +\object{EM Potential A}{A=A_\mu dx^\mu} +and the +\object{Current 1-form J}{J=j_\mu dx^\mu} + +The EM strength tensor +$F_{\alpha\beta}=\partial_\alpha A_\beta-\partial_\beta A_\alpha$ +\object{EM Tensor FT.a.b}{F_{ab}= +\partial_b\ipr\partial_a\ipr F} +where $F$ is the +\object{EM 2-form FF}{F} +which can be found {\tt From EM potential} +\begin{equation} +F=dA +\end{equation} +or {\tt From EM tensor} +\begin{equation} +F = \frac{1}{2}F_{ab}\,S^{ab} +\end{equation} + +The EM action $d$-form +\object{EM Action EMACT}{L_{\rm EM}= +-\frac{1}{8\pi}\,F\wedge *F} + +The {\tt Maxwell Equations} +\object{First Maxwell Equation MWFq}{d*F=-4\pi\sigma\,(-1)^{d}\,*J} +\object{Second Maxwell Equation MWSq}{dF=0} + +The current must satisfy the +\object{Continuity Equation COq}{d*J=0} + +The +\object{EM Energy-Momentum Tensor TEM.a.b}{T_{ab}^{\rm EM}} +is given by the equation +\begin{equation} +T^{\rm EM}_{ab} = \frac{\sigma}{4\pi} +F_{am}F_b{}^m +s\sigma\,g_{ab}\,*L_{\rm EM} +\end{equation} + +The rest of the section is valid in the dimension 4 only. + +In 4 dimensions the tensor $F_{ab}$ and its dual +$\stackrel{*}{F}_{ab}=\frac{1}{2}{\cal E}_{ab}{}^{mn}F_{mn}$ +are expressed via usual 3-dimensional vectors $\vec E$ and +$\vec H$ +\begin{eqnarray} +F_{ab}&=&-\sigma\left(\begin{array}{rrr} +E_1&E_2&E_3\\ +&-H_3&H_2\\ +&&-H_1\end{array}\right)\\[1.5mm] +\stackrel{*}{F}_{ab}&=&\sigma\left(\begin{array}{rrr} +H_1&H_2&H_3\\ +&E_3&-E_2\\ +&&E_1\end{array}\right) +\end{eqnarray} +Similarly for the current we have +\begin{equation} +J=\sigma(-\rho dt + \vec j\,d\vec x) +\end{equation} + +The {\tt EM scalars} +\object{First EM Scalar SCF}{I_1=\frac12F_{ab}F^{ab} +={\vec H}^2-{\vec E}^2} +\object{Second EM Scalar SCS}{I_2=\frac12\stackrel{*}{F}_{ab}F^{ab} +=2\vec E\cdot\vec H} +can be obtained as follows by {\tt Standard way} +\begin{equation} +I_1 = -*(F\wedge*F) +\end{equation} +\begin{equation} +I_2 = *(F\wedge F) +\end{equation} + +The +\object{Complex EM 2-form FFU}{\Phi} +can be found {\tt From EM 2-form} +\begin{equation} +\Phi=F-i*F +\end{equation} +or {\tt From EM Spinor} +\begin{equation} +\Phi = 2\Phi_{AB}\,S^{AB} +\end{equation} + +The 2-form $\Phi$ must obey the +\object{Selfduality Equation SDq.AB\cc}{\Phi\wedge S_{\dot A\dot B}} +and gives rise to the +\object{Complex Maxwell Equation MWUq}{d\Phi=-4i\sigma\pi\,*J} + +The EM 2-form $F$ can be restored {\tt From Complex EM 2-form} +\begin{equation} +F=\frac{1}{2}(\Phi+\overline\Phi) +\end{equation} + +The symmetric +\object{Undotted EM Spinor FIU.AB}{\Phi_{AB}} +is the spinorial analog of the tensor $F_{ab}$ +\begin{equation} + F_{ab} \tsst \epsilon_{AB} \Phi_{\dot A\dot B} ++ \epsilon_{\dot A\dot B} \Phi_{AB} +\end{equation} +It can be obtained either {\tt From complex EM 2-form} +\begin{equation} +\Phi_{AB} = -\frac{i}{2}*(\Phi\wedge S_{AB}) +\end{equation} +of {\tt From EM 2-form} +\begin{equation} +\Phi_{AB} = -i*(F\wedge S_{AB}) +\end{equation} + +The +\object{Complex EM Scalar SCU}{\iota=I_1-iI_2} +can be found {\tt From EM Spinor} +\begin{equation} +\iota = 2\Phi_{AB}\Phi^{AB} +\end{equation} +or {\tt From Complex EM 2-form} +\begin{equation} +\iota = -\frac{i}{2} *(\Phi\wedge\Phi) +\end{equation} + +Finally we have the +\object{EM Energy-Momentum Spinor TEMS.AB.CD\cc} +{T^{\rm EM}_{AB\dot A\dot B}=\frac{1}{2\pi}\Phi_{AB}\Phi_{\dot A\dot B}} + + +\section{Dirac Field} + +In this section upper sign corresponds to the +signature ${\scriptstyle(-,+,+,+)}$ and lower one to the +signature ${\scriptstyle(+,-,-,-)}$. + +The four component Dirac spinor consists of two 1-index spinors +\begin{equation} +\psi=\left(\begin{array}{c}\phi^A\\ \chi_{\dot A}\end{array}\right),\ \ +\overline\psi=\left(\chi_A\ \ \phi^{\dot A}\right) +\end{equation} +Thus we have the {\tt Dirac spinor} as the union of two objects +\object{Phi Spinor PHI.A}{\phi_A} +\object{Chi Spinor CHI.B}{\chi_B} + +The gamma-matrices are expressed via sigma-matrices as follows +\begin{equation} +\gamma^m=\sqrt2\left(\begin{array}{cc} +0&\sigma^{mA\dot B}\\ \sigma^m\!{}_{B\dot A}&0\end{array}\right) +\end{equation} + +Dirac field action 4-form +\begin{eqnarray} +&&\mbox{\tt Dirac Action 4-form DACT}=L_{\rm D}=\nonumber\\[1mm] +&&\quad=\left[\frac{i}2(\overline\psi\gamma^a +(\nabla_a+ieA_a)\psi-(\nabla_a-ieA_a)\overline\psi\gamma^a\psi) +-m_{\rm D}\overline\psi\psi\right]\upsilon +\end{eqnarray} + +The {\tt Standard way} to compute this quantity is +\begin{eqnarray} +L_{\rm D} &=& -\frac{i}{\sqrt2}\left[ +\phi_{\dot A}\theta^{A\dot A}\!\wedge*(D+ieA)\phi_A-{\rm c.c.} +-\chi_{\dot A} \theta^{A\dot A}\!\wedge*(D-ieA)\chi_A -{\rm c.c.}\right]- +\nonumber\\[1mm]&&\qquad\qquad\quad +-m_{\rm D}\left(\phi^A\chi_A+{\rm c.c.}\right)\upsilon +\end{eqnarray} + +The {\tt Dirac equation} is +\object{Phi Dirac Equation DPq.A\cc}{ +i\sqrt2\partial_{B\dot A}\ipr(D+ieA-\frac12Q)\phi^B-m_{\rm D}\chi_{\dot A}=0} +\object{Chi Dirac Equation DCq.A\cc}{ +i\sqrt2\partial_{B\dot A}\ipr(D-ieA-\frac12Q)\chi^B-m_{\rm D}\phi_{\dot A}=0} +where $Q$ is the torsion trace 1-form. Notice that terms with the +electromagnetic field $eA$ are included in equations iff +the value of $A$ is defined. The unit charge $e$ is given by the +constant \comm{ECONST}. + +The current 1-form can be computed {\tt From Dirac Spinor} +\begin{equation} +J=\mp\sqrt2e(\phi_A\phi_{\dot A}+\chi_A\chi_{\dot A})\theta^{A\dot A} +\end{equation} + +The symmetrized +\object{Dirac Energy-Momentum Tensor TDI.a.b}{T^{\rm D}_{ab}} +can be obtained as follows +\begin{eqnarray} +T^{\rm D}_{ab}&=& +*(\theta_{(a}\wedge T^{\rm D}_{b)})\nonumber\\[1mm] +T^{\rm D}_a&=&\mp\frac{i}{\sqrt2}\Big[ +*\theta^{A\dot A}\partial_a\ipr(D+ieA)\phi_A\phi_{\dot A} +-{\rm c.c.}\nonumber\\ +&&\qquad-*\theta^{A\dot A}\partial_a\ipr(D-ieA)\chi_A\chi_{\dot A} +-{\rm c.c.}\Big] +\pm\partial_a\ipr L_{\rm D} +\end{eqnarray} + +The +\object{Undotted Dirac Spin 3-Form SPDIU.AB}{s^{\rm D}_{AB}} +\begin{equation} +s^{\rm D}_{AB}=\frac{i}{2\sqrt2} +\left(*\theta_{(A|\dot A}\phi_{B)}\phi^{\dot A} +-*\theta_{(A|\dot A}\chi_{B)}\chi^{\dot A}\right) +\end{equation} + +The Dirac field mass $m_{\rm D}$ is given by the constant +\comm{DMASS}. + + +\section{Scalar Field} + +Formulas in this section are valid in any dimension +with the signature ${\scriptstyle(-,+,\dots,+)}$ and +${\scriptstyle(+,-,\dots,-)}$. +The sign factor $\sigma$ is $\sigma=-{\rm diag}_0$ +($+1$ for the first signature and $-1$ for the second). + +The scalar field +\object{Scalar Field FI}{\phi} + +The minimal scalar field action $d$-form +\object{Minimal Scalar Action SACTMIN}{ +L_{\rm Smin}= +-\frac{1}{2}\left[\sigma(\partial_\alpha\phi)^2+ +m_{\rm s}^2 \phi^2\right]\upsilon} + +The nonminimal scalar field action +\object{Scalar Action SACT}{ +L_{\rm S}= +-\frac{1}{2}\left[\sigma(\partial_\alpha\phi)^2+ +(m_{\rm s}^2+a_0R) \phi^2\right]\upsilon} + +The scalar field equation +\object{Scalar Equation SCq} +{s\sigma(-1)^d*d*d\phi-(m_{\rm s}^2+a_0R)\phi=0} +which gives +\[ +-\sigma\rim{\nabla}{}^\pi\rim{\nabla}_\pi\phi-(m_{\rm s}^2+a_0R)\phi=0 +\] + +The minimal energy-momentum tensor is +\begin{eqnarray} +&&\mbox{\tt Minimal Scalar Energy-Momentum Tensor TSCLMIN.a.b} +=T^{\rm Smin}_{ab}= \nonumber\\ +&&\qquad\qquad=\partial_a\phi\partial_b\phi+s\sigma\,g_{ab} +*L_{\rm Smin} +\end{eqnarray} +The nonminimal part of the scalar field energy-momentum +\seethis{See pages \pageref{graveq}\ and \pageref{metreq}.} +tensor can be taken into account in the left-hand side +of gravitational equations. + +The scalar field mass $m_{\rm s}$ are given by the +constant {\tt SMASS}. The nonminimal interaction +terms are included iff the switch \comm{NONMIN} \swind{NONMIN} +is turned on and the value of nonminimal interaction constant +$a_0$ is determined by the object +\object{A-Constants ACONST.i2}{a_i} +The default value of $a_0$ is the constant \comm{AC0}. + +\section{Yang-Mills Field} + +Formulas in this section are valid in any dimension +with the signature ${\scriptstyle(-,+,\dots,+)}$ and +${\scriptstyle(+,-,\dots,-)}$. +The sign factor $\sigma$ in the expressions below is +$\sigma=-{\rm diag}_0$ ($+1$ for the first signature and $-1$ +for the second). The indices $\scriptstyle i,j,k,l,m,n$ +are the internal space Yang-Mills indices and we a +assume that the internal Yang-Mills metric is $\delta_{ij}$. + +The Yang-Mills potential 1-form +\object{YM Potential AYM.i9}{A^i=A^i_\mu dx^\mu} + +The structural constants +\object{Structural Constants SCONST.i9.j9.k9}{c^i{}_{jk}=c^i{}_{[jk]}} + +The Yang-Mills strength 2-form +\object{YM 2-form FFYM.i9}{F^i} +and strength tensor +\object{YM Tensor FTYM.i9.a.b}{F^i{}_{ab}} + +The $F^i$ can be computed {\tt From YM potential} +\begin{equation} +F^i = dA^i + \frac12 c^i{}_{jk} \, A^j\wedge A^k +\end{equation} +or {\tt From YM tensor} +\begin{equation} +F^i = \frac12 F^i{}_{ab}\, S^{ab} +\end{equation} + +The {\tt Standard way} to find Yang-Mills strength tensor is +\begin{equation} +F^i{}_{ab}=\partial_b\ipr\partial_a\ipr F^i +\end{equation} + +The Yang-Mills action $d$-form +\object{YM Action YMACT}{L_{\rm YM}= +-\frac{1}{8\pi}F^i\wedge*F_i} + +The {\tt YM Equations} +\object{First YM Equation YMFq.i9}{d*F^i + c^i{}_{jk} \, A^j\wedge *F^k=0} +\object{Second YM Equation YMSq.i9}{dF^i + c^i{}_{jk} \, A^j\wedge F^k=0} + +The energy-momentum tensor +\object{YM Energy-Momentum Tensor TYM.a.b} +{\frac{\sigma}{4\pi}F^i{}_{am}F^i{}_b{}^m + s\sigma\,g_{ab}\, +*L_{\rm YM}} + + +\section{Geodesics} + +The geodesic equation +\object{Geodesic Equation GEOq\^m}{ +\frac{d^2x^\mu}{dt^2}+\{^\mu_{\pi\tau}\} +\frac{dx^\pi}{dt}\frac{dx^\tau}{dt}=0} +Here the parameter $t$ must be declared by the +\seethis{See page \pageref{affpar}.} +\cmdind{Affine Parameter} +{\tt Affine parameter} declaration. + +\section{Null Congruence and Optical Scalars} + +Let us consider the congruence defined by the vector field +$k^\alpha$ +\object{Congruence KV}{k=k^\mu\partial_\mu} + +This congruence is null iff +\object{Null Congruence Condition NCo}{k\cdot k=0} +holds. + +The congruence is geodesic iff the condition +\object{Geodesics Congruence Condition GCo'a}{k^\mu\rim{\nabla}_\mu k^a=0} +is fulfilled. + +For the null geodesic congruence one can calculate the +{\tt Optical scalars} +\object{Congruence Expansion thetaO}{\theta= +\frac{1}{2}\rim{\nabla}{}^\pi k_\pi} +\object{Congruence Squared Rotation omegaSQO}{\omega^2= +\frac{1}{2}(\rim{\nabla}_{[\alpha}k_{\beta]})^2} +\object{Congruence Squared Shear sigmaSQO}{\sigma\overline\sigma= +\frac{1}{2}\left[ (\rim{\nabla}_{(\alpha}k_{\beta)})^2 +-2\theta^2\right]} + +\section{Timelike Congruences and Kinematics} + +Let us consider the congruence determined by the velocity +vector $u^\alpha$ +\object{Velocity UU'a}{u^a} +\object{Velocity Vector UV}{u=u^a\partial_a} + +The velocity vector must be normalized and the quantity +\object{Velocity Square USQ}{u^2=u\cdot u} +must be constant but nonzero. + +If the frame metric coincides with its default +diagonal value \seethis{See \pref{defaultmetric}.} +$g_{ab}={\rm diag}(-1,\dots)$ +then {\tt By default} we have for the velocity +\begin{equation} +u^a=(1,0,\dots,0) +\end{equation} +which means that the congruence is comoving in the given frame. + +In general case the velocity can be obtained +{\tt From velocity vector} +\begin{equation} +u^a=u\ipr \theta^a +\end{equation} + +We introduce the auxiliary object +\object{Projector PR'a.b}{P^a{}_b= +\delta^a_b-\frac{1}{u^2}u^an_b} + +The following four quantities called {\tt Kinematics} +comprise the complete set of the congruence characteristics +\object{Acceleration accU'a}{A^a=\rim{\nabla}_uu^a} +\object{Vorticity omegaU.a.b}{\omega_{ab}= +P^m{}_aP^n{}_b \rim{\nabla}_{[m}u_{n]}} +\object{Volume Expansion thetaU}{\Theta=\rim{\nabla}_au^a} +\object{Shear sigmaU.a.b}{ +P^m{}_aP^n{}_b \rim{\nabla}_{(m}u_{n)}- +\frac{1}{(d-1)}P_{ab}\Theta} + + +\section{Ideal And Spin Fluid} + + +The ideal fluid is characterized by the +\object{Pressure PRES}{p} +and +\object{Energy Density ENER}{\varepsilon} + +The ideal fluid energy-momentum tensor is +\begin{eqnarray} +&&\mbox{\tt Ideal Fluid Energy-Momentum Tensor TIFL.a.b}= +T^{\rm IF}_{ab} = \nonumber\\ +&&\qquad\qquad=(\varepsilon+p)u_a u_b - u^2p g_{ab} +\end{eqnarray} + +The rest of the section requires the nonmetricity be zero +(\comm{NONMETR} is off). + +In addition spin-fluid is characterized by +\object{Spin Density SPFLT.a.b }{S^{\rm SF}_{ab}=S^{\rm SF}_{[ab]}} +or equivalently by +\object{Spin Density 2-form SPFL }{S^{\rm SF}} + +The spin 2-form can be obtained {\tt From spin density} +\begin{equation} +S^{\rm SF}=\frac{1}{2}S^{\rm SF}_{ab} \theta^a\wedge\theta^a +\end{equation} +and $s_{ab}$ is determined {\tt From spin density 2-form} +\begin{equation} +S^{\rm SF}_{ab}= \partial_b\ipr\partial_a\ipr S^{\rm SF} +\end{equation} + +The spin density must satisfy the Frenkel condition +\object{Frenkel Condition FCo}{u\ipr S^{\rm SF}=0} + +The spin fluid energy-momentum tensor is +\begin{eqnarray} +&&\mbox{\tt Spin Fluid Energy-Momentum Tensor TSFL.a.b}=T^{\rm SF}_{ab}= +\nonumber\\ +&&\qquad\qquad=(\varepsilon+p)u_a u_b - u^2p g_{ab}+\Delta_{(ab)} +\end{eqnarray} +where +\begin{equation} +\Delta_{ab}=-2(g^{cd}+u^{-2}\,u^cu^d) \nabla_c S^{\rm SF}_{(ab)d} +\end{equation} +\begin{equation} +s^{\rm SF}_{abc}=u_a\,S^{\rm SF}_{bc} +\end{equation} +if torsion is zero (\comm{TORSION} off) and +\begin{equation} +\Delta_{ab}=2u^{-2}\,u_au^d\,\nabla_u S^{\rm SF}_{bd} +\end{equation} +if torsion is nonzero (\comm{TORSION} on). + +Notice that the energy-momentum \seethis{See \pref{tsym}.} +tensor $T^{\rm SF}_{ab}$ is symmetrized. + +Finally yet another representation for the spin +is the undotted spin 3-form +\object{Undotted Fluid Spin 3-form SPFLU.AB }{s^{\rm SF}_{AB}} +which is given by the standard spinor $\tsst$ tensor correspondence rules +\begin{equation} + s^{\rm SF}_{mab}\,*\theta^m \tsst \epsilon_{AB} s^{\rm SF}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}s^{\rm SF}_{AB} +\end{equation} +according to Eq. (\ref{asys}). \seethis{See \pref{asys}.} +This quantity is used in the right-hand side of gravitational equations. + +\section{Total Energy-Momentum And Spin} +\label{totalc} + +\enlargethispage{4mm} + + +The total energy-momentum tensor +\object{Total Energy-Momentum Tensor TENMOM.a.b}{T_{ab}} +and the total undotted spin 3-form \seethis{See pages \pageref{graveq}\ and \pageref{metreq}.} +\object{Total Undotted Spin 3-form SPINU.AB}{s_{AB}} +play the role of sources in the right-hand side of the +gravitational equations. + +The expression for these quantities read +\begin{equation} +T_{ab} = +T^{\rm D}_{ab}+ +T^{\rm EM}_{ab}+ +T^{\rm YM}_{ab}+ +T^{\rm Smin}_{ab}+ +T^{\rm IF}_{ab}+ +T^{\rm SF}_{ab} \label{b1} +\end{equation} +\begin{equation} +s_{AB} = s_{AB}^{\rm D} + s_{AB}^{\rm SF} \label{b2} +\end{equation} +When $T_{ab}$ and +$s_{AB}$ are calculated \grg\ does not tries to find value +of all objects in the right-hand side of Eqs. (\ref{b1}), (\ref{b2}) +instead it adds only the quantities whose value are currently +defined. In particular if none of above tensors and spinors are +defined then $T_{ab}=s_{AB}=0$. + +Notice that $T_{ab}$ and all tensors in the right-hand side +of Eq. (\ref{b1}) are symmetric. +\seethis{See \pref{tsym}.} +They are the symmetric parts of the canonical energy-momentum tensors. + +In addition we introduce the +\object{Total Energy-Momentum Trace TENMOMT}{T=T^a{}_a} +and the spinor +\object{Total Energy-Momentum Spinor TENMOMS.AB.CD\cc}{T_{AB\dot C\dot D}} +is a spinorial equivalent of the traceless part of $T_{ab}$ +\begin{equation} +T_{ab}-\frac{1}{4}g_{ab}T \tsst T_{AB\dot A\dot B} +\end{equation} + + +\section{Einstein Equations} + +The Einstein equation +\object{Einstein Equation EEq.a.b} +{R_{ab}-\frac{1}{2}g_{ab}R +\Lambda R =8\pi G\, T_{ab}} + +And the {\tt Spinor Einstein equations} +\object{Traceless Einstein Equation CEEq.AB.CD\cc}{ +C_{AB\dot C\dot D} = 8\pi G\, T_{AB\dot C\dot D}} +\object{Trace of Einstein Equation TEEq} +{R-4\Lambda = -8\pi G\, T} + +The cosmological constant is included in these equations +iff the switch \comm{CCONST} is turned on \swind{CCONST} +and its value is given by the constant \comm{CCONST}. +The gravitational constant $G$ is given by the constant \comm{GCONST}. + + +\section{Gravitational Equations in Space With Torsion} + +Equations in this section are valid in dimension $d=4$ +with the signature ${\scriptstyle(-,+,+,+)}$ and +${\scriptstyle(+,-,-,-)}$ only. +The $\sigma=1$ for the first signature and $\sigma=-1$ +for the second. The nonmetricity must be zero and the +switch \comm{NONMETR} turned off. + +Let us consider the action +\begin{equation} +S=\int\left[\frac{\sigma}{16\pi G}L_{\rm g} ++L_{\rm m}\right] +\end{equation} +where +\object{Action LACT}{L_{\rm g}=\upsilon\,{\cal L}_{\rm g}} +is the gravitational action 4-form and +\begin{equation} +L_{\rm m} = \upsilon\,{\cal L}_{\rm m} +\end{equation} +is the matter action 4-form. + +Let us define the following variational derivatives +\begin{equation} +Z^\mu{}_{a} = \frac{1}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta h^a_\mu} +,\qquad +t^\mu{}_{a} = \frac{\sigma}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta h^a_\mu} +\end{equation} +\begin{equation} +V^\mu{}_{ab} = \frac{1}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta \omega^{ab}{}_\mu} +,\qquad +s^\mu{}_{ab} = \frac{\sigma}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta \omega^{ab}{}_\mu} +\end{equation} +Then the gravitational equations reads +\begin{eqnarray} +Z^\mu{}_a &=& -16\pi G\,t^\mu{}_a \label{zma} \\[2mm] +V^\mu{}_{ab} &=& -16\pi G\,s^\mu{}_{ab} \label{vab} +\end{eqnarray} +Here the first equation is an analog of Einstein equation +and has the canonical nonsymmetric energy-momentum +tensor $t^\mu{}_a$ as a source. The source in the second +equation is the spin tensor $s^\mu{}_{ab}$. + +Now we rewrite these equation in other equivalent form. +First let us define the following 3-forms +\begin{equation} +Z_a = Z^m{}_a\,*\theta_m,\qquad t_a = t^m{}_a\,*\theta_m +\end{equation} +\begin{equation} +V_{ab} = V^m{}_{ab}\,*\theta_m,\qquad s_{ab} = s^m{}_{ab}\,*\theta_m +\end{equation} +Notice that Eq. (\ref{zma}) is not symmetric but \label{tsym} +the antisymmetric part of this equation is expressed via second +Eq. (\ref{vab}) due to Bianchi identity. Therefore only the +symmetric part of Eq. (\ref{zma}) is essential. +Eq. (\ref{vab}) is +antisymmetric and we can consider its spinorial analog +using the standard relations +\begin{eqnarray} +V_{ab} &\tsst& V_{A\dot AB\dot B}= +\epsilon_{AB} V_{\dot A\dot B} + \epsilon_{\dot A\dot B}V_{AB} \\ +s_{ab} &\tsst& s_{A\dot AB\dot B}= +\epsilon_{AB} s_{\dot A\dot B} + \epsilon_{\dot A\dot B}s_{AB} +\end{eqnarray} \seethis{See \pref{asys}.} + +Finally we define the {\tt Gravitational equations} in the form \label{graveq} +\object{Metric Equation METRq.a.b}{-\frac12Z_{(ab)}=8\pi G\,T_{ab}} +\object{Torsion Equation TORSq.AB}{V_{AB}=-16\pi G\,s_{AB}} +where the currents in the right-hand side of equations are +\seethis{See \pref{totalc}.} +\object{Total Energy-Momentum Tensor TENMOM.a.b}{T_{ab}=t_{(ab)}} +\object{Total Undotted Spin 3-form SPINU.AB}{s_{AB}} + +Now let us consider the equations which are used in \grg\ to +compute the left-hand side of the gravitational equations +$Z_{(ab)}$ and $V_{AB}$. We have to emphasize that we use +\seethis{See \pref{spinors}.} +spinors and all restrictions imposed by the spinorial formalism +must be fulfilled. + +We consider the Lagrangian which is an arbitrary algebraic function +of the curvature and torsion tensors +\begin{equation} +{\cal L}_{\rm g} = {\cal L}_{\rm g}(R_{abcd},Q_{abc}) +\end{equation} +No derivatives of the torsion or curvature are permitted. +For such a Lagrangian we define so called curvature and torsion +momentums +\begin{equation} +\widetilde{R}{}^{abcd} = +2\frac{\partial{\cal L}_{\rm g}(R,Q)}{\partial R_{abcd}},\qquad +\widetilde{Q}{}^{abc} = +2\frac{\partial{\cal L}_{\rm g}(R,Q)}{\partial Q_{abc}},\qquad +\end{equation} + +The corresponding objects are +\object{Undotted Curvature Momentum POMEGAU.AB}{\widetilde{\Omega}_{AB}} +\object{Torsion Momentum PTHETA'a}{\widetilde{\Theta}{}^a} +where +\begin{eqnarray} +\widetilde{\Omega}_{ab} &=& \frac12 \widetilde{R}_{abcd}\,S^{cd} \\[1mm] +\widetilde{\Theta}{}^a &=& \frac12 \widetilde{Q}{}^a{}_{cd}\,S^{cd} +\end{eqnarray} +and +\begin{equation} +\widetilde{\Omega}_{ab} \tsst \widetilde{\Omega}_{A\dot AB\dot B}= +\epsilon_{AB} \widetilde{\Omega}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}\widetilde{\Omega}_{AB} +\end{equation} + +If value of three objects $L_{\rm g}$ ({\tt Action}), +$\widetilde{\Omega}_{AB}$ ({\tt Undotted curvature momentum}) +and $\widetilde{\Theta}{}^a$ are specified then the +{\tt Gravitational equations} can be calculated using equations +({\tt Standard way}) +\begin{eqnarray} +Z_{(ab)} &=& *(\theta_{(a}\wedge Z_{b)}),\nonumber\\[1mm] +Z_a &=& D\widetilde{\Theta}_a + + (\partial_a\ipr\Theta^b)\wedge\widetilde{\Theta}_b + +2(\partial_a\ipr\Omega^{MN})\wedge\widetilde{\Omega}_{MN} +\nonumber\\ +&& + {\rm c.c.}-\partial_a L_{\rm g} +\end{eqnarray} +\begin{eqnarray} +&&V_{AB} = -D\widetilde{\Omega}_{AB} - \widetilde{\Theta}_{AB},\nonumber\\[1mm] +&& +\theta_{[a}\wedge\widetilde{\Theta}_{b]} \tsst +\epsilon_{AB} \widetilde{\Theta}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}\widetilde{\Theta}_{AB} +\end{eqnarray} + +Since gravitational equations are computed in the +spinorial formalism with the standard null frame +\seethis{See pages \pageref{spinors}\ and \pageref{spinors1}.} +the metric equation is complex and components $\scriptstyle02$, +$\scriptstyle12$, $\scriptstyle22$ are conjugated to $\scriptstyle03$. +$\scriptstyle13$, $\scriptstyle33$. Since these components are not independent +For the sake of efficiency by default \grg\ computes only +the $\scriptstyle00$, $\scriptstyle01$, $\scriptstyle02$, +$\scriptstyle11$, $\scriptstyle12$, $\scriptstyle22$ and $\scriptstyle23$ +components of $Z_{(ab)}$ only. +If you want to have all components the switch \comm{FULL} must be +turned on. \swind{FULL} + +These equations allows one to compute field equations for +gravity theory with an arbitrary Lagrangian. +But the value of three quantities $L_{\rm g}$, +$\widetilde{\Omega}_{AB}$ and $\widetilde{\Theta}{}^a$ +must be specified by the user. In addition \grg\ has built-in +formulas for the most general quadratic in torsion and curvature +Lagrangian. The {\tt Standard way} for $L_{\rm g}$, +$\widetilde{\Omega}_{AB}$ and $\widetilde{\Theta}{}^a$ is \label{thetau} +\begin{eqnarray} +\widetilde{\Theta}{}^a &=& +i\mu_1 (\stackrel{\scriptscriptstyle\rm c}{\vartheta}{}^a -{\rm c.c.}) ++i\mu_2 (\stackrel{\scriptscriptstyle\rm t}{\vartheta}{}^a -{\rm c.c.}) ++i\mu_3 (\stackrel{\scriptscriptstyle\rm a}{\vartheta}\!{}^a -{\rm c.c.}), \\[2mm] +\widetilde{\Omega}_{AB} &=& +i(\lambda_0-\sigma\,8\pi G\, a_0\phi^2)\, S_{AB} \nonumber\\&& ++i\lambda_1 \OO{w}_{AB} +-i\lambda_2 \OO{c}_{AB} ++i\lambda_3 \OO{r}_{AB} \nonumber\\&& ++i\lambda_4 \OO{a}_{AB} +-i\lambda_5 \OO{b}_{AB} ++i\lambda_6 \OO{d}_{AB} , \\[2mm] +L_{\rm g} &=& (-2\Lambda +\frac{1}{2}\lambda_0R +-\sigma\,4\pi G a_0 \phi^2 R) \upsilon ++ \Omega^{AB}\wedge\widetilde{\Omega}_{AB} + {\rm c.c.} \nonumber\\&& ++ \frac{1}{2} \Theta^a\wedge\widetilde{\Theta}_a +\end{eqnarray} + +The cosmological term $\Lambda$ is included into +equations iff the switch \comm{CCONST} is turned on \swinda{CCONST} +and the value of $\Lambda$ is given by the constant \comm{CCONST}. +The term with the scalar field $\phi$ is included into +equations iff the switch \comm{NONMIN} is on. \swinda{NONMIN} +The gravitational constant $G$ is given by the constant \comm{GCONST}. +The parameters of the quadratic Lagrangian are given by the objects +\object{L-Constants LCONST.i6}{\lambda_i} +\object{M-Constants MCONST.i3}{\mu_i} +\object{A-Constants ACONST.i2}{a_i} +The default value of these objects ({\tt Standard way}) is +\begin{eqnarray} +\lambda_i &=& (\mbox{\tt LC0},\mbox{\tt LC1},\mbox{\tt LC2},\mbox{\tt LC3},\mbox{\tt LC4},\mbox{\tt LC5},\mbox{\tt LC6}), \\ +\mu_i &=& (0,\mbox{\tt MC1},\mbox{\tt MC2},\mbox{\tt MC32}), \\ +a_i &=& (\mbox{\tt AC0},0,0) +\end{eqnarray} + +\section{Gravitational Equations in Riemann Space} + +Equations in this section are valid in dimension $d=4$ +with the signature ${\scriptstyle(-,+,+,+)}$ and +${\scriptstyle(+,-,-,-)}$ only. +The $\sigma=1$ for the first signature and $\sigma=-1$ +for the second. The nonmetricity and torsion must be zero and the +switches \comm{NONMETR} and \comm{TORSION} must be turned off. + +Let us consider the action +\begin{equation} +S=\int\left[\frac{\sigma}{16\pi G}L_{\rm g} ++L_{\rm m}\right] +\end{equation} +where +\object{Action LACT}{L_{\rm g}=\upsilon\,{\cal L}_{\rm g}} +is the gravitational action 4-form and +\begin{equation} +L_{\rm m} = \upsilon\,{\cal L}_{\rm m} +\end{equation} +is the matter action 4-form. + +Let us define the following variational derivatives +\begin{equation} +Z^\mu{}_{a} = \frac{1}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta h^a_\mu} +,\qquad +T^\mu{}_{a} = \frac{\sigma}{\sqrt{-g}} +\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta h^a_\mu} +\end{equation} +Then the {\tt Metric equation} is \label{metreq} +\object{Metric Equation METRq.a.b}{-\frac12Z_{ab}=8\pi G\,T_{ab}} +Notice that $Z_{ab}$ and $T_{ab}$ are automatically symmetric. + +Let us define 3-form +\begin{equation} +Z_a = Z^m{}_a\,*\theta_m,\qquad t_a = t^m{}_a\,*\theta_m +\end{equation} + +Now we consider the equations which are used in \grg\ to +compute the left-hand side of the metric equation +$Z_{ab}$. We have to emphasize that we use +spinors and all restrictions imposed by the spinorial formalism +\seethis{See pages \pageref{spinors}\ or \pageref{spinors1}.} +must be fulfilled. + +We consider the Lagrangian which is an arbitrary algebraic function +of the curvature tensor +\begin{equation} +{\cal L}_{\rm g} = {\cal L}_{\rm g}(R_{abcd}) +\end{equation} +No derivatives of the curvature are permitted. +For such a Lagrangian we define so called curvature momentum +\begin{equation} +\widetilde{R}{}^{abcd} = +2\frac{\partial{\cal L}_{\rm g}(R)}{\partial R_{abcd}} +\end{equation} + +The corresponding \grg\ built-in object is +\object{Undotted Curvature Momentum POMEGAU.AB}{\widetilde{\Omega}_{AB}} +where +\begin{eqnarray} +\widetilde{\Omega}_{ab} &=& \frac12 \widetilde{R}_{abcd}\,S^{cd} \\[1mm] +\end{eqnarray} +and +\begin{equation} +\widetilde{\Omega}_{ab} \tsst \widetilde{\Omega}_{A\dot AB\dot B}= +\epsilon_{AB} \widetilde{\Omega}_{\dot A\dot B} ++ \epsilon_{\dot A\dot B}\widetilde{\Omega}_{AB} +\end{equation} + +If value of the objects $L_{\rm g}$ ({\tt Action}) and +$\widetilde{\Omega}_{AB}$ ({\tt Undotted curvature momentum}) is specified +then the {\tt Metric equation} can be calculated using equations +({\tt Standard way}) +\begin{eqnarray} +Z_{ab} &=& *(\theta_{(a}\wedge Z_{b)}),\nonumber\\[1mm] +Z_a &=& D [ +2\partial_m\ipr D\widetilde{\Omega}_a{}^{m} +-{\frac{1}{2}}\theta_a\!\wedge +(\partial_m\ipr\partial_n\ipr D\widetilde{\Omega}{}^{mn})] +\nonumber\\&& + +2(\partial_a\ipr\Omega^{MN})\wedge\widetilde{\Omega}_{MN} + + {\rm c.c.}-\partial_a L_{\rm g} +\end{eqnarray} + +Since gravitational equations are computed in the +spinorial formalism with the standard null frame +\seethis{See \pref{spinors}\ or \pref{spinors1}.} +the metric equation is complex and components $\scriptstyle02$, +$\scriptstyle12$, $\scriptstyle22$ are conjugated to $\scriptstyle03$, +$\scriptstyle13$, $\scriptstyle33$. +For the sake of efficiency by default \grg\ computes only +the components $\scriptstyle00$, $\scriptstyle01$, $\scriptstyle02$, +$\scriptstyle11$, $\scriptstyle12$, $\scriptstyle22$ and $\scriptstyle23$ +only. If you want to have all components the switch \comm{FULL} must be +turned on. \swinda{FULL} + +These equations allows one to compute field equations for +gravity theory with an arbitrary Lagrangian. +But the value of three quantities $L_{\rm g}$ and +$\widetilde{\Omega}_{AB}$ must be specified by user. +In addition \grg\ has built-in +formulas for the most general quadratic in the curvature +Lagrangian. The {\tt Standard way} for $L_{\rm g}$ and +$\widetilde{\Omega}_{AB}$ is +\begin{eqnarray} +\widetilde{\Omega}_{AB} &=& +i(\lambda_0-\sigma8\pi G\, a_0\phi^2)\, S_{AB} \nonumber\\&& ++i\lambda_1 \OO{w}_{AB} +-i\lambda_2 \OO{c}_{AB} ++i\lambda_3 \OO{r}_{AB}, \\[2mm] +L_{\rm g} &=& (-2\Lambda +{\frac{1}{2}}\lambda_0R +-\sigma4\pi G a_0 \phi^2 R) \upsilon ++ \Omega^{AB}\wedge\widetilde{\Omega}_{AB} + {\rm c.c.} +\end{eqnarray} + +The cosmological term is included into +equations iff the switch \comm{CCONST} is on \swinda{CCONST} +and the value of $\Lambda$ is given by the constant \comm{CCONST}. +The term with the scalar field $\phi$ is included into +equations iff the switch \comm{NONMIN} is on. \swinda{NONMIN} +The gravitational constant $G$ is given by the constant \comm{GCONST}. +The parameters of the quadratic lagrangian are given by the object +\object{L-Constants LCONST.i6}{\lambda_i} +\object{A-Constants ACONST.i2}{a_i} +The default value of these objects ({\tt Standard way}) is +\begin{eqnarray} +\lambda_i &=& (\mbox{\tt LC0},\mbox{\tt LC1},\mbox{\tt LC2},\mbox{\tt LC3},\mbox{\tt LC4},\mbox{\tt LC5},\mbox{\tt LC6}), \\ +a_i &=& (\mbox{\tt AC0},0,0) +\end{eqnarray} + + + +\appendix + +\chapter{\grg\ Switches}\vspace*{-6mm} +\index{Switches} + +\tabcolsep=1.5mm + +\begin{tabular}{|c|c|l|c|} +\hline +Switch & Default &\qquad Description & See \\ + & State & & page\\ +\hline +\tt AEVAL & Off & Use {\tt AEVAL} instead of {\tt REVAL}. &\pageref{AEVAL}\\ +\tt WRS & On & Re-simplify object before printing. &\pageref{WRS}\\ +\tt WMATR & Off & Write 2-index objects in matrix form. &\pageref{WMATR}\\ +\tt TORSION & Off & Torsion. &\pageref{TORSION}\\ +\tt NONMETR & Off & Nonmetricity. &\pageref{NONMETR}\\ +\tt UNLCORD & On & Save coordinates in {\tt Unload}. &\pageref{UNLCORD}\\ +\tt AUTO & On & Automatic object calculation in expressions. &\pageref{AUTO}\\ +\tt TRACE & On & Trace the calculation process. &\pageref{TRACE}\\ +\tt SHOWCOMMANDS & Off & Show compound command expansion. &\pageref{SHOWCOMMANDS}\\ +\tt EXPANDSYM & Off & Enable {\tt Sy Asy Cy} in expressions &\pageref{EXPANDSYM}\\ +\tt DFPCOMMUTE & On & Commutativity of {\tt DFP} derivatives. &\pageref{DFPCOMMUTE}\\ +\tt NONMIN & Off & Nonminimal interaction for scalar field. &\pageref{NONMIN}\\ +\tt NOFREEVARS & Off & Prohibit free variables in {\tt Print}. &\pageref{NOFREEVARS}\\ +\tt CCONST & Off & Include cosmological constant in equations. &\pageref{CCONST}\\ +\tt FULL & Off & Number of components in {\tt Metric Equation}. &\pageref{FULL}\\ +\tt LATEX & Off & \LaTeX\ output mode. &\pageref{LATEX}\\ +\tt GRG & Off & \grg\ output mode. &\pageref{GRG}\\ +\tt REDUCE & Off & \reduce\ output mode. &\pageref{REDUCE}\\ +\tt MAPLE & Off & {\sc Maple} output mode. &\pageref{MAPLE}\\ +\tt MATH & Off & {\sc Mathematica} output mode. &\pageref{MATH}\\ +\tt MACSYMA & Off & {\sc Macsyma} output mode. &\pageref{MACSYMA}\\ +\tt DFINDEXED & Off & Print {\tt DF} in index notation. &\pageref{DFINDEXED}\\ +\tt BATCH & Off & Batch mode. &\pageref{BATCH}\\ +\tt HOLONOMIC & On & Keep frame holonomic. &\pageref{HOLONOMIC}\\ +\tt SHOWEXPR & Off & Print expressions during algebraic &\pageref{SHOWEXPR}\\ +\tt & & classification. &\\ +\hline +\end{tabular} + +\chapter{Macro Objects} +\index{Macro Objects} + +Macro objects can be used in expression, in {\tt Write} and +{\tt Show} commands but not in the {\tt Find} command. +The notation for indices is the same as in the {\tt New Object} +declaration (see page \pageref{indices}). + +\begin{center} + +\section{Dimension and Signature} + +\begin{tabular}{|l|l|} +\hline +\tt dim & Dimension $d$ \\ +\hline +\tt sdiag.idim & {\tt sdiag(\parm{n})} is the $n$'th element of the \\ + & signature diag($-1,+1$\dots) \\ +\hline +\tt sign & Product of the signature specification \\ +\tt sgnt & elements $\prod_{n=0}^{d-1}\mbox{\tt sdiag(}n\mbox{\tt)}$ \\[1mm] +\hline +\tt mpsgn & {\tt sdiag(0)} \\ +\tt pmsgn & {\tt -sdiag(0)} \\ +\hline +\end{tabular} + +\section{Metric and Frame} + +\begin{tabular}{|l|l|} +\hline +\tt x\^m & $m$'th coordinate \\ +\tt X\^m & \\ +\hline +\tt h'a\_m & Frame coefficients \\ +\tt hi.a\^m & \\ +\hline +\tt g\_m\_n & Holonomic metric \\ +\tt gi\^m\^n & \\ +\hline +\end{tabular} + +\section{Delta and Epsilon Symbols} + +\begin{tabular}{|l|l|} +\hline +\tt del'a.b & Delta symbols \\ +\tt delh\^m\_n & \\ +\hline +\tt eps.a.b.c.d & Totally antisymmetric symbols \\ +\tt epsi'a'b'c'd & (number of indices depend on $d$) \\ +\tt epsh\_m\_n\_p\_q & \\ +\tt epsih\^m\^n\^p\^q & \\ +\hline +\end{tabular} + +\section{Spinors} + +\begin{tabular}{|l|l|} +\hline +\tt DEL'A.B & Delta symbol \\ +\hline +\tt EPS.A.B & Spinorial metric \\ +\tt EPSI'A'B & \\ +\hline +\tt sigma'a.A.B\cc & Sigma matrices \\ +\tt sigmai.a'A'B\cc & \\ +\hline +\tt cci.i3 & Frame index conjugation in standard null frame \\ + & {\tt cci(0)=0}\ {\tt cci(1)=1}\ {\tt cci(2)=3}\ {\tt cci(3)=2} \\ +\hline +\end{tabular} + +\section{Connection Coefficients} + +\begin{tabular}{|l|l|} +\hline +\tt CHR\^m\_n\_p & Christoffel symbols $\{{}^\mu_{\nu\pi}\}$ \\ +\tt CHRF\_m\_n\_p & and $[{}_{\mu},_{\nu\pi}]$ \\ +\tt CHRT\_m & Christoffel symbol trace $\{{}^\pi_{\pi\mu}\}$ \\ +\hline +\tt SPCOEF.AB.c & Spin coefficients $\omega_{AB\,c}$ \\ +\hline +\end{tabular} + +\section{NP Formalism} + +\begin{tabular}{|l|c|} +\hline +\tt PHINP.AB.CD~ & $\Phi_{AB\dot{c}\dot{D}}$ \\ +\tt PSINP.ABCD & $\Psi_{ABCD}$ \\ +\hline +\tt alphanp & $\alpha$ \\ +\tt betanp & $\beta$ \\ +\tt gammanp & $\gamma$ \\ +\tt epsilonnp & $\epsilon$ \\ +\tt kappanp & $\kappa$ \\ +\tt rhonp & $\rho$ \\ +\tt sigmanp & $\sigma$ \\ +\tt taunp & $\tau$ \\ +\tt munp & $\mu$ \\ +\tt nunp & $\nu$ \\ +\tt lambdanp & $\lambda$ \\ +\tt pinp & $\pi$ \\ +\hline +\tt DD & $D$ \\ +\tt DT & $\Delta$ \\ +\tt du & $\delta$ \\ +\tt dd & $\overline\delta$ \\ +\hline +\end{tabular} + +\end{center} + +\chapter{Objects} + +Here we present the complete list of built-in objects +with names and identifiers. +The notation for indices is the same as in the +{\tt New Object} declaration (see page \pageref{indices}). +Some names (group names) refer to a set of objects. +For example the group name {\tt Spinorial S - forms} below +denotes {\tt SU.AB} and {\tt SD.AB\cc} + +\begin{center} + + +\section{Metric, Frame, Basis, Volume \dots} +\begin{tabular}{|l|l|}\hline +\tt Frame &\tt T'a\\ +\tt Vector Frame &\tt D.a\\ +\hline +\tt Metric &\tt G.a.b\\ +\tt Inverse Metric &\tt GI'a'b\\ +\tt Det of Metric &\tt detG\\ +\tt Det of Holonomic Metric &\tt detg\\ +\tt Sqrt Det of Metric &\tt sdetG\\ +\hline +\tt Volume &\tt VOL\\ +\hline +\tt Basis &\tt b'idim \\ +\tt Vector Basis &\tt e.idim \\ +\hline +\tt S-forms &\tt S'a'b\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial S-forms} \\ +\tt Undotted S-forms &\tt SU.AB\\ +\tt Dotted S-forms &\tt SD.AB\cc\\ +\hline\end{tabular} + +\section{Rotation Matrices} +\begin{tabular}{|l|l|}\hline +\tt Frame Transformation &\tt L'a.b \\ +\tt Spinorial Transformation &\tt LS.A'B \\ +\hline\end{tabular} + +\section{Connection and related objects} +\begin{tabular}{|l|l|}\hline +\tt Frame Connection &\tt omega'a.b\\ +\tt Holonomic Connection &\tt GAMMA\^m\_n\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial Connection}\\ +\tt Undotted Connection &\tt omegau.AB\\ +\tt Dotted Connection &\tt omegad.AB\cc\\ +\hline +\tt Riemann Frame Connection &\tt romega'a.b\\ +\tt Riemann Holonomic Connection &\tt RGAMMA\^m\_n\\ +\hline +\multicolumn{2}{|c|}{\tt Riemann Spinorial Connection}\\ +\tt Riemann Undotted Connection &\tt romegau.AB\\ +\tt Riemann Dotted Connection &\tt romegad.AB\cc\\ +\hline +\tt Connection Defect &\tt K'a.b\\ +\hline\end{tabular} + +\section{Torsion} +\begin{tabular}{|l|l|}\hline +\tt Torsion &\tt THETA'a\\ +\tt Contorsion &\tt KQ'a.b\\ +\tt Torsion Trace 1-form &\tt QQ\\ +\tt Antisymmetric Torsion 3-form &\tt QQA\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial Contorsion}\\ +\tt Undotted Contorsion &\tt KU.AB\\ +\tt Dotted Contorsion &\tt KD.AB\cc\\ +\hline +\multicolumn{2}{|c|}{\tt Torsion Spinors }\\ +\multicolumn{2}{|c|}{\tt Torsion Components }\\ +\tt Torsion Trace &\tt QT'a\\ +\tt Torsion Pseudo Trace &\tt QP'a\\ +\tt Traceless Torsion Spinor &\tt QC.ABC.D\cc\\ +\hline +\multicolumn{2}{|c|}{\tt Torsion 2-forms}\\ +\tt Traceless Torsion 2-form &\tt THQC'a\\ +\tt Torsion Trace 2-form &\tt THQT'a\\ +\tt Antisymmetric Torsion 2-form &\tt THQA'a\\ +\hline +\multicolumn{2}{|c|}{\tt Undotted Torsion 2-forms}\\ +\tt Undotted Torsion Trace 2-form &\tt THQTU'a\\ +\tt Undotted Antisymmetric Torsion 2-form &\tt THQAU'a\\ +\tt Undotted Traceless Torsion 2-form &\tt THQCU'a\\ +\hline\end{tabular} + + +\section{Curvature} + +\label{curspincoll} +\begin{tabular}{|l|l|}\hline +\tt Curvature &\tt OMEGA'a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Spinorial Curvature}\\ +\tt Undotted Curvature &\tt OMEGAU.AB\\ +\tt Dotted Curvature &\tt OMEGAD.AB\cc\\ +\hline +\tt Riemann Tensor &\tt RIM'a.b.c.d\\ +\tt Ricci Tensor &\tt RIC.a.b\\ +\tt A-Ricci Tensor &\tt RICA.a.b\\ +\tt S-Ricci Tensor &\tt RICS.a.b\\ +\tt Homothetic Curvature &\tt OMEGAH\\ +\tt Einstein Tensor &\tt GT.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Curvature Spinors}\\ +\multicolumn{2}{|c|}{\tt Curvature Components}\\ +\tt Weyl Spinor &\tt RW.ABCD\\ +\tt Traceless Ricci Spinor &\tt RC.AB.CD\cc\\ +\tt Scalar Curvature &\tt RR\\ +\tt Ricanti Spinor &\tt RA.AB\\ +\tt Traceless Deviation Spinor &\tt RB.AB.CD\cc\\ +\tt Scalar Deviation &\tt RD\\ +\hline +\multicolumn{2}{|c|}{\tt Undotted Curvature 2-forms}\\ +\tt Undotted Weyl 2-form &\tt OMWU.AB \\ +\tt Undotted Traceless Ricci 2-form &\tt OMCU.AB \\ +\tt Undotted Scalar Curvature 2-form &\tt OMRU.AB \\ +\tt Undotted Ricanti 2-form &\tt OMAU.AB \\ +\tt Undotted Traceless Deviation 2-form &\tt OMBU.AB \\ +\tt Undotted Scalar Deviation 2-form &\tt OMDU.AB \\ +\hline +\multicolumn{2}{|c|}{\tt Curvature 2-forms}\\ +\tt Weyl 2-form &\tt OMW.a.b \\ +\tt Traceless Ricci 2-form &\tt OMC.a.b \\ +\tt Scalar Curvature 2-form &\tt OMR.a.b \\ +\tt Ricanti 2-form &\tt OMA.a.b \\ +\tt Traceless Deviation 2-form &\tt OMB.a.b \\ +\tt Antisymmetric Curvature 2-form &\tt OMD.a.b \\ +\tt Homothetic Curvature 2-form &\tt OSH.a.b \\ +\tt Antisymmetric S-Ricci 2-form &\tt OSA.a.b \\ +\tt Traceless S-Ricci 2-form &\tt OSC.a.b \\ +\tt Antisymmetric S-Curvature 2-form &\tt OSV.a.b \\ +\tt Symmetric S-Curvature 2-form &\tt OSU.a.b \\ +\hline +\end{tabular} + + +\section{Nonmetricity} +\begin{tabular}{|l|l|}\hline +\tt Nonmetricity &\tt N.a.b\\ +\tt Nonmetricity Defect &\tt KN'a.b\\ +\tt Weyl Vector &\tt NNW\\ +\tt Nonmetricity Trace &\tt NNT\\ +\hline +\multicolumn{2}{|c|}{\tt Nonmetricity 1-forms}\\ +\tt Symmetric Nonmetricity 1-form &\tt NC.a.b\\ +\tt Antisymmetric Nonmetricity 1-form &\tt NA.a.b\\ +\tt Nonmetricity Trace 1-form &\tt NT.a.b\\ +\tt Weyl Nonmetricity 1-form &\tt NW.a.b\\ +\hline\end{tabular} + + +\section{EM field} +\begin{tabular}{|l|l|}\hline +\tt EM Potential &\tt A\\ +\tt Current 1-form &\tt J\\ +\tt EM Action &\tt EMACT\\ +\tt EM 2-form &\tt FF\\ +\tt EM Tensor &\tt FT.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Maxwell Equations}\\ +\tt First Maxwell Equation &\tt MWFq\\ +\tt Second Maxwell Equation &\tt MWSq\\ +\hline +\tt Continuity Equation &\tt COq\\ +\tt EM Energy-Momentum Tensor &\tt TEM.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt EM Scalars}\\ +\tt First EM Scalar &\tt SCF\\ +\tt Second EM Scalar &\tt SCS\\ +\hline +\tt Selfduality Equation &\tt SDq.AB\cc\\ +\tt Complex EM 2-form &\tt FFU\\ +\tt Complex Maxwell Equation &\tt MWUq\\ +\tt Undotted EM Spinor &\tt FIU.AB\\ +\tt Complex EM Scalar &\tt SCU\\ +\tt EM Energy-Momentum Spinor &\tt TEMS.AB.CD\cc\\ +\hline\end{tabular} + +\section{Scalar field} +\begin{tabular}{|l|l|}\hline +\tt Scalar Equation &\tt SCq\\ +\tt Scalar Field &\tt FI\\ +\tt Scalar Action &\tt SACT\\ +\tt Minimal Scalar Action &\tt SACTMIN\\ +\tt Minimal Scalar Energy-Momentum Tensor &\tt TSCLMIN.a.b\\ +\hline\end{tabular} + + +\section{YM field} +\begin{tabular}{|l|l|}\hline +\tt YM Potential &\tt AYM.i9\\ +\tt Structural Constants &\tt SCONST.i9.j9.k9\\ +\tt YM Action &\tt YMACT\\ +\tt YM 2-form &\tt FFYM.i9\\ +\tt YM Tensor &\tt FTYM.i9.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt YM Equations}\\ +\tt First YM Equation &\tt YMFq.i9\\ +\tt Second YM Equation &\tt YMSq.i9\\ +\hline +\tt YM Energy-Momentum Tensor &\tt TYM.a.b\\ +\hline\end{tabular} + +\section{Dirac field} +\begin{tabular}{|l|l|}\hline +\multicolumn{2}{|c|}{\tt Dirac Spinor}\\ +\tt Phi Spinor &\tt PHI.A\\ +\tt Chi Spinor &\tt CHI.B\\ +\hline +\tt Dirac Action 4-form &\tt DACT\\ +\tt Undotted Dirac Spin 3-Form &\tt SPDIU.AB\\ +\tt Dirac Energy-Momentum Tensor &\tt TDI.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Dirac Equation}\\ +\tt Phi Dirac Equation &\tt DPq.A\cc\\ +\tt Chi Dirac Equation &\tt DCq.A\cc\\ +\hline\end{tabular} + +\section{Geodesics} +\begin{tabular}{|l|l|}\hline +\tt Geodesic Equation &\tt GEOq\^m\\ +\hline\end{tabular} + +\section{Null Congruence} +\begin{tabular}{|l|l|}\hline +\tt Congruence &\tt KV\\ +\tt Null Congruence Condition &\tt NCo\\ +\tt Geodesics Congruence Condition&\tt GCo'a\\ +\hline +\multicolumn{2}{|c|}{\tt Optical Scalars}\\ +\tt Congruence Expansion &\tt thetaO\\ +\tt Congruence Squared Rotation &\tt omegaSQO\\ +\tt Congruence Squared Shear &\tt sigmaSQO\\ +\hline\end{tabular} + +\section{Kinematics} +\begin{tabular}{|l|l|}\hline +\tt Velocity Vector &\tt UV\\ +\tt Velocity &\tt UU'a\\ +\tt Velocity Square &\tt USQ\\ +\tt Projector &\tt PR'a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Kinematics}\\ +\tt Acceleration &\tt accU'a\\ +\tt Vorticity &\tt omegaU.a.b\\ +\tt Volume Expansion &\tt thetaU\\ +\tt Shear &\tt sigmaU.a.b\\ +\hline\end{tabular} + +\section{Ideal and Spin Fluid} +\begin{tabular}{|l|l|}\hline +\tt Pressure &\tt PRES\\ +\tt Energy Density &\tt ENER\\ +\tt Ideal Fluid Energy-Momentum Tensor &\tt TIFL.a.b\\ +\hline +\tt Spin Fluid Energy-Momentum Tensor &\tt TSFL.a.b \\ +\tt Spin Density &\tt SPFLT.a.b \\ +\tt Spin Density 2-form &\tt SPFL \\ +\tt Undotted Fluid Spin 3-form &\tt SPFLU.AB \\ +\tt Frenkel Condition &\tt FCo \\ +\hline\end{tabular} + +\section{Total Energy-Momentum and Spin} +\begin{tabular}{|l|l|}\hline +\tt Total Energy-Momentum Tensor &\tt TENMOM.a.b\\ +\tt Total Energy-Momentum Spinor &\tt TENMOMS.AB.CD\cc\\ +\tt Total Energy-Momentum Trace &\tt TENMOMT\\ +\tt Total Undotted Spin 3-form &\tt SPINU.AB\\ +\hline\end{tabular} + +\section{Einstein Equations} +\begin{tabular}{|l|l|}\hline +\tt Einstein Equation &\tt EEq.a.b\\ +\hline +\multicolumn{2}{|c|}{\tt Spinor Einstein Equations}\\ +\tt Traceless Einstein Equation &\tt CEEq.AB.CD\cc\\ +\tt Trace of Einstein Equation &\tt TEEq\\ +\hline\end{tabular} + +\section{Constants} +\begin{tabular}{|l|l|}\hline +\tt A-Constants &\tt ACONST.i2\\ +\tt L-Constants &\tt LCONST.i6\\ +\tt M-Constants &\tt MCONST.i3\\ +\hline\end{tabular} + +\section{Gravitational Equations} +\begin{tabular}{|l|l|}\hline +\tt Action &\tt LACT\\ +\tt Undotted Curvature Momentum &\tt POMEGAU.AB\\ +\tt Torsion Momentum &\tt PTHETA'a\\ +\hline +\multicolumn{2}{|c|}{\tt Gravitational Equations}\\ +\tt Metric Equation &\tt METRq.a.b\\ +\tt Torsion Equation &\tt TORSq.AB\\ +\hline\end{tabular} + +\end{center} + + +\chapter{Standard Synonymy} +\index{Synonymy} + +Below we present the default synonymy as it is defined in the +global configuration file. See section \ref{tuning} to find out +how to change the default synonymy or define a new one. + +\begin{verbatim} + Affine Aff + Anholonomic Nonholonomic AMode ABasis + Antisymmetric Asy + Change Transform + Classify Class + Components Comp + Connection Con + Constants Const Constant + Coordinates Cord + Curvature Cur + Dimension Dim + Dotted Do + Equation Equations Eq + Erase Delete Del + Evaluate Eval Simplify + Find F Calculate Calc + Form Forms + Functions Fun Function + Generic Gen + Gravitational Gravity Gravitation Grav + Holonomic HMode HBasis + Inverse Inv + Load Restore + Next N + Normalize Normal + Object Obj + Output Out + Parameter Par + Rotation Rot + Scalar Scal + Show ? + Signature Sig + Solutions Solution Sol + Spinor Spin Spinorial Sp + standardlisp lisp + Switch Sw + Symmetries Sym Symmetric + Tensor Tensors Tens + Torsion Tors + Transformation Trans + Undotted Un + Unload Save + Vector Vec + Write W + Zero Nullify +\end{verbatim} + + +\makeatletter +\if@openright\cleardoublepage\else\clearpage\fi +\makeatother +\thispagestyle{empty} +\def\indexname{INDEX} +\printindex + +\end{document} + +%======== End of grg32.tex ==============================================% + DELETED grg32.tex Index: grg32.tex ================================================================== --- grg32.tex +++ /dev/null @@ -1,7015 +0,0 @@ -%==========================================================================% -% GRG 3.2 Manual (C) 1988-97 Vadim V. 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- - -\begin{document} - - -\begin{titlepage} -\hsize=150mm -\hrulefill -\vspace*{20mm} -\begin{center} -\Huge\bf GRG\\[1mm] -\normalsize Version 3.2 -\end{center} -\begin{center} -\Large Computer Algebra System for\\ -Differential Geometry,\\ -Gravitation and \\ -Field Theory -\vspace*{25mm}\\ -{\Large\itshape\bfseries Vadim V. Zhytnikov}\\ -\vfill -{\normalsize Moscow, 1992--1997 $\bullet$ Chung-Li, 1994} -\end{center} -\hrulefill -\end{titlepage} -\setcounter{page}{0}\thispagestyle{empty} - -\tableofcontents\thispagestyle{empty} - -\chapter{Introduction} - -Calculation of various geometrical and physical quantities and -equations is the usual technical problem which permanently arises -in geometry, field and gravity theory. Numerous indices, -contractions and components make these calculations very tedious -and error-prone. Since this calculus obeys the well defined rules the idea -to automate this kind of problems using computer is quite -natural. Now there are several computer algebra systems such as -\maple, \reduce, \mathematica\ or \macsyma\ which in principle -allow one to do this and it is not so hard -to write a program to calculate, for example, the -curvature tensor or connection. But suppose that we want to -make a non-trivial coordinate transformation or tetrad rotation, -calculate covariant or Lie derivative, compute a complicated -expression with numerous contraction or raise or lower some indices. -All these operations are typical in differential geometry -and field theory but their realization with the help of general -purpose computer algebra systems requires hard programming since -all these systems really know nothing about \emph{covariant properties} -of geometrical quantities. - -The computer algebra system \grg\ is designed in such a way -to make calculation in differential geometry and field theory -as simple and natural as possible. \grg\ is based on the -computer algebra system \reduce\ but \grg\ has its own simple -input language whose commands resembles English phrases. -Working with \grg\ no any knowledge of programming is required. - -\grg\ understands tensors, spinors, vectors, differential forms -and knows all standard operations with these quantities. -Input form for mathematical expressions is very close -to traditional mathematical notation including Einstein summation -rule. \grg\ knows the covariant properties of -these objects, you can easily raise and lower indices, -compute covariant and Lie derivatives, perform -coordinate and frame transformations. -\grg\ works in any dimension and allows one to represent tensor -quantities with respect to holonomic, orthogonal and even -any other arbitrary frame. - -One of the useful features of \grg\ is that it has a large -number of built-in standard field-theory -and geometrical quantities and formulas for their computation. -Thus \grg\ provides ready solutions to many standard problems. - -Another unique feature of \grg\ is that it can export -results of calculations into other computer algebra system. -You can save your data in to the file in the format of -\maple, \mathematica, \macsyma\ or \reduce\ in order to use -this system to proceed analysis of the data. -The \LaTeX\ output format is supported as well. -In addition \grg\ is compatible with \reduce\ graphics -shells providing niece book-quality output with Greek letters, -integral signs etc. - -The main built-in \grg\ capabilities are: -\begin{list}{$\bullet$}{\labelwidth=8mm\leftmargin=10mm} -\item Connection, torsion and nonmetricity. -\item Curvature. -\item Spinorial formalism. -\item Irreducible decomposition of the curvature, torsion, and - nonmetricity in any dimension. -\item Einstein equations. -\item Scalar field with minimal and non-minimal interaction. -\item Electromagnetic field. -\item Yang-Mills field. -\item Dirac spinor field. -\item Geodesic equation. -\item Null congruences and optical scalars. -\item Kinematics for time-like congruences. -\item Ideal and spin fluid. -\item Newman-Penrose formalism. -\item Gravitational equations for the theory with arbitrary - gravitational Lagrangian in Riemann and Riemann-Cartan - spaces. -\end{list} - -I would like to stress that current \grg\ version is -intended for calculations in a concrete coordinate map only. -It cannot operate with tensors as with objects having -abstract symbolic indices. - -This book consist of two main parts. First part -contains detailed description of \grg\ as a programming -system. Second part describes all built-in objects -and formulas for their computation. - - -\chapter{Programming in \grg} - -Throughout the chapter \comm{commands} are printed in -typewriter font. The slanted serif-less font is -used for command \parm{parameters}. -The optional parts of the commands are enclosed in -squared brackets \opt{option} and \rpt{\parm{id}} -stands for one or several repetitions of \parm{id}: -\parm{id} or \comm{\parm{id},\parm{id}} etc. -Examples are separated form the text by horizontal lines -$\stackrel{\rule{0.1mm}{1mm}\rule[1mm]{3mm}{0.1mm}} -{\rule{0.1mm}{1mm}\rule{3mm}{0.1mm}}$ and the user input -can be easily distinguished from the \grg\ output by the prompt -\comm{<-} which precedes every input line. - - -\section{Session, Tasks and Commands} - -To start \grg\ it is necessary to start \reduce\ and -\seethis{ -On some systems you have -to use {\tt\upshape load!\_package grg;}\newline since -{\tt\upshape load} is not defined.\newline -\newline -Sometimes it\newline is better to use two commands\newline -{\tt\upshape load grg32; grg;}\newline -or\newline -{\tt\upshape load grg; grg;}\newline -(See section \ref{configsect} for details.)} -enter the command {\tt load grg;} - -\begin{slisting} -REDUCE 3.5, 15 Oct 93, patched to 15 Jun 95 ... - -1: load grg; - -This is GRG 3.2 release 2 (Feb 9, 1997) ... - -System directory: c:{\bs}reduce{\bs}grg32{\bs} -System variables are upper-cased: E I PI SIN ... -Dimension is 4 with Signature (-,+,+,+) - -<- -\end{slisting} -Symbol \comm{<-} is the \grg\ prompt which shows that -now \grg\ waits for your input. The \grg\ \emph{task} (we prefer -this term instead of usual \emph{program}) consist of the -sequence of commands terminated by semicolon \comm{;}. -Reading the input \grg\ splits it on \emph{atoms}. -There are several types of atoms:\index{Atoms} -\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} -\item The identifier or symbol is a sequence of letters and digits -starting with a letter: -\begin{verbatim} - i I alpha1 beta ABC123D Find -\end{verbatim} -The identifiers in \grg\ may have trailing tilde character \cc. -Any other character may be incorporated in the identifier if -preceded by the exclamation sign:\index{Identifiers} -\begin{verbatim} - beta~ LIMIT!+ -\end{verbatim} -The identifiers in \grg\ play the role of the variables and -functions in mathematical expressions and words in commands. - -\item Integer numbers\index{Numbers} -\begin{verbatim} - 0 123 104341 -\end{verbatim} - -\item String is a sequence of characters enclosed in double quotes\index{Strings} -\begin{verbatim} - "file.txt" "This is a string" "dir *.doc" -\end{verbatim} -The strings in \grg\ are used for file names and operating system -commands. - -\item Nine special two-character atoms -\begin{verbatim} - ** _| /\ |= ~~ .. <= >= -> -\end{verbatim} - -\item Any other characters are considered as single-character atoms. -\end{list} - -The format of \grg\ commands is free. They can span one or several lines -and any number of spaces and tabulations can be inserted between two -neighbor atoms. - -\enlargethispage{3mm} - -The \grg\ session may consist of several independent tasks. -The command\index{Tasks}\cmdind{Quit} -\command{Quit;} -terminates both \grg\ and \reduce\ session and returns the control -to the operating system level. The command\cmdind{Stop} -\command{Stop;} -terminates current \grg\ task and brings -the session control menu:\index{Session control menu} - \begin{slisting} -<- Stop; - - Quit GRG - 0 - Start Task - 1 - Exit to REDUCE - 2 - - Type 0, 1 or 2: -\end{slisting} -\newpage - -\noindent -The option \comm{0} terminates \reduce\ session similarly to the -command \comm{Quit;}. -The choice \comm{1} starts new task by bringing -\grg\ to its initial state: all variables, declarations, substitutions -and results of calculations are cleared and all switches -resume their initial positions.\footnote{Usually -\grg\ does good job by resuming initial state and new task -turns out to be independent of previous ones. But on some -rare occasions the initial state cannot be completely recovered -and it is better to restart \reduce\ and \grg\ completely.} -Finally the option \comm{2} terminates \grg\ task and returns -control to the \reduce\ command level. In this case \grg\ can be -restarted later by the command \comm{grg;}. - -The commands in \grg\ are case insensitive, i.e. command -\comm{Quit;} is equivalent to \comm{quit;} and \comm{QUIT;} etc. -But notice that unlike \reduce\ variables and functions in -mathematical expressions in \grg\ \emph{are case sensitive}. - - -\subsection{Switches} -\index{Switches} - -Switches in \grg\ and \reduce\ are used to control various -system modes of operation. They are denoted by identifiers and -the commands\cmdind{On}\cmdind{Off} -\command{On \rpt{\parm{switch}};\\\tt -Off \rpt{\parm{switch}};} -turns the \parm{switch} on and off respectively. -Any switch defined by \reduce\ is available in \grg\ as well. -In addition \grg\ defines a couple of its own switches. -The full list of \grg\ switches is presented in appendix A. -The command\cmdind{Show Switch}\cmdind{Switch} -\command{\opt{Show} Switch \parm{switch};} -or equivalently -\command{Show \parm{switch};\\\tt -?~\parm{switch};} -prints current \parm{switch} position -\begin{slisting} -<- Show Switch TORSION; -TORSION is Off. -<- On torsion,gcd; -<- switch torsion; -TORSION is On. -<- switch exp; -GCD is On -\end{slisting} -Switches in \grg\ are case insensitive. - -\subsection{Batch File Execution} - -Usually \grg\ works in the interactive mode which -is not always convenient. The command\cmdind{Input}\index{Batch file execution} -\command{\opt{Input} "\parm{file}";} -reads the \parm{file} and executes commands stored in it. -The file names in \grg\ are always denoted by strings and exact -specification of \parm{file} is operating system dependent. -The word \comm{Input} is optional, thus in order to run batch -file it suffices to enter its name \comm{"\parm{file}";}. -The execution of batch file commands can be suspended by the -command\cmdind{Pause} -\command{Pause;} -After this command \grg\ enters the interactive mode. -One can enter one or several commands interactively and then -resume batch file execution by the command\cmdind{Next} -\command{Next;} - -In general no any special end-of-file symbol or command -is required in the \grg\ batch \parm{file} but is necessary -the symbol\index{end-of-file symbol \comm{\$}} -\comm{\$} is recognized by \grg\ as the end-of-file mark. - -If during the batch file execution an error occurs -\grg\ enter interactive mode and ask user -to input the command which is supposed to replace the -erroneous one. After the receiving of \emph{one} command -\grg\ automatically resumes the batch file execution. -The command \comm{Pause;} can be used if it is necessary -to execute \emph{several} commands instead of one. - -The command\cmdind{Output} -\command{Output "\parm{outfile}";} -redirects all \grg\ output into the \parm{outfile}. -The \parm{outfile} can be closed by the equivalent commands -\cmdind{EndO}\cmdind{End of Output} -\command{EndO;\\\tt -End of Output;} - -It is convenient to run long-time \grg\ tasks in background. -The way of doing this depend on the operating system. -For example to execute \grg\ task in background in UNIX it is -necessary to use the following command -\begin{listing} - reduce < task.grg > grg.out & -\end{listing} -Here we assume that the \reduce\ invoking command is \comm{reduce} -and the file \comm{task.grg} contains the \grg\ task commands: -\begin{listing} - load grg; - \parm{grg command}; - \parm{grg command}; - ... - \parm{grg command}; - quit; -\end{listing} -The output of the session will be written into the file \file{grg.out}. - -Since no proper reaction on errors is possible during the -background execution it is good idea to turn the switch -\comm{BATCH} on.\swind{BATCH} This makes \grg\ to terminate -the session immediately in the case of any error. - -\subsection{Operating System Commands} - -The command\cmdind{System} -\command{System "\parm{command}";} -executes the operating system \parm{command}. -The same command without parameters -\command{System;} -temporary suspends \grg\ session and passes the control to the -operating system command level. The details may depend -on the concrete operating system. In particular in UNIX -the command \comm{system;} may fail but UNIX has some -general mechanism for suspending running programs: -you can press \comm{\^Z} to suspend any program and \comm{\%+} -to resume its execution. - - -\subsection{Comments} - -%\reversemarginpar - -The comment commands\cmdind{Comment} -\command{Comment \parm{any text};\\\tt -\% \parm{any text};} -are used to supply additional information to \grg\ tasks -\seethis{See page \pageref{Unload} about the \comm{Unload} command.} -and data saved by the \comm{Unload} command. -The comment can be also attached to the end of any \grg\ command -\command{\parm{grg command} \% \parm{any text};} - -%\normalmarginpar - -\subsection{Timing} - -The command \cmdind{Time}\cmdind{Show Time} -\command{\opt{Show} Time;} -prints time elapsed since the beginning of current \grg\ task -including the percentage of so called garbage collections. -The garbage collection time can be also printed by the -command \cmdind{GC Time}\cmdind{Show GC Time} -\command{\opt{Show} GC Time;} - -If percentage of garbage collections grows and -exceeds say 30\% then memory of your system -is running short and you probably need more RAM. - - -\section{Declarations} - -Any object, variable or function in \grg\ must be declared. -This allows to locate misprints and makes the system more -reliable. Since \grg\ always work in some concrete -coordinate system (map) the coordinate declaration is the -most important one and must be present in every \grg\ task. - -\subsection{Dimension and Signature} - -During installation \grg\ always defines default value of -the dimension and signature.\index{Dimension!default}\index{Signature!default} -\seethis{See \pref{tuning} -to find out how to change the default dimension and signature.} -The information about this default value is printed\index{Dimension}\index{Signature} -upon \grg\ start in the form of the following (or similar) message line: -\begin{slisting} -Dimension is 4 with Signature (-,+,+,+) -\end{slisting} - - -The following command overrides the default dimension and signature\cmdind{Dimension} -\command{Dimension \parm{dim} with \opt{Signature} (\rpt{\parm{pm}});} -where \parm{dim} is the number \comm{2} or greater and \parm{pm} -is \comm{+} or \comm{-}. The \parm{pm} can be preceded or succeeded by -a number which denotes several repetitions of this \parm{pm}. -For example the declarations -\begin{listing} - Dimension 5 with Signature (+,+,-,-,-); - Dimension 5 with (2+,-3); -\end{listing} -are equivalent and defines 5-dimensional space with the -signature ${\rm diag}{\scriptstyle(+1,+1,-1,}$ ${\scriptstyle-1,-1)}$. - -The important point is that the dimension declaration must -be \emph{very first in the task} and goes before any other command. -Current dimension and signature can be printed by the command -\cmdind{Status}\cmdind{Show Status} -\command{\opt{Show} Status;} - - - -\subsection{Coordinates} - -The coordinate declaration command must be present in every -\grg\ task\cmdind{Coordinates} -\command{Coordinates \rpt{\parm{id}};} -Only few commands such as informational commands, other declarations, -switch changing commands may precede the coordinate declaration. -The only way to have a tusk without the coordinate declaration is -to load the file where coordinates where saved by the -\comm{Unload} command.\seethis{See \pref{UnloadLoad} -to find out how to save data and declarations into a file.} -but no any computation can be done before coordinates are -declared. Current coordinate list can be printed by the command\cmdindx{Write}{Coordinates} -\command{Write Coordinates;} - - -\begin{table} -\begin{center}\index{Constants!predefined} -\begin{tabular}{|l|l|} -\hline -\tt E I PI INFINITY & Mathematical constants $e,i,\pi$,$\infty$ \\ -\hline -\tt FAILED & \\ -\hline -\tt ECONST & Charge of the electron \\ -\tt DMASS & Dirac field mass \\ -\tt SMASS & Scalar field mass \\ -\hline -\tt GCONST & Gravitational constant \\ -\tt CCONST & Cosmological constants \\ -\hline -\tt LC0 LC1 LC2 LC3 & Parameters of the quadratic \\ -\tt LC4 LC5 LC6 & gravitational Lagrangian \\ -\tt MC1 MC2 MC3 & \\ -\hline -\tt AC0 & Nonminimal interaction constant \\ -\hline -\end{tabular} -\caption{Predefined constants}\label{predefconstants} -\end{center} -\end{table} - - -\subsection{Constants} -\index{Constants} - -Any constant must be declared by the command\cmdind{Constants} -\command{Constants \rpt{\parm{id}};} -The list of currently declared constants can be printed -by the command\cmdindx{Write}{Constants} -\command{Write Constants;} -There are also a number of built-in constants -which are listed in table \ref{predefconstants}. - -\subsection{Functions} - -Functions in \grg\ are the analogues of the \reduce\ \emph{operators} -but we prefer to use this traditional mathematical term. -The function must be declared by the command\cmdind{Functions} -\command{Functions \rpt{\parm{f}\opt{(\rpt{\parm{x}})}};} -Here \parm{f} is the function identifier. The optional list -of parameters \parm{x} defines function with \emph{implicit} -dependence. The \parm{x} must be either coordinate or constant. -The construction \comm{\parm{f}(*)} is a shortcut which -declares the function \parm{f} depending on \emph{all coordinates}. - -The following example declares three functions -\comm{fun1}, \comm{fun2} and \comm{fun3}. -The function \comm{fun1}, which was declared without implicit -coordinate list, must be always used in mathematical expressions -together with the explicit arguments like \comm{fun1(x+y)} etc. -The functions \comm{fun2} and \comm{fun3} can appear -in expressions in similar fashion but also as a single symbol -\comm{fun2} or \comm{fun3} -\begin{slisting} -<- Coordinates t, x, y, z; -<- Constant a; -<- Functions fun1, fun2(x,y), fun3(*); -<- Write functions; -Functions: - -fun1 fun2(x,y) fun3(t,x,y,z) - -<- d fun1(x+a); - -DF(fun1(a + x),x) d x - -<- d fun2; - -DF(fun2,x) d x + DF(fun2,y) d y - -<- d fun3; - -DF(fun3,t) d t + DF(fun3,x) d x + DF(fun3,y) d y + DF(fun3,z) d z -\end{slisting} - -The functions may have particular properties with respect -to their arguments permutation and sign. The corresponding -declarations are\cmdind{Symmetric}\cmdind{Antisymmetric}\cmdind{Odd}\cmdind{Even} -\command{Symmetric \rpt{\parm{f}};\\\tt -Antisymmetric \rpt{\parm{f}};\\\tt -Odd \rpt{\parm{f}};\\\tt -Even \rpt{\parm{f}};} -Notice that these commands are valid only after function \parm{f} -was declared by the command \comm{Function}. - -In addition to user-defined there is also large number of -functions predefined in \reduce. All these functions can be -used in \grg\ without declaration. The complete list of these -functions depends on \reduce\ versions. -Any function defined in the \reduce\ package (module) -is available too if the package is loaded before \grg\ was -started or during \grg\ session.\seethis{See \pref{packages} -to find out how to load the \reduce\ packages.} -For example the package \file{specfn} contains definitions -for various special functions. - -Finally there is also special declaration \cmdind{Generic Functions} -\command{Generic Functions \rpt{\parm{f}(\rpt{\parm{a}})};} -This command is valid iff the package \file{dfpart.red} is -installed on your \reduce\ system. Here unlike the usual -function declaration the list of parameters must be always -present and \parm{a} can be any identifier preferably -distinct from any other variable. -\seethis{See \pref{genfun} to find out about the generic functions.} -The role of \parm{a} is also completely different and is explained later. - -The list of declared functions can be printed by the command -\cmdindx{Write}{Functions} -\command{Write Functions;} -Generic functions in this output are marked by the label \comm{*}. - -\subsection{Affine Parameter} - -The variable which plays the role of affine parameter -in the geodesic equation must be declared by the command \label{affpar} -\command{Affine Parameter \parm{s};} -and can be printed by the command\cmdindx{Write}{Affine Parameter} -\command{Write Affine Parameter;} - -\vfill -\newpage - -\subsection{Case Sensitivity} -\label{case} - -Usually \reduce\ is case insensitive which means for example -that expression \comm{x-X} will be evaluated by \reduce\ as zero. -On the contrary all coordinates, constants and functions in \grg\ are -case sensitive, e.g. \comm{alpha}, \comm{Alpha} and \comm{ALPHA} -are all different. Notice that commands and switches in \grg\ -3.2 remain case insensitive. -\index{Internal \reduce\ case} - -Therefore all predefined by \grg\ constants and -all built-in objects must be used exactly as they -presented in this manual \comm{GCONST}, \comm{SMASS} etc. -The situation with the constants and functions which predefined -by \reduce\ is different. The point is that in spite of its default -case insensitivity internally \reduce\ converts everything -into some default case which may be upper or lower. -Therefore depending on the particular \reduce\ system they -must be typed either as -\begin{listing} - E I PI INFINITY SIN COS ATAN -\end{listing} -or in lower case -\begin{listing} - e i pi infinity sin cos atan -\end{listing} -For the sake of definiteness throughout this book we chose -the first upper case convention. - -When \grg\ starts it informs you about internal case of -your particular \reduce\ system by printing the message -\begin{slisting} -System variables are upper-cased: E I PI SIN ... -\end{slisting} -or -\begin{slisting} -System variables are lower-cased: e i pi sin ... -\end{slisting} -You can find out about the internal case -using the command\cmdind{Status}\cmdind{Show Status} -\command{\opt{Show} Status;} - -\vfill -\newpage - - -\subsection{Complex Conjugation} - -By default all variables and functions in \grg\ are considered to be -real excluding the imaginary unit constant \comm{I} (or \comm{i} as -explained above). But if two identifiers differ only by the trailing -character \comm{\cc} they are considered as a pair of -complex variables which are conjugated to each other. -In the following example coordinates -\comm{z} and \comm{z\cc} comprise such a pair: -\begin{slisting} -<- Coordinates u, v, z, z~; - -z & z~ - conjugated pair. - -<- Re(z); - - z + z~ --------- - 2 - -<- Im(z~); - - I*(z - z~) ------------- - 2 -\end{slisting} - - - -\section{Objects} - -Objects play a fundamental role in \grg. They represent -mathematical quantities such as metric, connection, curvature -and any other spinor or tensor geometrical and physical fields -and equations. \grg\ has quite large number of built-in -objects and knows many formulas for their calculation. -But you are not obliged to use the built-in quantities -and can declare your own. The purpose of the declaration is -to tell \grg\ basic properties of a new quantity. - - -\subsection{Built-in Objects} - -\noindent -An object is characterized by the following properties and attributes: -\index{Built-in objects} -\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent\parsep=0mm} -\item Name -\item Identifier or symbol -\item Type of the component -\item List of indices -\item Symmetries with respect to index permutation -\item Density and pseudo-tensor property -\item Built-in ways of calculation -\item Value -\end{list} - -The object \emph{name} is a sequence of words which are -usually the common English name of corresponding quantity. -The name is case insensitive and is used to denote -a particular object in commands. -So called \emph{group names}\index{Group names} -refer to a collection of closely related objects. In particular -the name {\tt Curvature Spinors} (see page \pageref{curspincoll}) -refers to the irreducible components of the curvature tensor in -spinorial representation. -Actual content of the group may depend on the environment. -In particular the group {\tt Curvature Spinors} includes -three objects in the Riemann space (Weyl spinor, traceless -Ricci spinor and scalar curvature) while in the space with -torsion we have six irreducible curvature spinors. - -The object \emph{identifier} or \emph{symbol} is an identifier -which denotes the object in mathematical expressions. Object -symbols are case sensitive. - -The object \emph{type} is the type of its component: objects can be -scalar, vector or $p$-form valued. The \emph{density} and -\emph{pseudo-tensor} properties of the object characterizes its -behaviour under coordinate and frame transformations. - -Objects can have the following types of indices: -\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} -\item Upper and lower holonomic coordinate indices. -\item Upper and lower frame indices. -\item Upper and lower spinorial indices. -\item Upper and lower conjugated spinorial indices. -\item Enumerating indices. -\end{list} -The major part of \grg\ built-in objects has frame indices. -\seethis{See page \pageref{metric} about the frame in \grg.} -The frame in \grg\ can be arbitrary but you can easily specify -the frame to be holonomic or say orthogonal. Then built-in -object indices become holonomic or orthogonal respectively. - -\grg\ deals only with the SL(2,C) spinors which are restricted -to the 4-dimensional spaces of Lorentzian signature. -\seethis{See \pref{spinors} about the spinorial formalism in \grg.} -The corresponding SL(2,C) indices take values 0 and 1. -The conjugated indices are transformed with the help -of the complex conjugated SL(2,C) matrix. -If some spinor is totally symmetric in the group of $n$ spinorial -indices (irreducible spinor) then these indices can be -replaced by a single so called \emph{summed spinorial index} -of rank $n$ which take values from 0 to $n$. -The summed spinorial indices provide the most economic -way to store the irreducible spinor components. - -Enumerating indices just label a collection of -values and have no any covariant meaning. Accordingly there is -no difference between upper and lower enumerating indices. - -Notice that an index of any type in \grg\ always runs from -0 up to some maximal value which depend on the index type -and dimensionality: $d-1$ for frame and coordinate indices,\index{Dimension} -and $n$ the spinor indices of the rank $n$. - -\grg\ understands various types of index symmetries: -symmetry, antisymmetry, cyclic symmetry and Hermitian -symmetry. These symmetries can apply not only to single -indices but to any group of indices as well. -\index{Index symmetries}\index{Canonical order of indices} -\grg\ uses object symmetries to decrease the amount of memory -required to store the object components. It stores only components -with the indices in certain \emph{canonical} order -and any other component are automatically -restored if necessary by appropriate index permutation. -The canonical order of indices is defined as follows: -for symmetry, antisymmetry or Hermitian symmetry indices -are sorted in such a way that index values grows from -left to the right. For cyclic symmetry indices are shifted to -minimize the numerical value of the whole list of indices. - -Finally there are two special types of objects: equations -and connection 1-forms. -\index{Equations} -Equations have all the same properties as any -other object but in addition they have left and right hand side -and are printed in the form of equalities. -The connections are used by \grg\ to construct covariant derivatives. -\index{Connections}\seethis{See \pref{conn2} about the connections.} -There are only four types of connections: holonomic -connection 1-form, frame connection 1-form, spinor connection -1-form and conjugated spinor connection 1-form. - -Almost all built-in objects have associated built-in \emph{ways of -calculation} (one or several). -\index{Ways of calculation} -Each way is nothing but a formula which can be used -to obtain the object value. - -Every object can be in two states. Initially when \grg\ starts -all objects are in \emph{indefinite} state, i.e. nothing is known -about their value. \index{Object value} -Since \grg\ always works in some concrete frame and coordinate -system the object value is a table of the components. -As soon as the value of certain object -is obtained either by direct assignment or using some built-in -formula (way of calculation) \grg\ remember this value -and store it in some internal table. Later this value -can be printed, re-evaluated used in expression etc. -The object can be returned to its initial indefinite state -using the command \comm{Erase}.\cmdind{Erase} -\grg\ uses object symmetries to reduce total number of -components to store. - -The complete list of built-in \grg\ objects is given in -appendix C. The chapter 3 also describes built-in objects -but in the usual mathematical style. The equivalent commands -\cmdind{Show \parm{object}} -\command{Show \parm{object};\\\tt% -?~\parm{object};} -prints detailed information about the object \parm{object} -including object name, identifier, list of indices, -type of the component, current state (is the value of an -object known or not), symmetries and ways of calculation. -Here \parm{object} is either object name or its identifier. - -The command\cmdind{Show *} -\command{Show *;} -prints complete list of built-in object names. This list -is quite long and the command -\command{Show \parm{c}*;} -gives list of objects whose names begin with the character -\parm{c} (\comm{a}--\comm{z}). - -Finally the command \cmdind{Show All} -\command{Show All;} -prints list of objects whose values are currently known. - -Notice that some built-in objects has limited scope. -In particular some objects exists only in certain dimensionality, -the quantities which are specific to spaces with torsion -are defined iff switch \comm{TORSION} is turned on etc. - -Let us consider some examples. We begin with the -curvature tensor $R^a{}_{bcd}$ -\begin{slisting} -<- Show Riemann Tensor; - -Riemann tensor RIM'a.b.c.d is Scalar - Value: unknown - Symmetries: a(3,4) - Ways of calculation: - Standard way (D,OMEGA) -\end{slisting} -This object has name {\tt Riemann Tensor} and identifier -{\tt RIM}. The object is {\tt Scalar} (0-form) valued and -has four frame indices. Frame indices are denoted by the -lower-case characters and their upper or lower position -are denoted by \comm{'} or \comm{.} respectively. -The Riemann tensor is antisymmetric in two last indices -which is denoted by \comm{a(3,4)}. - -The curvature 2-form $\Omega^a{}_b$ -\begin{slisting} -<- ? OMEGA; - -Curvature OMEGA'e.f is 2-form - Value: unknown - Ways of calculation: - Standard way (omega) - From spinorial curvature (OMEGAU*,OMEGAD) -\end{slisting} -has name {\tt Curvature} and the identifier {\tt OMEGA} -and is 2-form valued. - -The traceless Ricci spinor (the quantity which is usually -denoted in the Newman-Penrose formalism as $\Phi_{AB\dot{C}\dot{D}}$) -\begin{slisting} -<- ? Traceless Ricci Spinor; - -Traceless ricci spinor RC.AB.CD~ is Scalar - Value: unknown - Symmetries: h(1,2) - Ways of calculation: - From spinor curvature (OMEGAU,SD,VOL) -\end{slisting} -Spinorial indices -are denoted by upper case characters with the trailing \comm{\cc} -for conjugated indices. Usual spinorial indices are denoted -by a \emph{single} upper case letter while summed indices -are denoted by several characters. Thus, the traceless Ricci -spinor has two summed spinorial indices -of rank 2 each taking the values from 0 to 2. The spinor -is hermitian \comm{h(1,2)}. - -The Einstein equation is an example of equation -\begin{slisting} -<- ? Einstein Equation; - -Einstein equation EEq.g.h is Scalar Equation - Value: unknown - Symmetries: s(1,2) - Ways of calculation: - Standard way (G,RIC,RR,TENMOM) -\end{slisting} -and 1-form $\Gamma^\alpha{}_\beta$ is an example of the connection \enlargethispage{2mm} -\begin{slisting} -<- Show Holonomic Connection; - -\reversemarginpar - -Holonomic connection GAMMA^x_y is 1-form Holonomic Connection - Value: unknown - Ways of calculation: - From frame connection (T,D,omega) -\end{slisting} -The coordinate indices are denoted by the lower-case -letters with labels \comm{\^} and \comm{\_} denoting -upper and lower index position respectively. -Notice that above the first ``{\tt Holonomic connection}'' is the -name of the object while second ``{\tt Holonomic Connection}'' -means that \grg\ recognizes it as the connection and will -use \comm{GAMMA} to construct covariant derivatives for quantities -having the coordinate indices. \seethis{See \pref{cder} about the covariant derivatives.} -You can define any number of other holonomic -connections and use them in the covariant derivatives -on the equal footing with the built-in object \comm{GAMMA}. - -\normalmarginpar - -The notation in which command \comm{Show} prints -information about a particular object is the same as in the -new object declaration and is explained in details below. - - -\subsection{Macro Objects} -\index{Macro Objects}\label{macro} - -There is also another class of built-in objects which are -called \emph{macro objects}. The main difference between the -usual and macro objects is that macro quantities has no -permanent storage to their components instead they are calculated -dynamically only when its component is required in some expression. -In addition -they do not have names and are denoted only by the identifier only. -Usually macro objects play auxiliary role. The complete -list of macro objects can be found in appendix B. - -The example of macro objects are the Christoffel symbols -of second and first kind $\{{}^\alpha_{\beta\gamma}\}$ -and $[{}_{\alpha,\beta\gamma}]$ having identifiers -\comm{CHR} and \comm{CHRF} respectively -\begin{slisting} -<- Show CHR; - -CHR^x_y_z is Scalar Macro Object - Symmetries: s(2,3) - -<- ? CHRF; - -CHRF_u_v_w is Scalar Macro Object - Symmetries: s(2,3) -\end{slisting} - - -\subsection{New Object Declaration} - -\grg\ has very large number of built-in quantities -but you are not obliged to use them in your calculations -instead you can define new quantities. The command\cmdind{New Object} -\command{New Object \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}};} -declares a new object. The words \comm{New} or \comm{Object} are -optional (but not both) so the above command are equivalent to -\command{Object \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}};\\\tt -New \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}}; } -Here \parm{ID} is an identifier of a new object. The identifier can -contain letters \comm{a}--\comm{z}, \comm{A}--\comm{Z} but neither -digits nor any other symbols. The identifier must be unique and cannot -coincide with the identifier of any other built-in or user-defined object. - -The \parm{ilist} is the list of indices having the form \label{indices} -\command{\rpt{\parm{ipos}\ \parm{itype}}} -where \parm{ipos} defines the index position and \parm{itype} -specifies its type. The coordinate holonomic and frame indices -are denoted by single lower-case letters with \parm{ipos} -\command{{\tt '}\rm\ \ upper frame index -\\{\tt .}\rm\ \ lower frame index -\\{\tt \^}\rm\ \ upper holonomic index -\\{\tt \_}\rm\ \ lower holonomic index} -The frame and holonomic indices in \grg\ take values from 0 to -$d-1$ where $d$ is the current space dimensionality.\index{Dimension} - -Spinorial indices are denoted by upper case letters -with trailing \comm{\cc} for conjugated spinorial indices: -\comm{A}, \comm{B\cc} etc. Summed spinorial index of rank $n$ is -denoted by $n$ upper-case letters. For example \comm{ABC} denotes -summed spinorial index of the rank 3 (runs from 0 to 3) -and \comm{AB\cc} denotes conjugated summed index of the rank 2 -(values 0, 1, 2). The upper position for spinorial indices -are denoted either by \comm{'} or \comm{\^} and lower one by -\comm{.} or \comm{\_}. - -Finally the enumerating indices are denoted by a single -lower-case letter followed either by digits or by \comm{dim}. -For example the index declared as \comm{i2} runs from 0 -to 2 while specification \comm{a13} denotes index whose -values runs from 0 to 13. -The specification \comm{idim} denotes enumerating index -which takes the values from 0 to $d-1$. -Upper of lower position for enumerating indices are identical, -thus in this case symbols \comm{' . \^ \_} are equivalent. - -The \parm{ctype} defines the type of new object component: -\command{Scalar \opt{Density \parm{dens}}\\\tt -\parm{p}-form \opt{Density \parm{dens}}\\\tt -Vector \opt{Density \parm{dens}}} -This part of the declaration can be omitted and then the object -is assumed to be scalar-valued. The \parm{dens} defines pseudo-scalar -and density properties of the object with respect to -coordinate and frame transformations: -\command{\opt{sgnL}\opt{*sgnD}\opt{*L\^\parm{n}}\opt{*D\^\parm{m}}} -where \comm{D} and \comm{L} is the coordinate transformation -determinant ${\rm det}(\partial x^{\alpha'}/\partial x^\beta)$ and -frame transformation determinant ${\rm det}(L^a{}_b)$ respectively. -If \comm{sgnL} or \comm{sgnD} is specified then under appropriate -transformation the object must be multiplied on the -sign of the corresponding determinant (pseudo tensor). -The specification \comm{L\^\parm{n}} or \comm{D\^\parm{m}} means -that the quantity must be multiplied on the appropriate -degree of the corresponding determinant (tensor density). -The parameters \parm{p}, \parm{n} and \parm{m} may be given -by expressions (must be enclosed in brackets) but value -of these expressions must be always integer and positive -in the case of \parm{p}. - -The symmetry specification \parm{slst} is a list -\command{\rpt{\parm{slst1}}} -where each element \parm{slst1} describes symmetries -for one group of indices and has the form -\command{\parm{sym}(\rpt{\parm{slst2}})} -The \parm{sym} determines type of the symmetry -\command{% -\tt s \ \rm symmetry \\ -\tt a \ \rm antisymmetry \\ -\tt c \ \rm cyclic symmetry \\ -\tt h \ \rm Hermitian symmetry} -and \parm{slst2} is either index number \parm{i} or list of -index numbers \comm{(\rpt{\parm{i}})} or another symmetry -specification of the form \parm{slst1}. Notice that $n$th -object index can be present only in one of the \parm{slst1}. - -Let us consider an object having four indices. -Then the following symmetry specifications are possible - -\begin{tabular}{ll} -\comm{s(1,2,3,4)} & total symmetry \\[1mm] -\comm{a(1,2),s(3,4)} & antisymmetry in first pair of indices and \\ - & symmetry in second pair \\[1mm] -\comm{s((1,2),(3,4))} & symmetry in pair permutation \\[1mm] -\comm{s(a(1,2),a(3,4))} & antisymmetry in first and second pair of indices \\ - & and symmetry in pair permutation -\end{tabular}\newline -The last example is the well known symmetry of Riemann curvature tensor. -The specification \comm{a(1,2),s(2,3)} is erroneous since -second index present in both parts of the specification -which is not allowed. - -Declaration for new equations is completely similar\cmdind{New Equation} -\command{\opt{New} Equation \parm{ID}\,\opt{\parm{ilst}}\,\opt{is \parm{ctype}}\,\opt{with \opt{Symmetries}\,\parm{slst}};} - -\grg\ knows four types of connections:\cmdind{New Connection} \label{conn2} -\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} -\item Frame Connection 1-form $\omega^a{}_b$ having first upper and second lower frame indices -\item Holonomic Connection 1-form $\Gamma^\alpha{}_\beta$ having first upper and second lower coordinate indices -\item Spinor Connection 1-form $\omega_{AB}$ with lower spinor index of rank 2 -\item Conjugated Spinor Connection $\omega_{\dot{A}\dot{B}}$ 1-form with lower conjugated spinor index of rank 2 -\end{list} -Each of these connections are used to construct covariant derivatives -with respect to corresponding indices. In addition they are properly -transformed under the coordinate change and frame rotation. -There are complete set of built-in connections but you can declare -a new one by the command -\command{% -\opt{New} Connection \parm{ID}'a.b \opt{is 1-form};\\\tt -\opt{New} Connection \parm{ID}\^m\_n \opt{is 1-form};\\\tt -\opt{New} Connection \parm{ID}.AB\ \opt{is 1-form};\\\tt -\opt{New} Connection \parm{ID}.AB\cc\ \opt{is 1-form};} -Notice that any new connection must belong to one of the listed -above types and have indicated type and position of indices. This -representation of connection is chosen in \grg\ for the sake of -definiteness. - -There is one special case when new object can be declared -without explicit \comm{New Object} declaration. Let us -consider the following example: -\begin{slisting} -<- Coordinates t, x, y, z; -<- www=d x; -<- Show www; - -www is 1-form - Value: known -\end{slisting} -If we assign the value to some identifier \parm{id} -(\comm{www} in our example) -\seethis{See page \pageref{assig} about assignment command.} -and this identifier is not reserved yet by any other object then -\grg\ automatically declares a new object without indices -labeled by the identifier \parm{id} and having the type -of the expression in the right-hand side of the assignment -(1-form in our example). Notice that the \parm{id} must not include -digits since digits represent indices and any new object -with indices must be declared explicitly. - -The command -\command{Forget \parm{ID};} -completely removes the user-defined object with the -identifier \parm{ID}. - -Finally let us consider some examples: -\begin{slisting} -<- Coordinates t, x, y, z; -<- New RNEW'a.b_c_d is scalar density sgnD with a(3,4); -<- Show RNEW; - -RNEW'a.b_x_y is Scalar Density sgnD - Value: unknown - Symmetries: a(3,4) - -<- Null Metric; -<- Connection omnew.AA; -<- Show omnew; - -omnew.AB is 1-form Spinor Connection - Value: unknown -\end{slisting} -Here the first declaration defines a new scalar valued pseudo tensor -$\mbox{\comm{RNEW}}^a{}_{b\gamma\delta}$ which is antisymmetric -in the last pair of indices. Second declaration introduce new spinor -connection \comm{omnew}. Notice that new connection is automatically -declared 1-form and the type of connection is derived by the -type of new object indices (lower spinorial index of rank 2 in our -example). - - -\section{Assignment Command} -\index{Assignment (command)}\label{assig} - -The assignment command sets the value to the particular -components of the object. In general it has the form -\command{\opt{\parm{Name}} \rpt{\parm{comp} = \parm{expr}};} -or for equations -\command{\opt{\parm{Name}} \rpt{\parm{comp} = \parm{lhs}=\parm{rhs}};} -Here \parm{Name} is the optional object name. If the object -has no indices then \parm{comp} is the object identifier. -If the object has indices then \parm{comm} consist of identifier -with additional digits denoting indices. -For example the following command assigns standard spherical flat -value to the frame $\theta^a$ -\begin{listing} - Frame - T0 = d t, - T1 = d r, - T2 = r*d theta, - T3 = r*SIN(theta)*d phi; -\end{listing} -and the command -\begin{listing} - RIM0123 = 100; -\end{listing} -assigns the value to the $R^0{}_{123}$ component of the Riemann tensor. -Notice that in this notation each digit is considered as one index, -thus it does not work if the value of some index is greater than 9 -(e.g. if dimensionality is 10 or greater). In this case another -notation can be used in which indices are added to the object -identifier as a list of digits enclosed in brackets -\command{\opt{\parm{Name}} \parm{ID}(\rpt{\parm{n}})~= \parm{expr};} -In particular the command -\begin{listing} - RIM(0,1,2,3) = 100; -\end{listing} -is equivalent to the example above. - -The assignment set value only to the certain components of an object -leaving other components unchanged. But if before assignment -the object was in indefinite state (no value is known) then assignment -turns it to the definite state and all other components of the object -are assumed to be zero. - -The digits standing for object indices in the left-hand side -of an assignment can be replaced by identifiers -\index{Assignment (command)!tensorial} -\command{\opt{\parm{Name}} \parm{ID}(\rpt{\parm{id}})~= \parm{expr};} -Such assignment is called \emph{tensorial} one. -For example the following tensorial assignment set the value to the -curvature 2-form $\Omega^a{}_b$ -\begin{listing} - OMEGA(a,b) = d omega(a,b) + omega(a,m){\w}omega(m,b); -\end{listing} -This command is equivalent to $d\times d$ of assignments where \comm{a} -and \comm{b} take values from 0 to $d-1$ ($d$ is the space dimensionality).\index{Dimension} -Notice that identifiers in the left-hand side of tensorial assignment -must not coincide with any predefined or declared by the user -constant or coordinate. It is possible to mix digits and identifiers: -\begin{listing} - FT(0,a) = 0; -\end{listing} -Here \comm{FT} is identifier of the built-in object -{\tt EM Tensor} which is the electromagnetic strength tensor $F_{ab}$ -and this command sets the electric part of the tensor to zero. - -The assignment command takes into account symmetries of the -objects. For example {\tt EM Tensor} is antisymmetric -and in order to assign value say to the components $F_{01}=-F_{10}$ -it suffices to do this just for one of them -\begin{slisting} -<- Coordinates t, x, y, z; -<- EM Tensor FT01=111, FT(3,2)=222; -<- Write FT; -EM tensor: - -FT = 111 - t x - -FT = -222 - y z -\end{slisting} -We can see that \grg\ automatically transforms indices to the -\emph{canonical} order. This rule works in the case or -tensorial assignment as well -\begin{slisting} -<- Coordinates t, x, y, z; -<- Function ff; -<- EM Tensor FT(a,b)=ff(a,b); -<- Write FT; -EM tensor: - -FT = ff(0,1) - t x - -FT = ff(0,2) - t y - -FT = ff(0,3) - t z - -FT = ff(1,2) - x y - -FT = ff(1,3) - x z - -FT = ff(2,3) - y z - -<- FT(2,1); - - - ff(1,2) -\end{slisting} -In this case both parameters \comm{a} and \comm{b} runs from 0 to 3 -but \grg\ assigns the value only to the components -having indices in the canonical order \comm{a}$<$\comm{b}. -\grg\ follows this rule also if in the left-hand -side of tensorial assignment digits are mixed with -parameters which may sometimes produce unexpected result: -\begin{slisting} -<- Coordinates t, x, y, z; -<- Function ee; -<- FT(0,a)=ee(a); -<- Write FT; -EM tensor: - -FT = ee(1) - t x - -FT = ee(2) - t y - -FT = ee(3) - t z - -<- Erase FT; -<- FT(3,a)=ee(a); -<- Write FT; -EM tensor: - -0 -\end{slisting} -Observe the difference between these two assignments (the command -\comm{Erase FT;} destroys the previously assigned value). -In fact second assignment assigns no values since -\comm{3} and \comm{a} are not in the canonical order -\comm{3}$\geq$\comm{a} for \comm{a} running from 0 to 3. -Notice the difference from the case when all indices in -the left-hand side are given by the explicit numerical values. -In this case \grg\ automatically transforms the indices to their -canonical order and \comm{FT(3,2)=222;} is equivalent -to \comm{FT(2,3)=-222;}. - - -Finally there is one more form of the tensorial assignment -which can be applied to the summed spinorial indices. -\index{Assignment (command)!summed spinor indices} -Let us consider the spinorial analogue of electromagnetic strength -tensor $\Phi_{AB}$. This spinor is irreducible (i.e. symmetric in $\scriptstyle AB$). -The corresponding \grg\ built-in object {\tt Undotted EM Spinor} -(identifier \comm{FIU}) has one summed spinorial index of rank 2. -Let us consider two different assignment commands -\begin{slisting} -<- Coordinates u, v, z, z~; - -z & z~ - conjugated pair. - -<- Null Metric; -<- Function ee; -<- FIU(a)=ee(a); -<- Write FIU; -Undotted EM spinor: - - - -FIU = ee(0) - 0 - -FIU = ee(1) - 1 - -FIU = ee(2) - 2 - -<- Erase FIU; -<- FIU(a+b)=ee(a,b); -<- Write FIU; -Undotted EM spinor: - -FIU = ee(0,0) - 0 - -FIU = ee(0,1) - 1 - -FIU = ee(1,1) - 2 -\end{slisting} -In the first case \comm{a} is treated as a summed index -and runs from 0 to 2 but in the second case \comm{a} and \comm{b} -are considered as usual single SL(2,C) spinorial indices -each having values 0 and 1. - -The notation for the object components in the left-hand -side of assignment do not distinguishes upper and lower -indices. Actually the indices are always assumed to be in -the default position. -You can always check the default index types and positions -using the command \comm{Show \parm{object};}.\cmdind{Show \parm{object}} -For example the {\tt Riemann Tensor} has first upper and -three lower frame indices and the command \comm{RIM0123=100;} -and \comm{RIM(0,1,2,3)=100;} both assign value to the -$R^0{}_{123}$ component of the tensor where indices are -represented with respect to the current frame. - - -\section{Geometry} - -The number of built-in objects in \grg\ is rather large. -They all described in chapter 3 and appendices B and C. -In this section we consider only the most important ones. - -\subsection{Metric, Frame and Line-Element} -\index{Metric}\index{Frame} -\label{metric} - -The line-element in \grg\ is defined by the -following equation -\begin{equation} -ds^2 = g_{ab}\,\theta^a\!\otimes\theta^b -\end{equation} -where $\theta^a=h^a_\mu dx^\mu$ is the frame 1-form and $g_{ab}$ is the -frame metric. The corresponding built-in objects are -\comm{Frame} (identifier \comm{T}) and \comm{Metric} -(identifier \comm{G}). There are also the ``inverse'' -counterparts $\partial_a=h_a^\mu\partial_\mu$ ({\tt Vector Frame}, -identifier \comm{D}) and $g^{ab}$ ({\tt Inverse Metric}, identifier -\comm{GI}). To determine the metric properties of the space -you can assign some values to both the metric and the frame. -There are two well known special cases. First is the usual -coordinate formalism in which frame is holonomic $\theta^a=dx^\alpha$. -In this case there is no difference between frame and coordinate -indices. Another representation is known as the tetrad (in dimension 4) -formalism. In this case frame metric equals to some constant -matrix $g_{ab}=\eta_{ab}$ and significant information about -line-element ``is encoded'' in the frame. - -In general both metric and frame can be nontrivial but not -necessarily. If no any value is given by user to the frame -when \grg\ automatically assumes that frame is \emph{holonomic} -\index{Frame!default value} -\begin{equation} -\theta^a=dx^\alpha -\end{equation} -Thus if we assign the value to metric only we automatically -get standard coordinate formalism. On the contrary if -no value is assigned to the metric then \grg\ automatically -assumes\index{Signature} \label{defaultmetric} -\index{Metric!default value} -\begin{equation} -g_{ab} = {\rm diag}(+1,-1,\dots) -\end{equation} -where $+1$ and $-1$ on the diagonal of the matrix -correspond to the current signature specification. - -Notice that current signature is printed among other -information by the command\cmdind{Show Status}\cmdind{Status} -\command{\opt{Show} Status;} -and current line-element is printed by the command -\cmdind{ds2} -\command{ds2;} -or equivalently\cmdind{Line-Element} -\command{Line-Element;} - -Finally if neither frame nor metric are specified by user -then both these quantities acquire default value and we -automatically obtain flat space of the default signature: -\begin{slisting} -<- Dimension 4 with Signature(-,+,+,+); -<- Coordinates t, x, y, z; -<- ds2; -Assuming Default Metric. -Metric calculated By default. 0.05 sec -Assuming Default Holonomic Frame. -Frame calculated By default. 0.05 sec - - 2 2 2 2 2 - ds = - d t + d x + d y + d z - -\end{slisting} - - -\subsection{Spinors} -\label{spinors} - -Spinorial representations exist in spaces of various dimensions -and signatures but in \grg\ spinors are restricted -to the 4-dimensional spaces of Lorentzian signature ${\scriptstyle(-,+,+,+)}$ -or ${\scriptstyle(+,-,-,-)}$ only. Another restriction is that in the -spinorial formalism the metric must be the \index{Metric!Standard Null} -\emph{standard null metric}: -\index{Standard null metric}\index{Spinors}\index{Spinors!Standard null metric} -\begin{equation} -g_{ab}=g^{ab}=\pm\left(\begin{array}{rrrr} -0 & -1 & 0 & 0 \\ --1 & 0 & 0 & 0 \\ -0 & 0 & 0 & 1 \\ -0 & 0 & 1 & 0 -\end{array}\right) -\end{equation} -where upper sign correspond to the signature ${\scriptstyle(-,+,+,+)}$ and -lower sign to the signature ${\scriptstyle(+,-,-,-)}$. -There is special command\cmdind{Null Metric} -\command{Null Metric;} -which assigns this standard value to the metric. - -Thus spinorial frame (tetrad) in \grg\ must be null -\begin{equation} -ds^2 = \pm(-\theta^0\!\otimes\theta^1 --\theta^1\!\otimes\theta^0 -+\theta^2\!\otimes\theta^3 -+\theta^3\!\otimes\theta^2) -\end{equation} -and conjugation rules for this tetrad must be -\begin{equation} -\overline{\theta^0}=\theta^0,\quad -\overline{\theta^1}=\theta^1,\quad -\overline{\theta^2}=\theta^3,\quad -\overline{\theta^3}=\theta^2 -\end{equation} - -For the sake of efficiency the sigma-matrices $\sigma^a\!{}_{A\dot{B}}$ -for such a tetrad are chosen in the simplest form. The only -nonzero components of the matrices are\index{Sigma matrices} -\begin{eqnarray} -&&\sigma_0{}^{1\dot{1}}= -\sigma_1{}^{0\dot{0}}= -\sigma_2{}^{1\dot{0}}= -\sigma_3{}^{0\dot{1}}=1 \\[1mm] && -\sigma^0{}_{1\dot{1}}= -\sigma^1{}_{0\dot{0}}= -\sigma^2{}_{1\dot{0}}= -\sigma^3{}_{0\dot{1}}=\mp1 -\end{eqnarray} - - -\subsection{Connection, Torsion and Nonmetricity} -\label{conn} - -As was explained above \grg\ recognizes four types of connections: -holonomic $\Gamma^\alpha{}_\beta$, frame $\omega^a{}_b$, -spinorial $\omega_{AB}$ and conjugated spinorial -$\omega_{\dot{A}\dot{B}}$. Accordingly there are four -built-in objects: {\tt Holonomic Connection} (id. \comm{GAMMA}), -{\tt Frame Connection} (id. \comm{omega}), {\tt Undotted Connection} -(id. \comm{omegau}), {\tt Dotted Connection} (id. \comm{omegad}). -Connections are used in \grg\ in covariant derivatives. In addition -they are properly transformed under frame and coordinate -transformations. - -By default the connection in \grg\ are assumed to be Riemannian. -In particular in this case holonomic connection is nothing but -Christoffel symbols $\Gamma^\alpha{}_\beta= -\{{}^\alpha_{\beta\pi}\}dx^\pi$. -If it is necessary to work with torsion and/or nonmetricity -\swind{TORSION}\swind{NONMETR} -then the switches \comm{TORSION} and/or \comm{NONMETR} -must be turned on. \seethis{See \pref{conn2} about the built-in connections.} -In this case the Riemannian analogues -or the aforementioned four connections are available as well. - - -\section{Expressions} - -Expressions in \grg\ can be algebraic (scalar), vector or -p-form valued. \grg\ knows all the usual mathematical operations -on algebraic expressions, exterior forms and vectors. - -\subsection{Operations and Operators} - -The operations known to \grg\ are presented in the form of the table. -Operations are subdivided into six groups separated by horizontal -lines. Operations in each group have equal level of precedence and -the precedence level decreases from the top to the bottom of the table. -As in usual mathematical notation we can use brackets \verb"( )" -to change operation precedence. - -Other constructions which can be used in expression are -described below. - -\begin{table} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline -{\bf Operation} & {\bf Description} & {\bf Grouping} \\ -\hline -{\tt [$v_1$,$v_2$]} & Vector bracket & \\ -\hline -{\tt @} $x$ & Holonomic vector $\partial_x$ & \\ -\cline{1-2} -{\tt d} $a$ & Exterior differential & \\ -{\tt d} $\omega$ & & - {\tt d} \cc$a$ $\Leftrightarrow$ {\tt (d(}\cc$a${\tt))} \\ -\cline{1-2} -{\tt \dd} $a$ & Dualization & \\ -{\tt \dd} $\omega$ & & \\ -\cline{1-2} -{\tt \cc} $e$ & Complex conjugation & \\ -\hline -$a_1${\tt **}$a_2$ & Exponention & \\ -$a_1${\tt\^} $a_2$ & & \\ -\hline -$e$\ {\tt /}\ $a$ & Division & - $e${\tt /}$a_1${\tt /}$a_2$ $\Leftrightarrow$ -{\tt (}$e${\tt /}$a_1${\tt )/}$a_2$ \\ -\hline -$a$\ {\tt *}\ $e$ & Multiplication & \\ -\cline{1-2} -$v$\ {\tt |}\ $a$ & Vector acting on scalar & -$v$\ii$\omega_1$\w$\omega_2${\tt *}$a$ \\ -\cline{1-2} -$v$\ \ip\ $\omega$ & Interior product & $\Updownarrow$ \\ -\cline{1-2} -$v_1$\ {\tt.}\ $v_2$& Scalar product & -$v$\ii{\tt (}$\omega_1$\w{\tt(}$\omega_2${\tt *}$a${\tt ))} \\ -$v$\ {\tt.}\ $o$ & & \\ -$o_1$\ {\tt.}\ $o_2$& & \\ -\cline{1-2} -$\omega_1$\ \w\ $\omega_2$ & Exterior product & \\ -\hline -{\tt +}\ $e$ & Prefix plus & \\ -\cline{1-2} -{\tt -}\ $e$ & Prefix minus & \\ -\cline{1-2} -$e_1$\ {\tt +}\ $e_2$ & Addition & \\ -\cline{1-2} -$e_1$\ {\tt -}\ $e_2$ & Subtraction & \\ -\hline -\end{tabular} -\end{center} -\label{operators} -\caption{Operation and operators. Here: -$e$ is any expression, -$a$ is any scalar valued (algebraic) expressions, -$v$ is any vector valued expression, -$x$ is a coordinate, -$o$ is any 1-form valued expression, -$\omega$ is any form valued expression.} -\end{table} - - - -\subsection{Variables and Functions} - -Operator listed in the table 2.2 act on -the following types of the operands: -\begin{itemize} -\item[(i)] integer numbers (e.g. {\tt 0}, {\tt 123}), -\item[(ii)] symbols or identifiers (e.g. {\tt I}, {\tt phi}, {\tt RIM0103}), -\item[(iii)] functional expressions (e.g. {\tt SIN(x)}, {\tt G(0,1)} etc). -\end{itemize} - -Valid identifier must belong to one of the following types: -\begin{itemize} -\item Coordinate. -\item User-defined or built-in constant. -\item Function declared with the implicit dependence list. -\item Component of an object. -\end{itemize} - -Any valid functional expression must belong to one of the following types: -\itemsep=0.5mm -\begin{itemize} -\item User-defined function. -\item Function defined in \reduce\ (operator). -\item Component of built-in or user-defined object in functional notation. -\item Some special functional expressions listed below. -\end{itemize} - -\subsection{Derivatives} - -The derivatives in \grg\ and \reduce\ are written as -\command{DF(\parm{a},\rpt{\parm{x}\opt{,\parm{n}}})} -where \parm{a} is the differentiated expression, \parm{x} is -the differentiation variable and integer number \parm{n} is -the repetition of the differentiation. For example -\[ -\mbox{\tt DF(f(x,y),x,2,y)}=\frac{\partial^3f(x,y)}{\partial^2x\partial y} -\] - -There are also another type of derivatives -\command{DFP(\parm{a},\rpt{\parm{x}\opt{,\parm{n}}})} -\seethis{See section \ref{genfun} about the generic functions.} -They are valid only after {\tt Generic Function} -declaration if the package \file{dfpart} -is installed on your system. - -\subsection{Complex Conjugation} - -Symbol \comm{\cc\cc} in the sum of terms is an abbreviation: -\command{% -\tt $e$ + \cc\cc\ $=$\ $e$ + \cc$e$ \\ -\tt $e$ - \cc\cc\ $=$\ $e$ - \cc$e$ } - -Functions \comm{Re} and \comm{Im} gives real and imaginary -parts of an expression: -\command{% -\tt Re($e$)\ $=$\ ($e$+\cc$e$)/2 \\ -\tt Im($e$)\ $=$\ I*(-$e$+\cc$e$)/2} -\subsection{Sums and Products} -The following expressions represent sum and product -\command{Sum(\rpt{\parm{iter}},\parm{e})\\\tt -Prod(\rpt{\parm{iter}},\parm{e})} -where \parm{e} is the summed expression and \parm{iter} -defines summation variables. -The range of summation can be \label{iter} -specified by two methods. First ``long'' notation is -\command{\parm{id} = \parm{low}..\parm{up}} -and the identifier \parm{id} runs from \parm{low} up to -\parm{up}. Both \parm{low} and \parm{up} can be given -by arbitrary expressions but value of these expressions -must be integer. The \parm{low} can be omitted -\command{\parm{id} = \parm{up}} -and in this case \parm{id} runs from 0 to \parm{up}. -The identifier \parm{id} should not coincide with any -built-in or user-defined variable. - - -In ``short'' notation \parm{iter} is just identifier \label{siter} -\parm{id} and its range is determined using -the following rules -\begin{list}{$\bullet$}{\labelwidth=4mm\leftmargin=\parindent} -\item Mixed letter-digit \parm{id} runs from 0 to $d-1$ - where $d$ is the space dimensionality. -\begin{verbatim} - Aid j2s -\end{verbatim} -\item The \parm{id} consisting of lower-case letters runs from - $0$ to $d-1$ -\begin{verbatim} - j a abc kkk -\end{verbatim} -\item The \parm{id} consisting of upper-case letters runs from - $0$ to the number of letters in \parm{id}, e.g. the following - identifiers run from 0 to 1 and from 0 to 3 respectively -\begin{verbatim} - B ABC -\end{verbatim} -\item Letters with one trailing digit run from 0 to the value - of this digit. Both \parm{id} below runs from 0 to 3: -\begin{verbatim} - j3 A3 -\end{verbatim} -\item Letters with two digits run from the value of the - first digit to the value of the second digit. The \parm{id} below - run from 2 to 3: -\begin{verbatim} - j23 A23 -\end{verbatim} -\item Letters with 3 or more digits are incorrect -\begin{verbatim} - j123 -\end{verbatim} -\end{list} - -Two or more summation parameters are separated either -by commas or by one of the relational operators -\begin{listing} - < > <= => -\end{listing} -This means that only the terms satisfying these relations -will be included in the sum. For example -\[ -\mbox{\tt Sum(i24<=ABC,k=1..d-1,f(i24,ABC,k))} = -\sum_{i=2}^{4} \sum_{\scriptstyle a=0\atop\scriptstyle i\leq a}^{3} \sum^{d-1}_{k=1} f(i,a,k) -\] - -\enlargethispage{5mm} - -\grg's \comm{Sum} and \comm{Prod} -\seethis{Use \comm{SUM}, \comm{PROD} or \comm{sum}, \comm{prod} -depending on \reduce\ internal case as explained on page -\pageref{case}.} -should not be confused with \reduce's \comm{SUM} and \comm{PROD} -which are also available in \grg. \grg's \comm{Sum} apply -to any scalar, vector or form-valued expressions and always -expanded by \grg\ into the appropriate explicit sum of terms. On the -contrary \comm{SUM} defined in \reduce\ can be applied to the -algebraic expressions only. \grg\ leaves such expression unchanged -and passes -it to the \reduce\ algebraic evaluator. Unlike \comm{Sum} the -summation limits in \comm{SUM} can be given by algebraic -expressions. If value of these expressions is integer then -result of the \comm{SUM} will be the same as for \comm{Sum} -but if summation limits are symbolic sometimes \reduce\ is capable -to find a closed expression for such a sum but not always. -See the following example -\begin{slisting} -<- Coordinates t, x, y, z; -<- Function f; -<- Constants n, m; -<- Sum(k=1..3,f(k)); - -f(3) + f(2) + f(1) - -<- SUM(f(n),n,1,3); - -f(3) + f(2) + f(1) - -<- SUM(n,n,1,m); - - m*(m + 1) ------------ - 2 - -<- SUM(f(n),n,1,m); - -SUM(f(n),n,1,m) -\end{slisting} - -\newpage - -\subsection{Einstein Summation Rule} - -According to the Einstein summation rule if \grg\ encounters -some unknown repeated identifier \parm{id} then summation over this -\parm{id} is performed. The range of the summation variable -is determined according to the ``short'' notation explained in -the previous section. - - -\subsection{Object Components and Index Manipulation} - -The components of built-in or user-defined object can be -denoted in expressions by two methods which are -similar to the notation used in the left-hand side of the -assignment command. The first method uses the object identifier -with additional digits denoting the indices {\tt T0}, {\tt RIM0213}. -The second method uses the functional -notation {\tt T(0)}, {\tt RIM(0,2,1,3)}, {\tt OMEGA(j,k)}. - -In functional notation the default index type and position -\index{Index manipulations} -can be changed using the markers: {\tt '} upper frame, -{\tt .} lower frame, {\tt \^} upper holonomic, {\tt \_} lower -holonomic. For example expression {\tt RIM(a,b,m,n)} -gives components of Riemann tensor with the default indices -$R^a{}_{bmn}$ (first upper frame and three lower frame indices) -while expression {\tt RIM('a,'b,\_m,\_n)} gives -$R^{ab}{}_{\mu\nu}$ with two upper frame and two lower coordinate -indices. For enumerating indices position markers are ignored -and only {\tt '} and {\tt .} works for spinorial indices. - -In the spinorial formalism -\seethis{See \pref{spinors} about spinorial formalism.} -each frame index can be replaced by a pair if spinorial indices -according to the formulas: -\[ -A^a\sigma_a{}^{B\dot{D}}=A^{B\dot{D}},\qquad -B_a\sigma^a\!{}_{B\dot{D}}=B_{B\dot{D}} -\] -Accordingly any frame index can be replaced by a pair of -spinorial indices. -\label{sumspin} -Similarly one summed spinorial index or rank $n$ can be -replaced by $n$ single spinor indices. -There is only one restriction. If an object has several -frame and/or summed spinorial indices then \emph{all} -must be represented in such expanded form. -In the following example the null frame $\theta^a$ -is printed in the usual and spinorial $\theta^{B\dot C}$ -representations. The relationship -$\theta^a\sigma_a{}^{B\dot C}-\theta^{B\dot C}=0$ is -verifies as well -\begin{slisting} -<- Coordinates u, v, z, z~; - -z & z~ - conjugated pair. - -<- Null Metric; -<- Frame T(a)=d x(a); -<- ds2; -\newpage - 2 - ds = (-2) d u d v + 2 d z d z~ - -<- T(a); - -a=0 : d u - -a=1 : d v - -a=2 : d z - -a=3 : d z~ - -<- T(B,C); - -B=0 C=0 : d v - -B=0 C=1 : d z~ - -B=1 C=0 : d z - -B=1 C=1 : d u - -<- T(a)*sigmai(a,B,C)-T(B,C); - -0 -\end{slisting} - - -\subsection{Parts of Equations and Solutions} -\index{Equations!in expressions} - -The functional expressions -\command{LHS(\parm{eqcomp})\\\tt -RHS(\parm{eqcomp})} -give access to the left-hand and right-hand side of an -equation respectively. Here \parm{eqcomp} is the -component of the equation as explained in the -previous section. - -The \comm{LHS}, \comm{RHS} also provide access to the \parm{n}'th -\seethis{See page \pageref{solutions} about solutions.} -solution if \parm{eqcomp} is \comm{Sol(\parm{n})}. - - -\subsection{Lie Derivatives} -\index{Lie derivatives} - -The Lie derivative is given by the expression -\command{Lie(\parm{v},\parm{objcomp})} -where \parm{objcomp} is the component of an object in -functional notation. For example the following -expression is the Lie derivative of the metric $\pounds_vg_{ab}$ -\begin{listing} - Lie(vec,G(a,b)); -\end{listing} -The index manipulations in the Lie derivatives are permitted. -In particular the expression -\begin{listing} - Lie(vec,G(^m,b)); -\end{listing} -is the Lie derivative of the frame $\pounds_vg^\mu{}_{b} -\equiv \pounds_vh^\mu_a$ -and must vanish. - - - - -\subsection{Covariant Derivatives and Differentials} -\index{Covariant derivatives}\index{Covariant differentials} -\label{cder} - -The covariant differential -\command{Dc(\parm{objcomp}\opt{{\upshape\tt ,}\rpt{\parm{conn}}})} -and covariant derivative -\command{Dfc(\parm{v},\parm{objcomp}\opt{{\upshape\tt ,}\rpt{\parm{conn}}})} -Here \parm{objcomp} is an object component in functional notation -and \parm{v} is a vector-valued expression. -The optional parameters \parm{conn} are the identifiers of -connections. -\seethis{See page \pageref{conn} about the built-in connections.} -If \parm{conn} is omitted then \grg\ uses default -connection for each type of indices: frame, coordinate, -spinor and conjugated spinor. If \parm{conn} is indicated -then \grg\ uses this connection instead of default one -for appropriate type of indices. For example expression -\begin{listing} - Dc(OMEGA(a,b)) -\end{listing} -is the covariant differential of the curvature 2-form $D\Omega^a{}_b$. -This expression should vanish in Riemann space and should be -proportional to the torsion in Riemann-Cartan space. -Here \grg\ will use default object {\tt Frame connection} -(id. \comm{omega}). The expression -\begin{listing} - Dc(OMEGA(a,b),romega) -\end{listing} -is similar but it uses another built-in connection -{\tt Riemann frame connection } (id. \comm{romega}) which -are different if torsion or nonmetricity are nonzero. -The index manipulations are allowed in the covariant derivatives. -For example the expression -\begin{listing} - Dfc(v,RIC(\^m,\_n)) -\end{listing} -gives the covariant derivative of the curvature of the -Ricci tensor with first coordinate upper and second coordinate lower -indices $\nabla_vR^\mu{}_\nu$. - -\subsection{Symmetrization} - -The functional expressions works iff the switch \swind{EXPANDSYM} -\comm{EXPANDSYM} is on -\command{% -Asy(\rpt{\parm{i}},\parm{e})\\\tt -Sy(\rpt{\parm{i}},\parm{e})\\\tt -Cy(\rpt{\parm{i}},\parm{e})} -They produce antisymmetrization, symmetrization and cyclic symmetrization -of the expression \parm{e} with respect to \parm{i} without -corresponding $1/n$ or $1/n!$. - - -\subsection{Substitutions} -\index{Substitutions}\label{subs} - -The expression -\command{SUB(\rpt{\parm{sub}},\parm{e})} -is similar to the analogous expression in \reduce\ with two -generalizations: (i) it applies not only to algebraic -but to form and vector valued expression \parm{e} as well, -\seethis{See page \pageref{solutions} about solutions.} -(ii) as in {\tt Let} command \parm{sub} can be either -the relation {\tt \parm{l}\,=\,\parm{r}} or solution -{\tt Sub(\parm{n})}. - - -\subsection{Conditional Expressions} -\index{Conditional expressions}\index{Boolean expressions} - -The conditional expression -\command{If(\parm{cond},\parm{e1},\parm{e2})} -chooses \parm{e1} or \parm{e2} depending on the value of the -boolean expression \parm{cond}. - -Boolean expression appears in (i) the conditional expression -\label{bool} -{\tt If}, (ii) in {\tt For all Such That} substitutions. -Any nonzero expression is considered as {\bf true} and -vanishing expression as {\bf false}. Boolean expressions -may contain the following usual relations and logical -operations: {\tt < > <= >= = |= not and or}. They also may -contain the following predicates \vspace*{2mm} - -\begin{tabular}{|l|l|} -\hline -\tt OBJECT(\parm{obj}) & Is \parm{obj} an object identifier or not \\ -\hline -\tt ON(\parm{switch}) & Test position of the \parm{switch} \\ -\tt OFF(\parm{switch}) & \\ -\hline -\tt ZERO(\parm{object}) & Is the value of the \parm{object} zero or not \\ -\hline -\tt HASVALUE(\parm{object}) & Whether the \parm{object} has any value or not \\ -\hline -\tt NULLM(\parm{object}) & Is the \parm{object} the standard null metric \\ -\hline -\end{tabular} \vspace*{2mm} \newline -Here \parm{object} is an object identifier. - -The expression \comm{ERROR("\parm{message}")} causes an error -with the \comm{"\parm{message}"}. It can be used -to test any required conditions during the batch file execution. - - -\subsection{Functions in Expressions} - -Any function which appear in expression must be -either declared by the \comm{Function} declaration or -be defined in \reduce\ (in \reduce\ functions are called -operators). In general arguments of functions in \grg\ must be -algebraic expression with one exception. If one (and only one) -argument of some function $f$ is form-valued $\omega=a d x + b d y$ then -\grg\ applies $f$ to the algebraic -multipliers of the form $f(\omega) = f(a) d x+ f(b) d y$. -The same rule works for vector-valued arguments. -Let us consider the example in the \reduce\ -operator \comm{LIMIT} is applied to the -form-valued expression -\begin{slisting} -<- Coordinates t, x, y, z; -<- www=(x+y)\^2/(x\^2-1)*d x+(x+y)/(x-z)*d y; -<- www; - - 2 2 - x + 2*x*y + y x + y -(-----------------) d x + (-------) d y - 2 x - z - x - 1 - -<- LIMIT(www,x,INFINITY); - - d x + d y -\end{slisting} - -I would like to remind also that depending on the -particular \reduce\ system \reduce\ operators must be -used in \grg\ in upper \comm{LIMIT} or lower case \comm{limit}. -See page \pageref{case} for more details. - -Any function or operator defined in the \reduce\ package -can be used in \grg\ as well. Some examples are -considered in section \ref{packages}. - - -\subsection{Expression Evaluation} -\index{Expression evaluation} - -\grg\ evaluates expressions in several steps: - -(1) All \grg-specific constructions such as -\comm{Sum}, \comm{Prod}, \comm{Re}, \comm{Im} etc are -explicitly expanded. - -(2) If expression contains components of some built-in -or user defined object they are replaced by the appropriate value. -If the object is in indefinite state -\seethis{See page \pageref{find} about the \comm{Find} command.} -(no value of the object is known) then \grg\ tries to -calculate its value by the method used by the \comm{Find} command. -The automatic object calculation can be prevented by -\swind{AUTO} -turning the switch \comm{AUTO} off. -If due to some reason the object cannot be calculated then -expression evaluation is terminated with the error message. - -(3) After all object components are replaced by their -values \grg\ performs all ``geometrical'' operations: exterior -and interior products, scalar products etc. If expression is -form-valued when it is reduced to the form -$a\,dx^0\wedge dx^1\dots+b\,d x^1\wedge+\dots$ where $a$ and $b$ -are algebraic expressions (similarly for the vector-valued expressions). - -(4) The \reduce\ algebraic simplification routine -is applied to the algebraic expressions $a$, $b$. -\seethis{In the anholonomic mode the basis $b^i\wedge b^j\dots$ -is used instead. See section \ref{amode}.} -Final expression consist of exterior products of basis -coordinate differentials $dx^i\wedge dx^j\dots$ (or basis -vectors $\partial_{x^i}$) multiplied by the algebraic expressions. -The algebraic expressions contain only the coordinates, -constants and functions. - -\subsection{Controlling Expression Evaluation} - -There are many \reduce\ switches which control -algebraic expression evaluation. The number of these switches -and details of their work depend on the \reduce\ version. -Here we consider some of these switches. All examples below -are made with the \reduce\ 3.5. On other \reduce\ versions -result may be a bit different. - -Switches {\tt EXP} and {\tt MCD} control expansion and -reduction of rational expressions to a common denominator -respectively. -\begin{slisting} -<- (x+y)\^2; - - 2 2 -x + 2*x*y + y - -<- Off EXP; -<- (x+y)\^2; - - 2 -(x + y) - -<- On EXP; -<- 1/x+1/y; - - x + y -------- - x*y - -<- Off MCD; -<- 1/x+1/y; - - -1 -1 -x + y -\end{slisting} -These switches are normally on. - -Switches {\tt PRECISE} and {\tt REDUCED} control evaluation of -square roots:\label{PRECISE}\label{REDUCED} -\begin{slisting} -<- SQRT(-8*x\^2*y); - -2*SQRT( - 2*y)*x - -<- On REDUCED; -<- SQRT(-8*x\^2*y); - -2*SQRT(y)*SQRT(2)*I*x - -<- Off REDUCED; -<- On PRECISE; -<- SQRT(-8*x\^2*y); - -2*SQRT(y)*SQRT(2)*I*x - -<- On REDUCED, PRECISE; -<- SQRT(-8*x\^2*y); - -2*SQRT(y)*SQRT(2)*ABS(x) -\end{slisting} - - -Combining rational expressions the system by default -calculates the least common multiple of denominators but -turning the switch {\tt LCM} off prevents this calculation. - -Switch {\tt GCD} (normally off) makes the system -search and cancel the greatest common divisor of the -numerator and denominator of rational expressions. -Turning {\tt GCD} on may significantly slow down the -calculations. There is also another switch {\tt EZGCD} -which uses other algorithm for g.c.d. calculation. - - -Switches {\tt COMBINELOGS} and {\tt EXPANDLOGS} control -the evaluation of logarithms -\begin{slisting} -<- On EXPANDLOGS; -<- LOG(x*y); - -LOG(x) + LOG(y) - -<- LOG(x/y); - -LOG(x) - LOG(y) - -<- Off EXPANDLOGS; -<- On COMBINELOGS; -<- LOG(x)+LOG(y); - -LOG(x*y) -\end{slisting} - -By default all polynomials are considered by \reduce\ as -the polynomials with integer coefficients. The switches -{\tt RATIONAL} and {\tt COMPLEX} allow rational and -complex coefficients in polynomials respectively: -\begin{slisting} -<- (x\^2+y\^2+x*y/3)/(x-1/2); - - 2 2 - 2*(3*x + x*y + 3*y ) ------------------------ - 3*(2*x - 1) - -<- On RATIONAL; -<- (x\^2+y\^2+x*y/3)/(x-1/2); - - 2 1 2 - x + ---*x*y + y - 3 -------------------- - 1 - x - --- - 2 - -<- Off RATIONAL; -<- 1/I; - - 1 ---- - I - -<- (x\^2+y\^2)/(x+I*y); - - 2 2 - x + y ---------- - I*y + x - -<- On COMPLEX; -<- 1/I; - - - I - -<- (x\^2+y\^2)/(x+I*y); - -x - I*y -\end{slisting} -Switch {\tt RATIONALIZE} removes complex numbers from the -denominators of the expressions but it works even if -{\tt COMPLEX} is off. - -Turning off switch {\tt EXP} and on {\tt GCD} one can -make the system to factor expressions -\begin{slisting} -<- Off EXP; -<- On GCD; -<- x\^2+y\^2+2*x*y; - - 2 -(x + y) -\end{slisting} -Similar effect can be achieved by turning on switch {\tt FACTOR}. -Unfortunately this works only when \grg\ prints expressions and -internally expressions remain in the expanded form. -To make \grg\ to work with factored expressions internally one -must turn on {\tt FACTOR} and {\tt AEVAL}. -\swind{AEVAL} -The \grg\ switch {\tt AEVAL} make \grg\ to use an alternative -\reduce\ routine for algebraic expression evaluation and simplification. -This routine works well with {\tt FACTOR} on. -\seethis{See section \ref{tuning} about configuration files.} -Possibly it -is good idea to turn switch {\tt AEVAL} on by default. -This can be done using \grg\ configuration files. - -\subsection{Substitutions} -\index{Substitutions} - -The substitution commands in \grg\ are the same as the -corresponding \reduce\ instructions -\cmdind{Let}\cmdind{Match}\cmdind{For All Let} -\command{\opt{For All \rpt{\parm{x}}\,\opt{Such That \parm{cond}}} Let \rpt{\parm{sub}};\\\tt -\opt{For All \rpt{\parm{x}}\,\opt{Such That \parm{cond}}} Match \rpt{\parm{sub}};} -\seethis{See page \pageref{solutions} about solutions.} -where \parm{sub} is either relation {\tt \parm{l}\,=\,\parm{r}} -or the solution in the form \comm{Sol(\parm{n})}. -After the substitution is activated every appearance of \parm{l} will be -replaced by \parm{r}. The {\tt For All} substitutions have additional list -of parameters \parm{x} and will work for any value -of \parm{x}. The optional condition \parm{cond} imposes restrictions -on the value of the parameters \parm{x}. The \parm{cond} is -the boolean expression (see page \pageref{bool}). - -The substitution can be deactivated by the command -\cmdind{Clear} -\command{\opt{For All \rpt{\parm{x}}\,\opt{Such That \parm{cond}}} Clear \rpt{\parm{sub}};} -Notice that the variables \parm{x} must be exactly the same -as in the corresponding {\tt For All Let} command. - -The difference between \comm{Match} and \comm{Let} -is that the former matches the degrees of the -expressions exactly while \comm{Let} matches all powers which -are greater than one indicated in the substitution: -\begin{slisting} -<- Const a; -<- (a+1)\^8; - - 8 7 6 5 4 3 2 -a + 8*a + 28*a + 56*a + 70*a + 56*a + 28*a + 8*a + 1 - -<- Let a\^3=1; -<- (a+1)\^8; - - 2 -85*a + 86*a + 85 - -<- Clear a\^3; -<- Match a\^3=1; -<- (a+1)\^8; - - 8 7 6 5 4 2 -a + 8*a + 28*a + 56*a + 70*a + 28*a + 8*a + 57 -\end{slisting} - -Substitutions can be used for various purposes, for example: -(i) to define additional mathematical relations such as -trigonometric ones; -(ii) to ``assign'' value to the user-defined and built-in constants; -(iii) to define differentiation rules for functions. - -After some substitution is activated it applies to every -evaluated expression but value of the objects calculated -\emph{before} remain unchanged. -The command \comm{Evaluate} re-simplifies the value of the object -\cmdind{Evaluate} -\command{Evaluate \parm{object};} -here \parm{object} is the object name, or identifier, or the -group object name. -Let us consider a simple \grg\ task which -calculates the volume 4-form of some metric -\begin{slisting} -<- Coordinates t, x, y, z; -<- Constant a; -<- Tetrad T0=d t, T1=d x, T2=SIN(a)*d y+COS(a)*d z, - T3=-COS(a)*d y+SIN(a)* d z; -<- Find and Write Volume; -Volume : - - 2 2 -VOL = (SIN(a) + COS(a) ) d t \w\ d x \w\ d y \w\ d z -\end{slisting} -We see that \reduce\ do not know the -appropriate trigonometric rule. -Thus we are going to apply substitution -\begin{slisting} -<- For all x let SIN(x)\^2 = 1-COS(x)\^2; -<- Write Volume; -Volume : - -VOL = d t \w\ d x \w\ d y \w\ d z -\end{slisting} -The situation has been improved. -But actually, the \emph{internal} representation -of {\tt VOL} remains unchanged. {\tt Write} by default -re-simplifies expressions before printing. -\swinda{WRS} -By turning switch {\tt WRS} off we can prevent this -re-simplification: -\begin{slisting} -<- Off WRS; -<- Write Volume; -Volume : - 2 2 -VOL = (SIN(a) + COS(a) ) d t \w\ d x \w\ d y \w\ d z -\end{slisting} -Now we can apply \comm{Evaluate}: -\begin{slisting} -<- Evaluate Volume; -<- Write Volume; -Volume : - -VOL = d t \w\ d x \w\ d y \w\ d z -\end{slisting} -We see that the internal value of {\tt VOL} now has been -replaced by re-simplified expression. - -Notice that the command -\command{Evaluate All;} -applies \comm{Evaluate} to all objects whose value is -currently known. - -\subsection{Generic Functions} -\index{Generic Functions}\label{genfun} - -Unfortunately \reduce\ lacks the notion of partial derivative of a function. -The expression \comm{DF(f(x,y),x)} is treated by \reduce\ as the -``derivative of the expression \comm{f(x,y)} with respect to -the variable \comm{x}'' rather than the ``derivative of the function -\comm{f} with respect to its first argument''. -Due to this \reduce\ cannot handle -chain differentiation rule etc. This problem is fixed by the -package \file{dfpart} written by H.~Melenk. -This package introduces notion of generic function and -partial derivative \comm{DFP}. If \file{dfpart} is installed -on your \reduce\ system \grg\ provides the interface -to these facilities. - - - -Let us consider an example. First we declare -one usual and two generic functions -\begin{slisting} -<- Coordinates t, x, y, z; -<- Function f; -<- Generic Function g(a,b), h(b); -<- Write Functions; -Functions: - -g*(a,b) h*(b) f -\end{slisting} -Generic functions must be always declared with -the list of parameters (\comm{a} and \comm{b} in our example). -These parameters play the role of labels which denotes -arguments of the generic function and the partial -derivatives with respect to these arguments -are defined. Due to this generic functions allow the -chain differentiation rule -\begin{slisting} -<- DF(f(SIN(x),y),x); - -DF(f(SIN(x),y),x) - -<- DF(g(SIN(x),y),x); - -COS(x)*g (SIN(x),y) - a -\end{slisting} -Here subscript \comm{a} denotes -the derivative of the function \comm{g} with respect to the -first argument. \enlargethispage{5mm} -The operator \comm{DFP} is introduced to denotes such -derivatives in expressions: -\begin{slisting} -<- DF(g(x,y)*h(y),b); - -0 - -<- DFP(g(x,y)*h(y),b); - -g (x,y)*h(y) + h (y)*g(x,y) - b b -\end{slisting} - -\newpage - -If switch \swind{DFPCOMMUTE} -\comm{DFPCOMMUTE} is turned on then \comm{DFP} -derivatives commute. - - -\section{Using Built-in Formulas In Calculations} - -\grg\ has large number of built-in objects and almost -each object has built-in formulas or so called -\emph{ways of calculation} which can be used to find -the value of the object. This section explains how -these formulas (ways) can be used. - -\subsection{\comm{Find} Command} -\index{Ways of calculation}\cmdind{Find}\label{find} - -Almost each \grg\ built-in object has associated -\emph{ways of calculation}. Each way is nothing but -a formula or equation which allows to compute -the value of the object. All these formulas -are described in the usual mathematical style in -chapter 3. -The command\cmdind{Show \parm{object}} -\command{Show \parm{object};} -or equivalently -\command{?~\parm{object};} -prints information about object's ways of calculation. - -The command \comm{Find} applies built-in formulas to -calculate the object value -\command{Find \parm{object} \opt{\parm{way}};} -where \parm{object} is the object name, or identifier, or -group object name. -The optional specification \parm{way} indicates the -particular way if the \parm{object} has several built-in ways -of calculation. - -\enlargethispage{3mm} - -Consider the curvature 2-form $\Omega^a{}_b$ -(object \comm{Curvature}, id. \comm{OMEGA}): -\begin{slisting} -<- Show Curvature; - -Curvature OMEGA'a.b is 2-form - Value: unknown - Ways of calculation: - Standard way (omega) - From spinorial curvature (OMEGAU*,OMEGAD) -\end{slisting} - -\noindent -We can see that this object has two built in ways of -calculation. First way named {\tt Standard way} is the -usual equation -$\Omega^a{}_b=d\omega^a{}_b+\omega^a{}_m\wedge\omega^m{}_b$. -Second way under the name {\tt From spinorial curvature} -uses spinor $\tsst$ tensor relationship to compute the curvature 2-form -using its spinor analogues $\Omega_{AB}$ and -$\Omega_{\dot{A}\dot{B}}$ as the source data. -The ways of calculation are printed by the command {\tt Show} -in the form -\command{\parm{wayname} (\rpt{\parm{SI}})} -where \parm{wayname} is the way name and \seethis{See Eq. (\ref{omes}) on \pref{omes}.} -the \parm{SI} are the identifiers of the \emph{source} objects which are -present in the right-hand side of the equation. The value of -these objects must be known before the formula can be applied. - -%\enlargethispage{5mm} - -The \parm{way} in the \comm{Find} command allows one to -choose the particular way which can be done by two methods. -In the first form \parm{way} is just the name exactly as -it printed by the \comm{Show} command -\command{wayname} -or {\tt Using standard way} or {\tt By standard way} if the way -name is {\tt Standard way}. Another method to specify -the way is to indicate the appropriate source object -\command{From \parm{object}\\\tt% -Using \parm{object}} -where \parm{object} is the name or the identifier of the source object. -For example second (spinorial) way of calculation for the curvature -2-form can be chosen by the following equivalent commands \vspace{-1mm} -\begin{listing} - Find curvature from spinorial curvature; - Find curvature using OMEGAU; -\end{listing} -while first way is activated by the commands \vspace*{-1mm} -\begin{listing} - Find curvature by standard way; - Find curvature using omega; -\end{listing} -Recall that object identifiers are case sensitive -and \comm{omega} is the identifier -of the frame connection 1-form $\omega^a{}_b$ and should not be -confused with \comm{OMEGA}. - - -The \parm{way} specification in the \comm{Find} -can be omitted and in this case -\grg\ uses the following algorithm to choose -a particular way of calculation. Observe that the identifier -of the undotted curvature 2-form $\Omega_{AB}$ is marked -by the symbol $*$. This label marks so called \emph{main} -objects. If no way of calculation is specified when -\grg\ tries to choose the way, browsing the way list -form top to the bottom, for which the value of the \emph{main} -object is already known. If no switch way exists then -\grg\ just picks up the first way in the list. -Therefore in our example the command -\begin{listing} - Find curvature; -\end{listing} -will use the second way if the value of the object $\Omega_{AB}$ -(id. \comm{OMEGAU}) is known and second way otherwise. - -As soon as some way of calculation is chosen \grg\ tries to -calculate the values of the source objects which are present -in the right-hand side of corresponding equations. -\grg\ tries to do this by applying the \comm{Find} command without way -specification to these objects. Thus a single \comm{Find} -can cause quite long chain of calculations. -This recursive work is reflected by the appropriate -tracing messages. The tracing can be eliminated by turning off -switch \comm{TRACE}.\swind{TRACE} - -Here we present the sample \grg\ session which computes -curvature 2-form for the flat gravitational waves -\begin{slisting} - -<- Cord u, v, z, z~; - -z & z~ - conjugated pair. - -<- Null Metric; -<- Function H(u,z,z~); -<- Frame T0=d u, T1=d v+H*d u, T2=d z, T3=d z~; -<- ds2; - - 2 2 - ds = ( - 2*H) d u + (-2) d u d v + 2 d z d z~ - -<- Find Curvature; -Sqrt det of metric calculated. 0.16 sec -Volume calculated. 0.16 sec -Vector frame calculated From frame. 0.16 sec -Inverse metric calculated From metric. 0.16 sec -Frame connection calculated. 0.22 sec -Curvature calculated. 0.22 sec -<- Write Curvature; -Curvature: - - 1 -OMEGA = ( - DF(H,z,2)) d u \w d z + ( - DF(H,z,z~)) d u \w d z~ - 2 - - 1 -OMEGA = ( - DF(H,z,z~)) d u \w d z + ( - DF(H,z~,2)) d u \w d z~ - 3 - - 2 -OMEGA = ( - DF(H,z,z~)) d u \w d z + ( - DF(H,z~,2)) d u \w d z~ - 0 -\newpage - 3 -OMEGA = ( - DF(H,z,2)) d u \w d z + ( - DF(H,z,z~)) d u \w d z~ - 0 -\end{slisting} - - -Finally we want to emphasize that ways associated -with some object may depend on the concrete environment. -In particular the {\tt Standard way} for -the curvature 2-form is always available but second -way which is essentially related to spinors works -\seethis{See \pref{spinors} about the spinorial formalism.} -only in the 4-dimensional spaces of Lorentzian signature -and iff the metric is null. -If some way is not valid in the current environment -it simply disappears from the way list printed by the \comm{Show}. - -It should be noted also that the \comm{Find \parm{object};} -command works only if the \parm{object} is in the indefinite state -and is rejected if the value of the \parm{object} is already known. -If you want to re-calculate the object then previous value must be -cleared by the \comm{Erase} command. - -\subsection{\comm{Erase} command} -\cmdind{Erase} - -The command -\command{Erase \parm{object};} -destroys the \parm{object} value and returns it to initial -indefinite state. It can be used also to free the -memory. - -\subsection{\comm{Zero} command} -\cmdind{Zero} - -Command -\command{Zero \parm{object};} -assigns zero values to all \parm{object} components. - -\subsection{\comm{Normalize} command} -\cmdind{Normalize} - -Command -\command{Normalize \parm{object};} -applies to equations. It replaces equalities -of the form $l=r$ by the equalities $l-r=0$ -and re-simplifies the result. - -\subsection{\comm{Evaluate} command} -\cmdind{Evaluate} - -The command -\command{Evaluate \parm{object};} -re-simplifies existing value of the \parm{object}. -This command is useful if we want to apply new substitutions -\seethis{See page \pageref{subs} about substitutions.} -to the object whose value is already known. -The command -\command{Evaluate All;} -re-simplifies all objects whose value is currently known. - - -\section{Printing Result of Calculations} - -\subsection{\comm{Write} Command} -\cmdind{Write} - -The command -\command{Write \parm{object};} -prints value of the \parm{object}. Here \parm{object} -id the object name or identifier.\index{Group name} -Group names denoting a collection of several objects -\seethis{See page \pageref{macro} about macro objects.} -and macro object identifiers can be used in the \comm{Write} -command as well. In addition word \comm{All} -can be used to print all currently known objects. - - -The command \comm{Write} can print declarations as well if -\parm{object} is {\tt functions}, {\tt constants}, or -{\tt affine parameter}. - - -The command -\command{Write \rpt{\parm{object}}~to~"\parm{file}";} -or equivalently -\command{Write \rpt{\parm{object}}~>~"\parm{file}";} -writes result into the \comm{"\parm{file}"}. Notice -that \comm{Write} always destroys previous contents of the -file. Therefore we have another command -\command{Write to "\parm{file}";\\\tt% -Write > "\parm{file}";} -which redirects all output into the file. The standard output -can be restored by the commands\cmdind{End of Write}\cmdind{EndW} -\command{EndW;\\\tt% -End of Write;} - -\enlargethispage{3mm} - -By default \comm{Write} re-simplifies the expressions -before printing them. \swind{WRS} -\seethis{See page \pageref{subs} about substitutions.} -This is convenient when substitutions are activated -but slows down the printing especially for very large -expressions. The re-simplification can be abolished -by turning off switch \comm{WRS}. -If switch \comm{WMATR} is turned on then -\swind{WMATR} -\grg\ prints all 2-index scalar-valued objects in -the matrix form -\begin{slisting} -<- Coordinates t, x, y, z; -<- On wmatr; -<- Find and Write metric; -Assuming Default Metric. -Metric calculated By default. 0.06 sec -Metric: - -[-1 0 0 0] -[ ] -[0 1 0 0] -[ ] -[0 0 1 0] -[ ] -[0 0 0 1] -\end{slisting} - - -\comm{Write} prints frame, spinor and enumerating indices as -numerical subscripts while holonomic indices are printed as -the coordinate identifiers. If frame is holonomic -and there is no difference between frame and coordinate indices then -by default all frame indices are also labelled by the -appropriate identifiers. But is switch \comm{HOLONOMIC} \swinda{HOLONOMIC} -is turned off they are still printed as numbers. - -\subsection{\comm{Print} Command} -\cmdind{Print} - -The \comm{Write} command described in the previous section -prints value of an object. This value must be -calculated beforehand by the \comm{Find} command -or established by the assignment. -The command \comm{Print} evaluates expression and -immediately prints its value. It has several forms -\command{% -\opt{Print} \parm{expr} \opt{For \parm{iter}};\\\tt -For \parm{iter} Print \parm{expr};} -Here \parm{expr} is expression to be evaluated and -\parm{iter} indicates that expression must be -evaluated for several value of some variable. -The specification \parm{iter} is completely the same as -is the \comm{Sum} expression and is described in details -in section \ref{iter} on page \pageref{iter}. -It consists of the list of parameters -separated by commas \comm{,} or relational operators -{\tt < > => =<}. For example the command -\begin{listing} - G(a,b) for a "\parm{file}";\\\tt -Unload \parm{object} To "\parm{file}";} -writes \parm{object} value into \comm{"\parm{file}"} in some -special format. -Here \parm{object} is name or identifier of an object. - -The data can be later restored with help of the command\cmdind{Load} -\command{Load "\parm{file}";} - -The command {\tt Unload} always overwrites previous \comm{"\parm{file}"} -contents. To save several objects in one file one must use -the following sequence of commands\cmdind{EndU}\cmdind{End of Unload} -\begin{listing} - Unload > "\parm{file}"; - Unload \parm{object}; - Unload \parm{object}; - ... - Unload \parm{object}; - End Of Unload; -\end{listing} -Here command \comm{Unload > "\parm{file}";} opens -\comm{"\parm{file}"} and {\tt End Of Unload;} closes it. -The last command has the short form -\command{EndU;} -In fact presented above sequence of commands can be -abbreviated as -\command{Unload \rpt{\parm{object}}~>~"\parm{file}";} - -One needs to stress that only the commands {\tt Unload \dots;} -can be used between {\tt Unload > \dots} and -{\tt End Of Unload;}. If this rule does not hold then {\tt Load} -may fail to restore the file. -The only additional command which can be used among these -{\tt Unload \parm{object};} commands is the comment -{\tt \% \parm{text};}. This command insertes -the comment \parm{text} into the \comm{"\parm{file}"}. -Later when \comm{"\parm{file}"} will be restored by the -{\tt Load} the \parm{text} message will be printed. -This allows one to attach comments to unreadable files -produced by {\tt Unload} command. - -As in other commands \parm{object} in \comm{Unload} command -is either the name or identifier of an object. Names {\tt Coordinates}, -{\tt Constants} and {\tt Functions} can also be used to -save declarations. And finally, the command -\command{Unload All > "\parm{file}";} -saves all objects whose value is currently known -\seethis{See section \ref{amode} about anholonomic basis.} -and all declarations. Moreover, in the anholonomic basis mode this -command saves full information about an anholonomic basis. - -When data or coordinates declarations are restored from a file -they replace current values. Function and constant declarations -are added to current declarations. - -One should realize that serious troubles may appear when different -coordinates are used in the current session and in the restored file. -Even the order of coordinates is extremely important. -We strongly recommend saving all declarations (especially coordinates) -in addition to other objects. It ensures at least that will \grg\ print a -warning message if some contradictions are detected between -current declarations and declarations stored into a file. -The best way to avoid these troubles is to use the command -\command{Unload All > "\parm{file}";} -Loading the file saved by this command at the very beginning of -a new \grg\ task completely restores the previous \grg\ state -with all data and declarations. - -Sometimes one needs to prevent the {\tt Load}/{\tt Unload} operations -with coordinates.\swind{UNLCORD} -If switch {\tt UNLCORD} is turned off (normally on) -then all {\tt Load} and {\tt Unload} operations -with coordinates are blocked. - -Since {\tt Unload} writes data in human-unreadable form there -is the command\cmdind{Show File}\cmdind{File}\cmdind{Show {"\parm{file}"}} -\command{Show \opt{File} "\parm{file}";} -or equivalently -\command{?~\opt{File}~"\parm{file}";\\\tt -File "\parm{file}";} -which prints short information about objects and declarations -contained in the \comm{"\parm{file}"}. -It also prints comments contained in the file. - - -\subsection{Coordinate Transformations} -\index{Coordinate transformations} - -Command\cmdind{New Coordinates} -\command{New Coordinates \rpt{\parm{new}} with \rpt{\parm{old}=\parm{expr}};} -introduces new coordinates \parm{new} and -defines how old coordinates \parm{old} are expressed in terms -of new ones. If the specified transformation is nonsingular -\grg\ converts all existing objects to the new coordinate system. - - -The {\tt New Coordinates} command properly transforms all -objects having coordinate indices. The transformation -of frame indices depend on the switch \comm{HOLONOMIC}. \swind{HOLONOMIC} -In general case when frame is not holonomic then objects -having frame indices remain unchanged and only their components -are transformed into the new coordinate system. But if frame -is holonomic then by default all frame indices are transformed -similarly to the coordinate ones. Notice that in such situation -the frame after transformation once again will be holonomic -in the new coordinate system. -But if switch \comm{HOLONOMIC} is turned off the system -distinguishes frame and coordinate indices in spite of the current -frame type. In such situation the holonomic frame -ceases to be holonomic after coordinate transformation. - -\subsection{Frame Transformations} -\index{Frame transformations} - -Spinorial rotations are performed by -the command\cmdind{Make Spinorial Rotation}\cmdind{Spinorial Rotation} -\command{\opt{Make} Spinorial Rotation \opt{ -((\parm{expr}${}_{00}$,\parm{expr}${}_{01}$), -(\parm{expr}${}_{10}$,\parm{expr}${}_{11}$))};} -where expressions $\mbox{\parm{expr}}_{AB}$ comprise the SL(2,C) -transformation matrix -\[ -\phi'_A=L_A{}^B\phi_B,\ \ -\mbox{\parm{expr}}_{AB}=L_A{}^B -\] - -If the specified matrix is really a SL(2,C) one then \grg\ -performs appropriate transformation on all objects whose -value is currently known. - -Matrix specification in the command can be omitted -\command{\opt{Make} Spinorial Rotation;} -In this case the SL(2,C) matrix $L_A{}^B$ must be specified as -the value of a special object {\tt Spinorial Transformation LS.A'B} -(identifier {\tt LS}). - -Command for frame rotation is analogously\cmdind{Make Rotation}\cmdind{Rotation} -\command{\opt{Make} Rotation \opt{ -((\parm{expr}${}_{00}$,\parm{expr}${}_{01}$,...), -(\parm{expr}${}_{10}$,\parm{expr}${}_{11}$,...),...)};} -with the nonsingular $d\times d$ rotation matrix -\[ -A'^a=L^a{}_bA^b,\ \ \mbox{\parm{expr}}_{ab}=L^a{}_b -\] -\grg\ verifies that this matrix is a valid \emph{rotation} -by checking that frame metric $g_{ab}$ \emph{remains unchanged} -under this transformation -\[ -g'_{ab} = L^m{}_a L^n{}_b g_{mn} = g_{ab} -\] - -Once again the matrix specification -can be omitted and transformation $L^a{}_b$ can be specified as the value -of the object {\tt Frame Transformation L'a.b} (identifier {\tt L}) -\command{\opt{Make} Rotation;} - -Frame rotation commands correctly transform frame and -spinor connection 1-forms. - - -Finally, there is a special form of the frame -transformation command\cmdind{Change Metric} -\command{Change Metric \opt{ -((\parm{expr}${}_{00}$,\parm{expr}${}_{01}$,...), -(\parm{expr}${}_{10}$,\parm{expr}${}_{11}$,...),...)};} -The only difference between this command and {\tt Make Rotation} -is that {\tt Change Metric} does not impose -any restriction on the transformation matrix and -transformed metric does not necessary coincides -with the original one. - -Sometimes it is convenient to keep some object unchanged -under the frame transformation. The command\cmdind{Hold} -\command{Hold \parm{object};} -makes the system to keep the \parm{object} unchanged -during frame and spinor transformations. The command\cmdind{Release} -\command{Release \parm{object};} -discards the action of the \comm{Hold} command. - - -\subsection{Algebraic Classification} -\index{Algebraic classification} - -The command\cmdind{Classify} -\command{Classify \parm{object};} -performs algebraic classification of the \parm{object} -specified by its name or identifier. -Currently \grg\ knows algorithms for classifying -the following irreducible spinors - -\begin{tabular}{ll} -$X_{ABCD}$ & Weyl spinor type \\ -$X_{AB\dot{C}\dot{D}}$ & Traceless Ricci spinor type \\ -$X_{AB}$ & Electromagnetic stress spinor type \\ -$X_{A\dot{B}}$ & Vector in the spinorial representation -\end{tabular} \newline - -\reversemarginpar - -The {\tt Classify} command can be applied to any built-in or -user-defined object having one of the listed above -\seethis{See page \pageref{sumspin} about summed spinor indices.} -types of indices. Notice that all spinors must be irreducible -(totally symmetric in dotted and undotted indices) -and $X_{AB\dot{C}\dot{D}}$, $X_{A\dot{B}}$ must be Hermitian. -Groups of the irreducible indices must be represented -as a single summed index. - -\normalmarginpar - -\grg\ uses the algorithm by F.~W.~Letniowski and R.~G.~McLenaghan -[Gen. Rel. Grav. 20 (1988) 463-483] for Petrov-Penrose -classification of Weyl spinor $X_{ABCD}$. The obvious -simplification of this algorithm is applied to -the spinor analog of electromagnetic strength tensor $X_{AB}$. -The spinor $X_{AB\dot{C}\dot{D}}$ is classified by the algorithm -by G.~C.~Joly, M.~A.~H.~McCallum and W.~Seixas -[Class. Quantum Grav. 7 (1990) 541-556, -Class. Quantum Grav. 8 (1991) 1577-1585]. - -The classification process is accompanied by the -tracing messages which can be eliminated by turning \swinda{TRACE} -off the switch \comm{TRACE}. -On the contrary if one turns on \swind{SHOWEXPR} -the switch \comm{SHOWEXPR} then \grg\ prints -all expressions which appear during the classification -to let you check whether the decision about -nonvanishing of these expressions is really correct or not. -This facility is important also in classifying -$X_{AB\dot{C}\dot{D}}$ and $X_{A\dot{B}}$ -since algebraic type for this objects may depend on -the \emph{sign} of some expressions which -cannot be determined by \grg\ correctly. - - -\subsection{\reduce\ Packages and Functions in \grg} -\index{Using \reduce\ packages} -\label{packages} - -Any procedure or function defined -in \reduce\ package can be used in \grg. -The package must be loaded either before -\grg\ is started or during \grg\ session by one of the -equivalent commands -\cmdind{Package}\cmdind{Use Package}\cmdind{Load} -\command{\opt{Use} Package \parm{package};\\\tt -Load \parm{package};} -where \parm{package} is the package name. Notice that an -identifier must be used for the package name unlike -the \comm{Load "\parm{file}";} command described in \enlargethispage{5mm} -section \ref{UnloadLoad}. Let us consider some examples. -The \reduce\ package \file{specfn} contains -definitions of various special functions and -below we demonstrate 11th Legendre polynomial -\begin{slisting} -<- Coordinates t, x, y, z; -<- package specfn; -<- LEGENDREP(11,x); - - 10 8 6 4 2 - x*(88179*x - 230945*x + 218790*x - 90090*x + 15015*x - 693) -------------------------------------------------------------------- - 256 -\end{slisting} - -\newpage - -Another example demonstrates the \file{taylor} package -\begin{slisting} -<- Coordinates t, x, y, z; -<- www=d(E^(x+y)*SIN(x)); -<- www; - - x + y x + y -(E *(COS(x) + SIN(x))) d x + (E *SIN(x)) d y - -<- load taylor; -<- TAYLOR(www,x,0,5); - - y y - y y y 2 E 4 E 5 6 y y 2 -(E + 2*E *x + E *x - ----*x - ----*x + O(x )) d x + (E *x + E *x - 6 15 - - y y - E 3 E 5 6 - + ----*x - ----*x + O(x )) d y - 3 30 -\end{slisting} - -You can also define your own operators and procedures -in \reduce\ and later use them in \grg. -In the following example file \file{lasym.red} contains -a definition of little \reduce\ procedure -which computes a leading term of asymptotic expansion -of the rational function at large values of some -variable. This file is inputted in \reduce\ before -\grg\ is started -\begin{slisting} - -1: in "lasym.red"; - -procedure leadingterm(w,x); - lterm(num(w),x)/lterm(den(w),x); - -leadingterm - -end; - -2: load grg; - -This is GRG 3.2 release 2 (Feb 9, 1997) ... - -System directory: c:{\bs}red35{\bs}grg32{\bs} -System variables are upper-cased: E I PI SIN ... -Dimension is 4 with Signature (-,+,+,+) - -<- Coordinates t, r, theta, phi; -<- OMEGA01=(123*r^3+2*r+t)/(r+t)^5*d theta{\w}d phi; -<- OMEGA01; - - 3 - 123*r + 2*r + t -(-------------------------------------------------) d theta \w d phi - 5 4 3 2 2 3 4 5 - r + 5*r *t + 10*r *t + 10*r *t + 5*r*t + t - -<- LEADINGTERM(OMEGA01,r); - - 123 -(-----) d theta \w d phi - 2 - r -\end{slisting} - - -\subsection{Anholonomic Basis Mode} -\index{Anholonomic basis mode}\index{Basis}\label{amode} - -\grg\ may work in both holonomic and anholonomic basis modes. -In the first default case, values of all expressions are -represented in a natural holonomic (coordinate) basis: -$d x^\mu,~d x^\mu\wedge x^\nu\dots$ for exterior -forms and $\partial_\mu=\partial/\partial x^\mu$ -for vectors. In the second case an -arbitrary basis $b^i=b^i_\mu d x^\mu$ is used for -forms and inverse vector basis $e_i=e_i^\mu\partial_\mu$ for vectors -($b^i_\mu e^\mu_j=\delta^i_j$). You can specify this basis -assigning a value to built-in object -{\tt Basis} (identifier {\tt b}). If {\tt Basis} is not -specified by user then \grg\ assumes that it coincides -with the frame $b^i=\theta^i$. - -Frame should not be confused with basis. Frame $\theta^a$ is used -only for ``external'' purposes to represent tensor indices -while basis $b^i$ and vector basis $e_i$ is used for ``internal'' -purposes to represent form and vector valued object components. - -The command\cmdind{Anholonomic} -\command{Anholonomic;} -switches the system to the anholonomic basis mode and -the command\cmdind{Holonomic} -\command{Holonomic;} -switches it back to the standard holonomic mode. - -Working in anholonomic mode \grg\ creates some internal tables -for efficient calculation of exterior differentiation and -complex conjugation. In anholonomic mode the command -\cmdind{Unload} -\begin{listing} - Unload All > "\parm{file}"; -\end{listing} -automatically saves these tables into the \comm{"\parm{file}"}. -Subsequent\cmdind{Load} -\begin{listing} - Load "\parm{file}"; -\end{listing} -restores the tables and automatically switches the current mode to -anholonomic one. Note that automatic anholonomic mode -saving/restoring works only if {\tt All} is used in -{\tt Unload} command. - -One can find out the current mode with the help of the command -\cmdind{Show Status}\cmdind{Status} -\command{\opt{Show} Status;} - - -\subsection{Synonymy} -\index{Synonymy} - -Sometimes \grg\ commands may be rather long. For -instance, in order to find the curvature 2-form $\Omega_{ab}$ -from the spinorial curvature $\Omega_{AB}$ and $\Omega_{\dot{A}\dot{B}}$ -the following command should be used -\begin{listing} - Find Curvature From Spinorial Curvature; -\end{listing} -Certainly, this command is clear but typing of such long -phrases may be very dull. \grg\ has synonymy mechanism -which allows one to make input much shorter. - -The synonymous words in commands and object names -are considered to be equivalent. The complete list -of predefined \grg\ synonymy is given in appendix D. -Here we present just the most important ones -\begin{verbatim} - Connection Con - Constants Const Constant - Coordinates Cord - Curvature Cur - Dotted Do - Equation Equations Eq - Find F Calculate Calc - Functions Fun Function - Next N - Show ? - Spinor Spin Spinorial Sp - Switch Sw - Symmetries Sym Symmetric - Undotted Un - Write W -\end{verbatim} -Words in each line are considered as equivalent -in all commands. Thus the above command can be abbreviated as -\begin{listing} - F cur from sp cur; -\end{listing} - -Section \ref{tuning} explains how to change built-in synonymy -and how to define a new one. - - -\subsection{Compound Commands} -\index{Compound commands} - -Sometime one may need to perform several consecutive actions -with one object. In this case we can use so called -\emph{compound commands} to shorten the input. -Internally \grg\ replaces each compound command by several usual -ones. For example the compound command -\begin{listing} - Find and Write Einstein Equation; -\end{listing} -to a pair of usual ones -\begin{listing} - Find Einstein Equation; - Write Einstein Equation; -\end{listing} -Actions (commands) can be attached to the end of the -compound command as well: -\begin{listing} - Find, Write Curvature and Erase It; -\qquad\qquad \udr - Find \& Write \& Erase Curvature; -\qquad\qquad \udr - Find Curvature; - Write Curvature; - Erase Curvature; -\end{listing} -Note that we have used {\tt ,} and {\tt \&} instead of {\tt and} -in this example. All these separators are equivalent in compound -commands. - -Now let us consider the case when one needs to perform a single action -with several objects. The command -\begin{listing} - Write Frame, Vector Frame and Metric; -\end{listing} -is equivalent to -\begin{listing} - Write Frame; - Write Vector Frame; - Write Metric; -\end{listing} -Way specification can be attached to the {\tt Find} command: -\begin{listing} - Find QT, QP From Torsion using spinors; -\qquad\qquad \udr - Find QT From Torsion using spinors; - Find QP From Torsion using spinors; -\end{listing} -One can combine several actions and several objects. -For example, the command -\begin{listing} - Find omega, Curvature by Standard Way and Write and Erase Them; -\end{listing} -is equivalent to the sequence of -$(2{\rm\ objects})\times(3{\rm\ commands}) =6$ -commands -\begin{listing} - Find omega by Standard Way; - Find Curvature by Standard Way; - Write omega; - Write Curvature; - Erase omega; - Erase Curvature; -\end{listing} -Note that the way specification is attached only to ``left'' -commands ({\tt Find} in our case). - -The compound commands mechanism works only with -{\tt Find}, {\tt Erase}, {\tt Write} and {\tt Evaluate} commands. - -And finally, \grg\ always replaces {\tt Re-\parm{command};} by -{\tt Erase and \parm{command};}. For example -\begin{listing} - Re-Calculate Maxwell Equations; -\qquad\qquad \udr - Erase and Calculate Maxwell Equations; -\end{listing} - -You can see how \grg\ expand compound commands into the -\swind{SHOWCOMMANDS} -usual ones by turning switch \comm{SHOWCOMMANDS} on. - - -\section{Tuning \grg} -\label{tuning} - -\grg\ can be tuned according to your needs and preferences. -The configuration files allow one to change some default settings -and the environment variable \comm{grg} defines the system -directory which can be used as the depository for -frequently used files. - -\subsection{Configuration Files} -\label{configsect} - -The configuration files allows one to establish -\begin{list}{$\bullet$}{\labelwidth=8mm\leftmargin=10mm} -\item Default dimension and signature. -\item Initial position of switches. -\item \reduce\ packages which must be preloaded. -\item Synonymy. -\item Default \grg\ start up method. -\end{list} - -There are two configuration files. First \emph{global} -configuration file \file{grgcfg.sl} defines the settings -\index{Global configuration file} -during system installation when \grg\ is compiled. -These global settings become permanent and can be changed only -if \grg\ is recompiled. The \emph{local} -configuration file \file{grg.cfg} allows one to override -global settings locally. -\index{Local configuration file} -When \grg\ starts it search the file \file{grg.cfg} -in current directory (folder) and if it is present -uses the corresponding settings. - -Below we are going to explain how to change settings in -both global and local configuration files but before -doing this we must emphasize that this need some care. -First, the configuration files use LISP command format -which differs from usual \grg\ commands. -Second, is something is wrong with configuration file -then no clear diagnostic is provided. -Finally, if global configuration is damaged you will -not be able to compile \grg. The best strategy is to -make a back-up copy of the configuration files before start -editing them. -Notice that lines preceded by the percent sign -\comm{\%} are ignored by the system (comments). - -Both local \file{grg.cfg} and global \file{grgcfg.sl} -configuration files have similar structure and can include -the following commands. - -Command\index{Signature!default}\index{Dimension!default} -\begin{listing} - (signature!> - + + + +) -\end{listing} -establishes default dimension 5 with the signature -$\scriptstyle(-,+,+,+,+)$. Do not forget \comm{!} and spaces between -\comm{+} and \comm{-}. This command \emph{must be present} -in the global configuration file \file{grgcfg.sl} -otherwise \grg\ cannot be compiled. - -The commands -\begin{listing} - (on!> page) - (off!> allfac) -\end{listing} -change default switch position. In this example we -turn on the switch \comm{PAGE} (this switch is defined -in DOS \reduce\ only and allows one to scroll back and forth -through input and output) and turn off switch -\comm{ALLFAC}. - -The command -\begin{listing} - (package!> taylor) -\end{listing} -makes the system to load \reduce\ package \file{taylor} -during \grg\ start. - -The command of the form\index{Synonymy} -\begin{listing} - (synonymous!> - ( affine aff ) - ( antisymmetric asy ) - ( components comp ) - ( unload save ) - ) -\end{listing} -defines synonymous words. The words in each line will be -equivalent in all \grg\ commands. - -Finally the command -\begin{listing} - (setq ![autostart!] nil) -\end{listing} -alters default \grg\ start up method. It makes sense only -in the global configuration file \file{grgcfg.sl}. -By default \grg\ is launched by single command -\begin{listing} - load grg; -\end{listing} -which firstly load the program into memory and then -automatically starts it. Unfortunately on some systems -this short method does not work properly: \grg\ shows wrong -timing during computations, the \comm{quit;} command returns -the control to \reduce\ session instead of terminating the -whole program. If the aforementioned option is activated then -\grg\ must be launched by two commands -\begin{listing} - load grg; - grg; -\end{listing} -which fixes the problems. Here first command just loads the program -into memory and second one starts it manually. Notice that -one can always use commands -\begin{listing} - load grg32; - grg; -\end{listing} -to start \grg\ manually. Command \comm{load grg32;} always -loads \grg\ into memory without starting it independently -on the option under consideration. - - -\subsection{System Directory} -\index{System directory} - -The environment variable \comm{grg} or \comm{GRG} -defines so called \grg\ system directory (folder). -The way of setting this variable is operating system -dependent. For example the following commands -can be used to set \comm{grg} variable in DOS, UNIX and -VAX/VMS respectively: -\begin{listing} - set grg=d:{\bs}xxx{\bs}yyy{\bs} {\rm DOS} - setenv grg /xxx/yyy/ {\rm UNIX} - define grg SYS$USER:[xxx.yyy] {\rm VAX/VMS} -\end{listing} -The value of the variable \comm{grg} must point -out to some directory. -In DOS and UNIX the directory -name must include trailing \comm{\bs} or \comm{/} -respectively. The command\cmdind{Show Status}\cmdind{Status} -\command{\opt{Show} Status;} -prints current system directory. - -When \grg\ tries to input some batch file containing -\grg\ commands it first searches it in the current working -directory and if the file is absent then it tries -to find it in the system directory. Therefore if you have -some frequently used files you can define the system directory -and move these files there. In this case it is not necessary -to keep them in each working directory. Notice \grg\ uses -the same strategy when opening local configuration file -\file{grg.cfg}. Thus if system directory is defined and it -contains the file \file{grg.cfg} the settings contained in -this file effectively overrides global settings without -recompiling \grg. - - -\section{Examples} - -In this section we want to demonstrate how \grg\ can be applied -to solve simple but realistic problem. -We want to calculate the Ricci tensor for the Robertson-Walker -metric by three different methods. - -First \grg\ task (program) -\begin{listing} - Coordinates t,r,theta,phi; - Function a(t); - Frame T0=d t, T1=a*d r, T2=a*r*d theta, T3=a*r*SIN(theta)*d phi; - ds2; - Find and Write Ricci Tensor; - RIC(\_j,\_k); -\end{listing} -defines the Robertson-Walker metric using the tetrad -formalism with the orthonormal Lorentzian tetrad $\theta^a$. -Using built-in formulas for the Ricci tensor the only one command -is required to accomplish out goal -{\tt Find and Write Ricci Tensor;}. The command {\tt ds2;} -just shows the metric we are dealing with. Notice that -command {\tt Find ...} gives the \emph{tetrad} components of the Ricci -tensor $R_{ab}$. Thus, in addition we print coordinate -components of the tensor $R_{\mu\nu}$ by the command -{\tt RIC(\_j,\_k);}. The hard-copy of the corresponding -\grg\ session is presented below \enlargethispage{4mm} -\begin{slisting} -<- Coordinates t, r, theta, phi; -<- Function a(t); -<- Frame T0=d t, T1=a*d r, T2=a*r*d theta, T3=a*r*SIN(theta)*d phi; -<- ds2; -Assuming Default Metric. -Metric calculated By default. 0.16 sec - - 2 2 2 2 2 2 2 2 2 2 2 - ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi - -<- Find and Write Ricci Tensor; -Sqrt det of metric calculated. 0.21 sec -Volume calculated. 0.21 sec -Vector frame calculated From frame. 0.21 sec -Inverse metric calculated From metric. 0.21 sec -Frame connection calculated. 0.38 sec -Curvature calculated. 0.49 sec -Ricci tensor calculated From curvature. 0.54 sec -Ricci tensor: - - - 3*DF(a,t,2) -RIC = ---------------- - 00 a -\newpage - 2 - DF(a,t,2)*a + 2*DF(a,t) -RIC = -------------------------- - 11 2 - a - - 2 - DF(a,t,2)*a + 2*DF(a,t) -RIC = -------------------------- - 22 2 - a - - 2 - DF(a,t,2)*a + 2*DF(a,t) -RIC = -------------------------- - 33 2 - a - -<- RIC(_j,_k); - - - 3*DF(a,t,2) -j=0 k=0 : ---------------- - a - - 2 -j=1 k=1 : DF(a,t,2)*a + 2*DF(a,t) - - 2 2 -j=2 k=2 : r *(DF(a,t,2)*a + 2*DF(a,t) ) - - 2 2 2 -j=3 k=3 : SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) -\end{slisting} -Tracing messages demonstrate that \grg\ automatically -applied several built-in equations to obtain required value of -$R_{ab}$. The metric is automatically assumed to be -Lorentzian $g_{ab}={\rm diag}(-1,1,1,1)$. -First \grg\ computed the frame connection 1-form $\omega^a{}_b$. -Next the curvature 2-form $\Omega^a{}_b$ was computed using -standard equation (\ref{omes}) on page \pageref{omes}. -Finally the Ricci tensor was obtained using -relation (\ref{rics}) on page \pageref{rics}. - -Second \grg\ task is similar to the first one: -\begin{listing} - Coordinates t,r,theta,phi; - Function a(t); - Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; - ds2; - Find and Write Ricci Tensor; -\end{listing} -The only difference is that now we work in the coordinate -formalism by assigning value to the metric rather than -frame. The frame is assumed to be holonomic automatically. -\begin{slisting} -<- Coordinates t, r, theta, phi; -<- Function a(t); -<- Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; -<- ds2; -Assuming Default Holonomic Frame. -Frame calculated By default. 0.11 sec - - 2 2 2 2 2 2 2 2 2 2 2 - ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi - -<- Find and Write Ricci Tensor; -Sqrt det of metric calculated. 0.22 sec -Volume calculated. 0.22 sec -Vector frame calculated From frame. 0.22 sec -Inverse metric calculated From metric. 0.27 sec -Frame connection calculated. 0.33 sec -Curvature calculated. 0.60 sec -Ricci tensor calculated From curvature. 0.60 sec -Ricci tensor: - - - 3*DF(a,t,2) -RIC = ---------------- - t t a - - 2 -RIC = DF(a,t,2)*a + 2*DF(a,t) - r r - - 2 2 -RIC = r *(DF(a,t,2)*a + 2*DF(a,t) ) - theta theta - - 2 2 2 -RIC = SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) - phi phi -\end{slisting} -Once again \grg\ uses the same built-in formulas to compute -the Ricci tensor but now all quantities have holonomic -indices instead of tetrad ones. - -Finally the third task demonstrate how \grg\ can be used -without built-in equations. Once again we use coordinate -formalism and declare two new objects the Christoffel symbols -\comm{Chr} and Ricci tensor \comm{Ric} -(since \grg\ is case sensitive they are different from the built-in -objects \comm{CHR} and \comm{RIC}). Next we use -well-known equations to compute these quantities -\begin{listing} - Coordinates t,r,theta,phi; - Function a(t); - Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; - ds2; - New Chr^a_b_c with s(2,3); - Chr(j,k,l)= 1/2*GI(j,m)*(@x(k)|G(l,m)+@x(l)|G(k,m)-@x(m)|G(k,l)); - New Ric_a_b with s(1,2); - Ric(j,k) = @x(n)|Chr(n,j,k) - @x(k)|Chr(n,j,n) - + Chr(n,m,n)*Chr(m,j,k) - Chr(n,m,k)*Chr(m,n,j); - Write Ric; -\end{listing} -The hard-copy of the corresponding session is -\begin{slisting} -<- Coordinates t, r, theta, phi; -<- Function a(t); -<- Metric G00=-1, G11=a^2, G22=(a*r)^2, G33=(a*r*SIN(theta))^2; -<- ds2; -Assuming Default Holonomic Frame. -Frame calculated By default. 0.16 sec - - 2 2 2 2 2 2 2 2 2 2 2 - ds = - d t + (a ) d r + (a *r ) d theta + (SIN(theta) *a *r ) d phi - -<- New Chr^a_b_c with s(2,3); -<- Chr(j,k,l)=1/2*GI(j,m)*(@x(k)|G(l,m)+@x(l)|G(k,m)-@x(m)|G(k,l)); -Inverse metric calculated From metric. 0.27 sec -<- New Ric_a_b with s(1,2); -<- Ric(j,k)=@x(n)|Chr(n,j,k)-@x(k)|Chr(n,j,n)+Chr(n,m,n)*Chr(m,j,k) - -Chr(n,m,k)*Chr(m,n,j); -<- Write Ric; -The Ric: - - - 3*DF(a,t,2) -Ric = ---------------- - t t a - - 2 -Ric = DF(a,t,2)*a + 2*DF(a,t) - r r -\newpage - 2 2 -Ric = r *(DF(a,t,2)*a + 2*DF(a,t) ) - theta theta - - 2 2 2 -Ric = SIN(theta) *r *(DF(a,t,2)*a + 2*DF(a,t) ) - phi phi -\end{slisting} - - - -\chapter{Formulas} -\parindent=0pt -\arraycolsep=1pt -\parskip=1.6mm plus 1mm minus 1mm - -This chapter describes in usual mathematical manner all \grg\ -built-in objects and formulas. The description is extremely short -since it is intended for reference only. -If not stated explicitly we use lower case greek letters -${\scriptstyle \alpha,\beta,\dots}$ for -holonomic (coordinate) indices; ${\scriptstyle a,b,c,d,m,n}$ for -anholonomic frame indices and ${\scriptstyle i,j,k,l}$ -for enumerating indices. - -To establish the relationship between \grg\ built-in object6s -and mathematical quantities we use the following notation -\[ -\mbox{\tt Frame Connection omega'a.b} = \omega^a{}_b -\] -This equality means that there is built-in object named -{\tt Frame Connection} having identifier {\tt omega} -which represent the frame connection 1-form $\omega^a{}_b$. -If the name is omitted then we deal with \emph{macro} object -(see page \pageref{macro}). The notation for indices -in the left-hand side of such equalities is the same -as in the {\tt New object} declaration and -is explained on page \pageref{indices}. - -This chapter contains not only definitions of all built-in -objects but all formulas which \grg\ knows and can apply -to find their value. If an object has -several formulas for its computation when each formula -is given together with the corresponding name which is printed -in the typewriter font. -In the case then an object has only one associated -formula the way name is usually omitted. - - -\section{Dimension and Signature} - -Let us denote the space-time dimensionality by $d$ -and $n$'th element of the signature specification -${\rm diag}{\scriptstyle(+1,-1,\dots)}$ by ${\rm diag}_n$ -($n$ runs from 0 to $d-1$). - -There are several macro objects which gives access to -the dimension and signature -\object{dim}{d} -\object{sdiag.idim}{{\rm diag}_i} -\object{sgnt \mbox{=} sign}{s=\prod^{d-1}_{i=0}{\rm diag}_i} -\object{mpsgn}{{\rm diag}_0} -\object{pmsgn}{-{\rm diag}_0} - -The macros (two equivalent ones) which give access to -coordinates -\object{X\^m \mbox{=} x\^m}{x^\mu} - - -\section{Metric, Frame and Basis} - -Frame $\theta^a$ and metric $g_{ab}$ plays the -fundamental role in \grg. Together they determine the -space-time line element -\begin{equation} -ds^2 = g_{ab}\,\theta^a\!\otimes\theta^b = - g_{\mu\nu}\,dx^\mu\!\otimes dx^\nu -\end{equation} - -The corresponding objects are -\object{Frame T'a}{\theta^a=h^a_\mu dx^\mu} -\object{Metric G.a.b}{g_{ab}} -and ``inverse'' objects are -\object{Vector Frame D.a}{\partial_a=h^\mu_a\partial_\mu} -\object{Inverse Metric GI'a'b}{g^{ab}} - -The frame can be computed by two ways. First, {\tt By default} -frame is assumed to be holonomic -\begin{equation} -\theta^a = dx^\alpha -\end{equation} -and {\tt From vector frame} -\begin{equation} -\theta^a= |h_a^\mu|^{-1} d x^\mu -\end{equation} - -The vector frame can be obtained {\tt From frame} -\begin{equation} -\partial_a= |h^a_\mu|^{-1} \partial_\mu -\end{equation} - -The metric can be computed {\tt By default} \index{Metric!default value} -\begin{equation} -g_{ab} = {\rm if}\ a=b\ {\rm then}\ {\rm diag}_a\ {\rm else}\ 0 -\end{equation} -or {\tt From inverse metric} -\begin{equation} -g_{ab} = |g^{ab}|^{-1} -\end{equation} - -The inverse metric can be computed {\tt From metric} -\begin{equation} -g^{ab} = |g_{ab}|^{-1} -\end{equation} - -The holonomic metric $g_{\mu\nu}$ and frame $h^a_\mu$ -are given by the macro objects: -\object{g\_m\_n}{g_{\mu\nu}} -\object{gi\^m\^n}{g^{\mu\nu}} -\object{h'a\_m}{h^a_\mu} -\object{hi.a\^m}{h_a^\mu} - -The metric determinants and related densities -\object{Det of Metric detG}{g={\rm det}|g_{ab}|} -\object{Det of Holonomic Metric detg}{{\rm det}|g_{\mu\nu}|} -\object{Sqrt Det of Metric sdetG}{\sqrt{sg}} - -The volume $d$-form -\object{Volume VOL}{\upsilon = \sqrt{sg}\,\theta^0\wedge\dots\wedge\,\theta^{d-1} -=\frac{1}{d!}{\cal E}_{a_0\dots a_{d-1}}\,\theta^{a_0}\wedge\dots\wedge\,\theta^{a_{d-1}}} - -The so called s-forms play the role of basis in the space of the -2-forms -\object{S-forms S'a'b}{S^{ab}=\theta^a\wedge\theta^b} - -The basis and corresponding inverse vector basis are used -when \grg\ works in the anholonomic mode -\seethis{See page \pageref{amode}.} -\object{Basis b'idim }{b^i=b^i_\mu dx^\mu} -\object{Vector Basis e.idim }{e_i=b_i^\mu\partial_\mu} -The basis can be computed {\tt From frame} -\begin{equation} -b^i=\theta^i -\end{equation} -or {\tt From vector basis} -\begin{equation} -b^i = |b_i^\mu|^{-1}dx^\mu -\end{equation} -The vector basis can be computed {\tt From basis} -\begin{equation} -e_i = |b^i_\mu|^{-1}\partial_\mu -\end{equation} - - -\section{Delta and Epsilon Symbols} - -Macro objects for Kronecker delta symbols -\object{del\^m\_n}{\delta^\mu_\nu} -\object{delh'a.b}{\delta^a_b} -and totally antisymmetric tensors -\object{eps.a.b.c.d}{{\cal E}_{abcd},\quad{\cal E}_{0123}=\sqrt{sg}} -\object{epsi'a'b'c'd}{{\cal E}^{abcd},\quad{\cal E}_{0123}=\frac{s}{\sqrt{sg}}} -\object{epsh\_m\_n\_k\_l}{{\cal E}_{\mu\nu\kappa\lambda},\quad{\cal E}_{0123}=\sqrt{s\,{\rm det}|g_{\mu\nu}|}} -\object{epsih\^m\^n\^k\^l}{{\cal E}^{\mu\nu\kappa\lambda},\quad{\cal E}_{0123}=\frac{s}{\sqrt{s\,{\rm det}|g_{\mu\nu}|}}} -The definition for epsilon-tensors is given for dimension 4. -The generalization to other dimensions is obvious. - - -\section{Dualization} - -We use the following definition for the dualization -operation. For any $p$-form -\begin{equation} -\omega_p=\frac{1}{p!}\omega_{\alpha_1\dots\alpha_p}dx^{\alpha_1}\wedge -\dots\wedge dx^{\alpha_p} -\end{equation} -the dual $(d-p)$-form is -\begin{equation} -*\omega_p=\frac{1}{p!(d-p)!}{\cal E}_{\alpha_1\dots\alpha_{d-p}} -{}^{\beta_1\dots\beta_p}\,\omega_{\beta_1\dots\beta_p}\, -dx^{\alpha_1}\wedge\dots\wedge dx^{\alpha_{d-p}} -\end{equation} - -The equivalent relation which also uniquely defines the $*$ -operation is -\begin{equation} -*(\theta^{a_1}\wedge\dots\wedge \theta^{a_p}) = -(-1)^{p(d-p)} \partial_{a_p}\ipr\dots\partial_{a_1}\ipr\,\upsilon -\end{equation} - -With such convention we have the following identities -\begin{eqnarray} -**\omega_p &=& s(-1)^{p(d-p)}\,\omega_p \\[0.5mm] -*\upsilon &=& s \\[0.5mm] -*1 &=& \upsilon -\end{eqnarray} - - -\section{Spinors} -\label{spinors1} - -The notion of spinors in \grg\ is restricted to - 4-dimensional spaces of Lorentzian signature ${\scriptstyle(-,+,+,+)}$ -or ${\scriptstyle(+,-,-,-)}$ only. In this section the upper sign relates to the -signature ${\scriptstyle(-,+,+,+)}$ and lower one to -${\scriptstyle(+,-,-,-)}$. - -In addition to work with spinors the metric must have the following -form which we call the \emph{standard null metric} \index{Metric!Standard Null} -\index{Standard null metric}\index{Spinors}\index{Spinors!Standard null metric} -\begin{equation} -g_{ab}=g^{ab}=\pm\left(\begin{array}{rrrr} -0 & -1 & 0 & 0 \\ --1 & 0 & 0 & 0 \\ -0 & 0 & 0 & 1 \\ -0 & 0 & 1 & 0 -\end{array}\right) -\end{equation} -Such value of the metric can be established by the command -\cmdind{Null Metric} -{\tt Null metric;}. - -Therefore the line-element for spinorial formalism has the form -\begin{equation} -ds^2 = \pm(-\theta^0\!\otimes\theta^1 --\theta^1\!\otimes\theta^0 -+\theta^2\!\otimes\theta^3 -+\theta^3\!\otimes\theta^2) -\end{equation} - -We require also the conjugation rules for this null tetrad (frame) be -\begin{equation} -\overline{\theta^0}=\theta^0,\quad -\overline{\theta^1}=\theta^1,\quad -\overline{\theta^2}=\theta^3,\quad -\overline{\theta^3}=\theta^2 -\end{equation} - -For such a metric and frame we fix sigma-matrices in the -following form \index{Sigma matrices} -\begin{eqnarray} \label{sigma} -&&\sigma_0{}^{1\dot{1}}= -\sigma_1{}^{0\dot{0}}= -\sigma_2{}^{1\dot{0}}= -\sigma_3{}^{0\dot{1}}=1 \\[1mm] && -\sigma^0{}_{1\dot{1}}= -\sigma^1{}_{0\dot{0}}= -\sigma^2{}_{1\dot{0}}= -\sigma^3{}_{0\dot{1}}=\mp1 -\end{eqnarray} - -The sigma-matrices obey the rules -\begin{eqnarray} -g_{mn}\sigma^m\!{}_{A\dot B}\sigma^n\!{}_{C\dot D} &=& -\mp \epsilon_{AC}\epsilon_{\dot B\dot D} \\[1mm] -\sigma^{aM\dot N}\sigma^b\!{}_{M\dot N} &=& \mp g^{ab} -\end{eqnarray} - -The antisymmetric SL(2,C) spinor metric -\begin{equation} -\epsilon_{AB}=\epsilon^{AB} -=\epsilon_{\dot A\dot B} -=\epsilon^{\dot A\dot B}= -\left(\begin{array}{rr} -0 & 1 \\ --1 & 0 -\end{array}\right) -\end{equation} -can be used to raise and lower spinor indices -\begin{equation} -\varphi^A=\varphi_B\,\epsilon^{BA},\qquad -\varphi_A=\epsilon_{AB}\,\varphi^B -\end{equation} - -The following macro objects represent standard -spinorial quantities -\object{DEL'A.B}{\delta^A_B} -\object{EPS.A.B}{\epsilon_{AB}} -\object{EPSI'A'B}{\epsilon^{AB}} -\object{sigma'a.A.B\cc}{\sigma^a\!{}_{A\dot B}} -\object{sigmai.a'A'B\cc}{\sigma_a{}^{A\dot B}} - -The relationship between tensors and spinors -is established by the sigma-matrices -\begin{eqnarray} -X^a &\tsst& X^{A\dot A}=A^a\sigma_a{}^{A\dot A} \\ -X_a &\tsst& X_{A\dot A}=A_a\sigma^a\!{}_{A\dot A} -\end{eqnarray} -where sigma-matrices are given by Eq. (\ref{sigma}) -We shall denote similar equations by the sign $\tsst$ -conserving alphabetical relationship between tensor indices in the -left-hand side and spinorial one in the right-hand side: -$\scriptstyle a\tsst A\dot A$, $\scriptstyle b\tsst B\dot B$. - -There is one quite important special case. Any real -antisymmetric tensor $X_{ab}$ are equivalent to the -pair of conjugated irreducible (symmetric) spinors -\begin{eqnarray} -&& X_{ab}=X_{[ab]} \tsst X_{A\dot AB\dot B}= -\epsilon_{AB} X_{\dot A\dot B} + \epsilon_{\dot A\dot B}X_{AB} -\nonumber\\[1mm] -&& X_{AB}=\frac{1}{2}X_{A\dot AB\dot B}\epsilon^{\dot A\dot B},\ - X_{\dot A\dot B}=\frac{1}{2}X_{A\dot AB\dot B}\epsilon^{AB} -\end{eqnarray} -The explicit form of these relations for the sigma-matrices -(\ref{sigma}) is -\begin{equation} -\begin{array}{rclrcl} -X_0 &=& X_{13} & X_{\dot0} &=& X_{12} \\[1mm] -X_1 &=&-\frac{1}{2}(X_{01}-X_{23})\qquad & X_{\dot1} &=& --\frac{1}{2}(X_{01}+X_{23}) \\[1mm] -X_2 &=& -X_{02} & X_{\dot2} &=& -X_{03} -\end{array}\label{asys} -\end{equation} -and the ``inverse'' relation -\begin{equation} -\begin{array}{rclrcl} -X_{01} &=& -X_1-X_{\dot1},\qquad & X_{23} &=& X_1-X_{\dot1}, \\[1mm] -X_{02} &=& -X_2, & X_{12} &=& X_{\dot0}, \\[1mm] -X_{03} &=& -X_{\dot 2}, & X_{13} &=& X_0 -\end{array}\label{asyt} -\end{equation} - -We shall apply the relations (\ref{asys}) and (\ref{asyt}) to various -antisymmetric quantities. In particular the {\tt Spinorial S-forms} -\object{Undotted S-forms SU.AB}{S_{AB}} -\object{Dotted S-forms SD.AB\cc}{S_{\dot A\dot B}} -The {\tt Standard way} to compute these quantities uses -relations (\ref{asys}) -\begin{equation} - S_{ab}=\theta_a\wedge\theta_b \tsst -\epsilon_{AB} S_{\dot A\dot B} + \epsilon_{\dot A\dot B}S_{AB} -\end{equation} -Spinorial S-forms are self dual -\begin{equation} -*S_{AB}=iS_{AB},\qquad -*S_{\dot A\dot B}=-iS_{\dot A\dot B} -\end{equation} -and exteriorly orthogonal -\begin{equation} -S_{AB}\wedge S_{CD}=-\frac{i}2\upsilon(\epsilon_{AC}\epsilon_{BD}+ -\epsilon_{AD}\epsilon_{BC}),\quad S_{AB}\wedge S_{\dot C\dot D}=0 -\end{equation} - -There is one subtle pint concerning tensor quantities in the -spinorial formalism. Since spinorial null tetrad is complex -with the conjugation rule $\overline{\theta^2}=\theta^3$ -all tensor quantities represented in this frame also becomes -complex with similar conjugation rules for any tensor index. -There is special macro object {\tt cci} which performs such -``index conjugation'': {\tt cci{0}=0}, {\tt cci(1)=1}, -{\tt cci{2}=3}, {\tt cci(3)=2}. Therefore the correct expression -for the $\overline{\theta^a}$ is {\tt \cc T(cci(a))} but not -{\tt \cc T(a)}. - - - -\section{Connection, Torsion and Nonmetricity} -\label{conn1} - -Covariant derivatives and differentials for -quantities having frame and coordinate indices are -\begin{eqnarray} -DX^a{}_b &=& dX^a{}_b -+ \omega^a{}_m\wedge X^m{}_b - \omega^m{}_b\wedge X^a{}_m \\[1mm] -DX^\mu{}_\nu &=& dX^\mu{}_\nu -+ \Gamma^\mu{}_\pi\wedge X^\pi{}_\nu - \Gamma^\pi{}_\nu\wedge X^\mu{}_\pi -\end{eqnarray} - -The corresponding built-in connection 1-forms are -\object{Frame Connection omega'a.b}{\omega^a{}_b=\omega^a{}_{b\mu}dx^\mu} -\object{Holonomic Connection GAMMA\^m\_n} -{\Gamma^\mu{}_\nu=\Gamma^\mu{}_{\nu\pi}dx^\pi} - -Frame connection can be computed {\tt From holonomic connection} -\begin{equation} -\omega^a{}_b = \Gamma^a{}_b + dh^\mu_b\,h^a_\mu -\end{equation} -and inversely holonomic connection can be obtained -{\tt From frame connection} -\begin{equation} -\Gamma^\mu{}_\nu=\omega^\mu{}_\nu + dh^b_\nu\,h^\mu_b -\end{equation} - -By default these connections are Riemannian (i.e. symmetric and -metric compatible). To work with nonsymmetric -connection with torsion the switch \comm{TORSION}\swinda{TORSION} -must be turned on. Then the torsion 2-form is -\object{Torsion THETA'a}{\Theta^a=\frac12Q^a{}_{pq}S^{pq},\quad -Q^a{}_{bc}=\Gamma^a{}_{bc}-\Gamma^a_{cb}} -Finally to work with non metric-compatible -spaces with nonmetricity the switch \comm{NONMETR}\swinda{NONMETR} -must be turned on. The nonmetricity 1-form is -\object{Nonmetricity N.a.b}{N_{ab}=N_{ab\mu}dx^\mu, -\quad N_{ab\mu}=-\nabla_\mu g_{ab}} -In general any torsion or nonmetricity related object is -defined iff the corresponding switch is on. - -If either \comm{TORSION} or \comm{NONMETR} is on then Riemannian -versions of the connection 1-forms are available as well -\object{Riemann Frame Connection romega'a.b} -{\rim{\omega}{}^a{}_b} -\object{Riemann Holonomic Connection RGAMMA\^m\_n} -{\rim{\Gamma}{}^\mu{}_\nu} - -The Riemann holonomic connection can be obtained -{\tt From Riemann frame connection} -\begin{equation} -\rim{\Gamma}{}^\mu{}_\nu=\rim{\omega}{}^\mu{}_\nu + dh^b_\nu\,h^\mu_b -\end{equation} - - - -If torsion is nonzero but nonmetricity vanishes -(\comm{TORSION} is on, \comm{NONMETR} is off) then -the difference between the connection and Riemann connection -is called the contorsion 1-form -\object{Contorsion KQ'a.b}{\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b= -\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_{b\mu}dx^\mu= -\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} - -If nonmetricity is nonzero but torsion vanishes -(\comm{TORSION} is off, \comm{NONMETR} is on) then -the difference between the connection and Riemann connection -is called the nonmetricity defect -\object{Nonmetricity Defect KN'a.b} -{\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b= -\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_{b\mu}dx^\mu= -\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} - -Finally if both torsion and nonmetricity are nonzero -(\comm{TORSION} and \comm{NONMETR} are on) then we -\object{Connection Defect K'a.b} -{K^a{}_b=K^a{}_{b\mu}dx^\mu= -\Gamma^a{}_b-\rim{\Gamma}{}^a{}_b} -\begin{equation} -K^a{}_b = \stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b -+ \stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b -\end{equation} - - -For the sake of convenience we introduce also macro objects -which compute the usual Christoffel symbols -\object{CHR\^m\_n\_p }{ \{{}^\mu_{\nu\pi}\} = -\frac{1}{2}g^{\mu\tau}(\partial_\pi g_{\nu\tau} -+\partial_\nu g_{\pi\tau} --\partial_\tau g_{\nu\pi})} -\object{CHRF\_m\_n\_p }{ [{}_{\mu},_{\nu\pi}] = -\frac{1}{2}(\partial_\pi g_{\nu\mu} -+\partial_\nu g_{\pi\mu} --\partial_\mu g_{\nu\pi})} -\object{CHRT\_m }{ \{{}^\pi_{\pi\mu}\} = -\frac{1}{2{\rm det}|g_{\alpha\beta}|}\partial_\mu\left( -{\rm det}|g_{\alpha\beta}|\right)} - -The connection, frame, metric, torsion and nonmetricity are -related to each other by the so called structural equations -which in the most general case read -\begin{eqnarray} -&& D\theta^a + \Theta^a = 0 \nonumber\\[2mm] -&& Dg_{ab} + N_{ab} = 0 \label{str0} -\end{eqnarray} -or in the equivalent ``explicit'' form -\begin{equation} -\begin{array}{ll} -\omega^a{}_b\wedge\theta^b = -t^a,\qquad & t^a=d\theta^a+\Theta^a,\\[2mm] -\omega_{ab}+\omega_{ba} = n_{ab},\qquad & n_{ab}=dg_{ab}+N_{ab} \label{str} -\end{array} -\end{equation} - -The solution to equations (\ref{str}) are given by the relation -\begin{equation} -\omega^a{}_b = -\frac{1}{2}\left[ -\partial^a\ipr t_b + \partial_b\ipr t^a + n^a{}_b -+\big(\partial^a\ipr(\partial_b\ipr t_c-n_{bc}) -+\partial_b\ipr n^a{}_c\big)\theta^c\right] \label{solstr} -\end{equation} - -For various specific values of $n_{ab}$ and $t^a$ equations -(\ref{str}) and (\ref{solstr}) can be used for different purposes. - -In the most general case (\ref{solstr}) is the {\tt Standard way} to -compute connection 1-form $\omega^a{}_b$. -The torsion and nonmetricity are included in -these equations depending on the switches \comm{TORSION} and -\comm{NONMETR}. - -The same equation (\ref{solstr}) with $n_{ab}=dg_{ab}$ and -$t^a=d\theta^a$ is the {\tt Standard way} to find Riemann -frame connection $\rim{\omega}{}^a{}_b$. - -If torsion is nonzero then $\omega^a{}_b$ can be computed -{\tt From contorsion} -\begin{equation} -\omega^a{}_b = \rim{\omega}{}^a{}_b -+ \stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b \label{a1} -\end{equation} -where $\rim{\omega}{}^a{}_b$ is given by Eq. (\ref{solstr}). - -Similarly if nonmetricity is nonzero then $\omega^a{}_b$ can be computed -{\tt From nonmetricity defect} -\begin{equation} -\omega^a{}_b = \rim{\omega}{}^a{}_b -+ \stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b \label{a2} -\end{equation} -where $\rim{\omega}{}^a{}_b$ is given by Eq. (\ref{solstr}). - -Finally if both torsion and nonmetricity are -nonzero then $\omega^a{}_b$ can be computed -{\tt From connection defect} -\begin{equation} -\omega^a{}_b = \rim{\omega}{}^a{}_b + K^a{}_b \label{a3} -\end{equation} -where $\rim{\omega}{}^a{}_b$ is given by Eq. (\ref{solstr}). - -The Riemannian part of connection in Eqs. (\ref{a1}), -(\ref{a2}), (\ref{a3}) are directly computed by Eq. (\ref{solstr}) -(not via the object \comm{romega}). - -The contorsion $\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b$ -is obtained {\tt From torsion} by (\ref{solstr}) -with $t^a=\Theta^a$, $n_{ab}=0$. - -The nonmetricity defect $\stackrel{\scriptscriptstyle N}{K}\!{}^a{}_b$ -is obtained {\tt From nonmetricity} by (\ref{solstr}) -with $t^a=0$, $n_{ab}=N_{ab}$. - -Analogously (\ref{solstr}) with $t^a=\Theta^a$, $n_{ab}=N_{ab}$ -is the {\tt Standard way} to compute the connection defect $K^a{}_b$. - -The torsion $\Theta^a$ can be calculated {\tt From contorsion} -\begin{equation} -\Theta^a = -\stackrel{\scriptscriptstyle Q}{K}\!{}^a{}_b\wedge\theta^b -\end{equation} -or {\tt From connection defect} -\begin{equation} -\Theta^a = -K^a{}_b\wedge\theta^b -\end{equation} - -The nonmetricity $N_{ab}$ can be computed {\tt From nonmetricity defect} -\begin{equation} -N_{ab} = \stackrel{\scriptscriptstyle N}{K}_{ab}+ -\stackrel{\scriptscriptstyle N}{K}_{ba} -\end{equation} -or {\tt From connection defect} -\begin{equation} -N_{ab} = K_{ab}+K_{ba} -\end{equation} - - -\section{Spinorial Connection and Torsion} - -Spinorial connection is defined in \grg\ iff nonmetricity -is zero and switch \comm{NONMETR} is turned off. -The upper sign in this section correspond to the signature -${\scriptstyle(-,+,+,+)}$ while lower one to the signature -${\scriptstyle(+,-,-,-)}$. - -Spinorial connection is defined by the equation -\begin{equation} -DX^A_{\dot B} = dX^A{}_{\dot B} -\mp\omega^A{}_M\,X^M{}_{\dot B} -\pm\omega^{\dot M}{}_{\dot B}\,X^A{}_{\dot M} -\end{equation} -where due to antisymmetry of the frame connection -$\omega_{ab}=\omega_{[ab]}$ we have {\tt Spinorial connection} -1-forms -\begin{equation} -\omega_{ab} \tsst -\epsilon_{AB} \omega_{\dot A\dot B} -+ \epsilon_{\dot A\dot B} \omega_{AB} -\end{equation} -\object{Undotted Connection omegau.AB}{\omega_{AB}} -\object{Dotted Connection omegad.AB\cc}{\omega_{\dot A\dot B}} - -The spinorial connection 1-forms -$\omega_{AB}$ and $\omega_{\dot A\dot B}$ -can be calculated {\tt From frame connection} by the -standard spinor $\tsst$ tensor relation (\ref{asys}). - -Inversely the frame connection $\omega_{ab}$ can be -found {\tt From spinorial connection} by relation (\ref{asyt}). - -Since $\omega_{ab}$ is real the spinorial equivalents -$\omega_{AB}$ and $\omega_{\dot A\dot B}$ can be computed from -each other {\tt By conjugation} -\begin{equation} -\omega_{\dot A\dot B}=\overline{\omega_{AB}},\qquad -\omega_{AB}=\overline{\omega_{\dot A\dot B}} -\end{equation} - -If torsion is nonzero (\comm{TORSION} is on) when we have -in addition the {\tt Riemann spinorial connection} -\object{Riemann Undotted Connection romegau.AB}{\rim{\omega}_{AB}} -\object{Riemann Dotted Connection romegad.AB\cc}{\rim{\omega}_{\dot A\dot B}} - -The Riemann spinorial connection $\rim{\omega}_{AB}$ -can be calculated by {\tt Standard way} -\begin{equation} -\stackrel{{\scriptscriptstyle\{\}}}{\omega}_{AB}= \label{ssolver} -\pm i*[ d S_{AB}\wedge\theta_{C\dot C} - -\epsilon_{C(A} d S_{B)M}\wedge \theta^M_{\ \ \dot C}]\theta^{C\dot C} -\end{equation} -The conjugated relation is used for $\rim{\omega}_{\dot A\dot B}$. - -The {\tt Spinorial contorsion} 1-forms -\object{Undotted Contorsion KU.AB}{\stackrel{\scriptscriptstyle Q}{K}\!{}_{AB}} -\object{Dotted Contorsion KD.AB\cc}{\stackrel{\scriptscriptstyle Q}{K}\!{}_{\dot A\dot B}} -are the spinorial analogues of the contorsion 1-form -\begin{equation} -\stackrel{\scriptscriptstyle Q}{K}_{ab} \tsst -\epsilon_{AB} \stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B} -+ \epsilon_{\dot A\dot B} \stackrel{\scriptscriptstyle Q}{K}_{AB} -\end{equation} - -The spinorial contorsion 1-forms -$\stackrel{\scriptscriptstyle Q}{K}_{AB}$ and $\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}$ -can be calculated {\tt From contorsion} by the -standard spinor $\tsst$ tensor relation (\ref{asys}). - -Inversely the contorsion $\stackrel{\scriptscriptstyle Q}{K}_{ab}$ can be -found {\tt From spinorial contorsion} by relation (\ref{asyt}). - -The spinorial equivalents -$\stackrel{\scriptscriptstyle Q}{K}_{AB}$ and $\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}$ -can be computed from -each other {\tt By conjugation} -\begin{equation} -\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}=\overline{\stackrel{\scriptscriptstyle Q}{K}_{AB}},\qquad -\stackrel{\scriptscriptstyle Q}{K}_{AB}=\overline{\stackrel{\scriptscriptstyle Q}{K}_{\dot A\dot B}} -\end{equation} - -The {\tt Standard way} to find $\omega_{AB}$ is -\begin{equation} -\omega_{AB} = \rim{\omega}_{AB}+\stackrel{\scriptscriptstyle Q}{K}_{AB} -\end{equation} -where $\rim{\omega}_{AB}$ is given directly by Eq. (\ref{ssolver}). -The conjugated Eq. is used for $\omega_{\dot A\dot B}$. - - -\section{Curvature} - -The curvature 2-form -\object{Curvature OMEGA'a.b}{\Omega^a{}_b= -\frac{1}{2}R^a_{bcd}\,S^{cd}} -can be computed {\tt By standard way} -\begin{equation} -\Omega^a{}_b = d\omega^a{}_b + \omega^a{}_n \wedge \omega^n{}_b \label{omes} -\end{equation} - -The Riemann curvature tensor is given by the relation -\object{Riemann Tensor RIM'a.b.c.d}{R^a{}_{bcd}= -\partial_d\ipr\partial_c\ipr\Omega^a{}_b} - -The Ricci tensor -\object{Ricci Tensor RIC.a.b}{R_{ab}} -can be computed {\tt From Curvature} -\begin{equation} -R_{ab} = \partial_b\ipr\partial_m\ipr\Omega^m{}_a \label{rics} -\end{equation} -or {\tt From Riemann tensor} -\begin{equation} -R_{ab} = R^m{}_{amb} -\end{equation} - -The -\object{Scalar Curvature RR}{R} -can be computed {\tt From Ricci Tensor} -\begin{equation} -R = R_{mn}\,g^{mn} -\end{equation} - -The Einstein tensor is given by the relation -\object{Einstein Tensor GT.a.b}{G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R} - -If nonmetricity is nonzero (\comm{NONMETR} is on) then we have -\object{Homothetic Curvature OMEGAH}{\OO{h}} -\object{A-Ricci Tensor RICA.a.b}{\RR{A}_{ab}} -\object{S-Ricci Tensor RICS.a.b}{\RR{S}_{ab}} - -They can be calculated {\tt From curvature} by the -relations -\begin{equation} -\OO{h}=\Omega^n{}_n -\end{equation} -\begin{equation} -\RR{A}_{ab}= \partial_b\ipr\partial^m\ipr\Omega_{[ma]} -\end{equation} -\begin{equation} -\RR{S}_{ab}= \partial_b\ipr\partial^m\ipr\Omega_{(ma)} -\end{equation} -and the scalar curvature can be computed {\tt From A-Ricci tensor} -\begin{equation} -R = \RR{A}_{mn}g^{mn} -\end{equation} - - -\section{Spinorial Curvature} - -Spinorial curvature is defined in \grg\ iff nonmetricity -is zero and switch \comm{NONMETR} is turned off. -The upper sign in this section correspond to the signature -${\scriptstyle(-,+,+,+)}$ while lower one to the signature -${\scriptstyle(+,-,-,-)}$. - -The {\tt Spinorial curvature} 2-forms -\object{Undotted Curvature OMEGAU.AB}{\Omega_{AB}} -\object{Dotted Curvature OMEGAD.AB\cc}{\Omega_{\dot A\dot B}} -is related to the curvature 2-form $\Omega_{ab}$ by the standard -relation -\begin{equation} -\Omega_{ab} \tsst -\epsilon_{AB} \Omega_{\dot A\dot B} -+ \epsilon_{\dot A\dot B} \Omega_{AB} -\end{equation} - -The spinorial curvature 1-forms -$\Omega_{AB}$ and $\Omega_{\dot A\dot B}$ -can be calculated {\tt From curvature} by the -relation (\ref{asys}). - -The frame curvature $\Omega_{ab}$ can be -found {\tt From spinorial curvature} by relation (\ref{asyt}). - -The $\Omega_{AB}$ and $\Omega_{\dot A\dot B}$ can be -computed from each other {\tt By conjugation} -\begin{equation} -\Omega_{\dot A\dot B}=\overline{\Omega_{AB}},\qquad -\Omega_{AB}=\overline{\Omega_{\dot A\dot B}} -\end{equation} - -The {\tt Standard way} to calculate $\Omega_{AB}$ is -\begin{equation} -\Omega_{AB} = d\omega_{AB} \pm \omega_A{}^M\wedge\omega_{MB} -\end{equation} -The conjugated relation is used for $\Omega_{\dot A\dot B}$. - - -\section{Curvature Decomposition} - -In general curvature consists of 11 irreducible pieces. -If nonmetricity is nonzero then one can -perform decomposition -\begin{equation} -R_{abcd}=\RR{A}_{abcd}+\RR{S}_{abcd},\qquad -\RR{A}_{abcd}=R_{[ab]cd},\qquad -\RR{S}_{abcd}=R_{(ab)cd} -\end{equation} -Here the S-part of the curvature vanishes identically if -nonmetricity is zero and we consider further decomposition -of A and S parts independently. - -First we consider the A-part of the curvature. It can be -decomposed into 6 pieces -\begin{equation} -\RR{A}_{abcd} = -\RR{w}_{abcd}+ -\RR{c}_{abcd}+ -\RR{r}_{abcd}+ -\RR{a}_{abcd}+ -\RR{b}_{abcd}+ -\RR{d}_{abcd} -\end{equation} -Here first three terms are the well-known irreducible pieces -of the Riemannian curvature while last three terms vanish if -torsion is zero. The corresponding 2-forms are -\object{Weyl 2-form OMW.a.b } -{\OO{w}_{ab} = \frac12 \RR{w}_{abcd}\,S^{cd}} -\object{Traceless Ricci 2-form OMC.a.b } -{\OO{c}_{ab} = \frac12 \RR{c}_{abcd}\,S^{cd}} -\object{Scalar Curvature 2-form OMR.a.b } -{\OO{r}_{ab} = \frac12 \RR{r}_{abcd}\,S^{cd}} -\object{Ricanti 2-form OMA.a.b } -{\OO{a}_{ab} = \frac12 \RR{a}_{abcd}\,S^{cd}} -\object{Traceless Deviation 2-form OMB.a.b } -{\OO{b}_{ab} = \frac12 \RR{b}_{abcd}\,S^{cd}} -\object{Antisymmetric Curvature 2-form OMD.a.b } -{\OO{d}_{ab} = \frac12 \RR{d}_{abcd}\,S^{cd}} - -The {\tt Standard way} to find these quantities is given -by the following formulas. -\begin{equation} -\OO{r}_{ab} = \frac{1}{d(d-1)}R\,S_{ab} -\end{equation} -\begin{equation} -\OO{c}_{ab} = \frac{1}{(d-2)}\left[ -C_{am}\,\theta^m\!\wedge\theta_b --C_{bm}\,\theta^m\!\wedge\theta_a\right],\quad -C_{ab}=\RR{A}_{(ab)}-\frac{1}{d}g_{ab}R -\end{equation} -\begin{equation} -\OO{a}_{ab} = \frac{1}{(d-2)}\left[ -A_{am}\,\theta^m\!\wedge\theta_b --A_{bm}\,\theta^m\!\wedge\theta_a\right],\quad -A_{ab}=\RR{A}_{[ab]} -\end{equation} -\begin{equation} -\OO{d}_{ab} = \frac{1}{12}\partial_b\ipr\partial_a\ipr -(\OO{A}_{mn}\wedge\theta^m\!\wedge\theta^n) -\end{equation} -\begin{equation} -\OO{b}_{ab} =\frac{1}{2}\left[ -\partial_b\ipr(\theta^m\!\wedge\OO{A0}_{am}) --\partial_a\ipr(\theta^m\!\wedge\OO{A0}_{bm}) -\right] -\end{equation} -where -\[ -\OO{A0}_{ab} = -\OO{A}_{ab} --\OO{c}_{ab} --\OO{r}_{ab} --\OO{a}_{ab} --\OO{d}_{ab} -\] -And finally -\begin{equation} -\OO{w}_{ab} = -\OO{A}_{ab} --\OO{c}_{ab} --\OO{r}_{ab} --\OO{a}_{ab} --\OO{b}_{ab} --\OO{d}_{ab} -\end{equation} - -If $d=2$ then $\OO{A}_{ab}$ turns out to be irreducible and -coincides with the scalar curvature irreducible piece -\begin{equation} -\OO{A}_{ab} = \OO{r}_{ab} -\end{equation} - -Now we consider the decomposition of the S curvature part which -is nonzero iff nonmetricity is nonzero. First we consider -the case $d\geq3$. In this case we have 5 irreducible components -\begin{equation} -\RR{S}_{abcd} = -\RR{h}_{abcd}+ -\RR{sc}_{abcd}+ -\RR{sa}_{abcd}+ -\RR{v}_{abcd}+ -\RR{u}_{abcd} -\end{equation} - -The corresponding 2-forms are -\object{Homothetic Curvature 2-form OSH.a.b } -{\OO{h}_{ab} = \frac12 \RR{h}_{abcd}\,S^{cd}} -\object{Antisymmetric S-Ricci 2-form OSA.a.b } -{\OO{sa}_{ab} = \frac12 \RR{sa}_{abcd}\,S^{cd}} -\object{Traceless S-Ricci 2-form OSC.a.b } -{\OO{sc}_{ab} = \frac12 \RR{sc}_{abcd}\,S^{cd}} -\object{Antisymmetric S-Curvature 2-form OSV.a.b } -{\OO{v}_{ab} = \frac12 \RR{v}_{abcd}\,S^{cd}} -\object{Symmetric S-Curvature 2-form OSU.a.b } -{\OO{u}_{ab} = \frac12 \RR{u}_{abcd}\,S^{cd}} - - -The {\tt Standard way} to compute the decomposition is -\begin{equation} -\OO{h}_{ab}=-\frac{1}{(d^2-4)}\left[ -\theta_a\wedge\partial_b\ipr\OO{h}{} -+\theta_b\wedge\partial_a\ipr\OO{h}{} --g_{ab}\OO{h}{}d\right] -\end{equation} -\begin{equation} -\OO{sa}_{ab} =\frac{d}{(d^2-4)}\left[ -\theta_a\wedge(\RR{S}_{[bm]}\wedge\theta^m) -+\theta_b\wedge(\RR{S}_{[am]}\wedge\theta^m) --\frac{2}{d}g_{ab}\,\RR{S}_{cd}S^{cd}\right] -\end{equation} -\begin{equation} -\OO{sc}_{ab} =\frac{1}{d}\left[ -\theta_a\wedge(\RR{S}_{(bm)}\wedge\theta^m) -+\theta_b\wedge(\RR{S}_{(am)}\wedge\theta^m)\right] \label{ccc} -\end{equation} -\begin{equation} -\OO{v}_{ab} = \frac{1}{4}\left[ -\partial_a\ipr(\OO{S0}_{bm}\wedge\theta^m) -+\partial_b\ipr(\OO{S0}_{am}\wedge\theta^m)\right] -\end{equation} -where -\[ -\OO{S0}_{ab} = -\OO{S}_{ab} --\OO{h}_{ab} --\OO{sa}_{ab} --\OO{sc}_{ab} -\] -And finally -\begin{equation} -\OO{u}_{ab} = -\OO{S}_{ab} --\OO{h}_{ab} --\OO{sa}_{ab} --\OO{sc}_{ab} --\OO{v}_{ab} -\end{equation} - -If $d=2$ then only the h- and sc-components are nonzero. -The $\OO{sc}_{ab}$ are given by (\ref{ccc}) and -\begin{equation} -\OO{h}_{ab} = \OO{S}_{ab}-\OO{sc}_{ab} -\end{equation} - -\begin{center} -\begin{tabular}{|c|c|c|} -\hline object & exists if & and has $n$ components \\ -\hline -\vv$R_{abcd}$ & & $\frac{d^3(d-1)}{2}$ \\[1mm] -\hline\vv$\rim{R}{}_{abcd}$ & & $\frac{d^2(d^2-1)}{12}$ \\[1mm] -\hline\vv$\RR{A}_{abcd}$ & & $\frac{d^2(d-1)^2}{4}$ \\[1mm] -\hline\vv$\RR{S}_{abcd}$ & & $\frac{d^2(d^2-1)}{4}$ \\[1mm] -\hline\vv$\RR{w}_{abcd}$ & $d\geq4$ & $\frac{d(d+1)(d+2)(d-3)}{12}$ \\ -\vv$\RR{c}_{abcd}$ & $d\geq3$ & $\frac{(d+2)(d-1)}{2}$ \\ -\vv$\RR{r}_{abcd}$ & & $1$ \\[1mm] -\hline\vv$\RR{a}_{abcd}$ & $d\geq3$ & $\frac{d(d-1)}{2}$ \\ -\vv$\RR{b}_{abcd}$ & $d\geq4$ & $\frac{d(d-1)(d+2)(d-3)}{8}$ \\ -\vv$\RR{d}_{abcd}$ & $d\geq4$ & $\frac{d(d-1)(d-2)(d-3)}{24}$ \\[1mm] -\hline\vv$\RR{h}_{abcd}$ & & $\frac{d(d-1)}{2}$ \\ -\vv$\RR{sa}_{abcd}$ & $d\geq3$ & $\frac{d(d-1)}{2}$ \\ -\vv$\RR{sc}_{abcd}$ & & $\frac{(d+2)(d-1)}{2}$ \\ -\vv$\RR{v}_{abcd}$ & $d\geq4$ & $\frac{d(d+2)(d-1)(d-3)}{8}$ \\ -\vv$\RR{u}_{abcd}$ & $d\geq3$ & $\frac{(d-2)(d+4)(d^2-1)}{8}$ \\[1mm] -\hline -\end{tabular} -\end{center} - - - -\section{Spinorial Curvature Decomposition} - -Spinorial curvature is defined in \grg\ iff nonmetricity -is zero and switch \comm{NONMETR} is turned off. -The upper sign in this section correspond to the signature -${\scriptstyle(-,+,+,+)}$ while lower one to the signature -${\scriptstyle(+,-,-,-)}$. - -Let us introduce the spinorial analog of the curvature tensor -\begin{eqnarray} -R_{abcd}&\tsst& -\ \ R_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}} -+R_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD} \nonumber\\[1mm] -&&+R_{AB\dot{C}\dot{D}}\epsilon_{\dot{A}\dot{B}}\epsilon_{CD} -+R_{\dot{A}\dot{B} CD}\epsilon_{AB}\epsilon_{\dot{C}\dot{D}}, \\[1.5mm] -R_{ABCD}&=&-i*(\Omega_{AB}\wedge S_{CD}),\ \ -R_{AB\dot{C}\dot{D}}\ =\ i*(\Omega_{AB}\wedge S_{\dot{C}\dot{D}})\\[1.5mm] -R_{\dot{A}\dot{B}\dot{C}\dot{D}}&=&\overline{R_{ABCD}},\ \ -R_{\dot{A}\dot{B} CD}\ =\ \overline{R_{AB\dot{C}\dot{D}}} -\end{eqnarray} - -The quantities $R_{ABCD}$ and $R_{AB\dot C\dot D}$ can be used to compute -the {\tt Curvature spinors} ($\equiv$ {\tt Curvature components}) -\object{Weyl Spinor RW.ABCD}{C_{ABCD}} -\object{Traceless Ricci Spinor RC.AB.CD\cc}{C_{AB\dot C\dot D}} -\object{Scalar Curvature RR}{R} -\object{Ricanti Spinor RA.AB}{A_{AB}} -\object{Traceless Deviation Spinor RB.AB.CD\cc}{B_{AB\dot C\dot D}} -\object{Scalar Deviation RD}{D} -All these spinors are irreducible (totally symmetric). - -Weyl spinor $C_{ABCD}$ {\tt From spinor curvature} is -\begin{eqnarray} -C_{abcd}&\tsst& C_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}} - +C_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD} \\[1mm] -C_{ABCD}&=&R_{(ABCD)} \label{RW} -\end{eqnarray} - -Traceless Ricci spinor $C_{AB\dot{A}\dot{B}}$ {\tt From spinor curvature} is -\begin{eqnarray} -C_{ab}&\tsst&C_{AB\dot{A}\dot{B}}\\[2mm] -C_{AB\dot{C}\dot{D}}&=&\pm(R_{AB\dot{C}\dot{D}}+R_{\dot{C}\dot{D} AB}) -\end{eqnarray} - -Scalar curvature {\tt From spinor curvature} is -\begin{equation} R=2(R^{MN}_{\ \ \ \ MN}+R^{\dot{M}\dot{N}}_{\ \ \ \ \dot{M}\dot{N}}) -\end{equation} - -Antisymmetric Ricci spinor $A_{AB}$ {\tt From spinor curvature} is -\begin{eqnarray} -A_{ab}&\tsst& A_{AB}\epsilon_{\dot{A}\dot{B}}+A_{\dot{A}\dot{B}}\epsilon_{AB}\\[1mm] -A_{AB}&=&\mp R^{\ \ \ \,M}_{(A|\ \ M|B)} -\end{eqnarray} - -Traceless deviation spinor $B_{AB\dot{A}\dot{B}}$ {\tt From spinor curvature} is -\begin{eqnarray} -B_{ab}&\tsst&B_{AB\dot{A}\dot{B}}\\[1mm] -B_{AB\dot{C}\dot{D}}&=&\pm i(R_{AB\dot{C}\dot{D}}-R_{\dot{C}\dot{D} AB}) -\end{eqnarray} - -Deviation trace {\tt From spinor curvature} is -\begin{equation} -D=-2i(R^{MN}_{\ \ \ \ MN}-R^{\dot{M}\dot{N}}_{\ \ \ \ \dot{M}\dot{N}}) -\end{equation} - -Note that spinors $C_{AB\dot{A}\dot{B}},B_{AB\dot{A}\dot{B}}$ are Hermitian -\begin{equation} -C_{AB\dot{C}\dot{D}}=\overline{C_{CD\dot{A}\dot{B}}},\ \ -B_{AB\dot{C}\dot{D}}=\overline{B_{CD\dot{A}\dot{B}}} -\end{equation} - -Finally we introduce the decomposition for the spinorial -curvature 2-form -\begin{equation} -\Omega_{AB}= -\OO{w}_{AB}+\OO{c}_{AB}+\OO{r}_{AB} -+\OO{a}_{AB}+\OO{b}_{AB}+\OO{c}_{AB} -\end{equation} -where the {\tt Undotted curvature 2-forms} -\object{Undotted Weyl 2-form OMWU.AB }{\OO{w}_{AB}} -\object{Undotted Traceless Ricci 2-form OMCU.AB }{\OO{c}_{AB}} -\object{Undotted Scalar Curvature 2-form OMRU.AB }{\OO{r}_{AB}} -\object{Undotted Ricanti 2-form OMAU.AB }{\OO{a}_{AB}} -\object{Undotted Traceless Deviation 2-form OMBU.AB }{\OO{b}_{AB}} -\object{Undotted Scalar Deviation 2-form OMDU.AB }{\OO{d}_{AB}} -are given by -\begin{eqnarray} -\OO{w}_{AB}&=&C_{ABCD}S^{CD} \\[1mm] -\OO{c}_{AB}&=&\pm\frac12 C_{AB\dot{C}\dot{D}}S^{\dot{C}\dot{D}} \\[1mm] -\OO{r}_{AB}&=&\frac1{12}S_{AB}R \\[1mm] -\OO{a}_{AB}&=&\pm A_{(A}^{\ \ \ M}S_{M|B)} \\[1mm] -\OO{b}_{AB}&=&\mp\frac{i}2 B_{AB\dot{C}\dot{D}}S^{\dot{C}\dot{D}} \\[1mm] -\OO{d}_{AB}&=&\frac{i}{12}S_{AB}D -\end{eqnarray} - - - - - - - -\section{Torsion Decomposition} - -The torsion tensor -\begin{equation} -Q_{abc}=Q_{a[bc]},\qquad -\Theta^a=\frac{1}{2}Q^a{}_{bc}\,S^{bc} -\end{equation} -consists of three irreducible pieces -\begin{equation} -Q_{abc} = -\stackrel{\rm c}{Q}_{abc} -+\stackrel{\rm t}{Q}_{abc} -+\stackrel{\rm a}{Q}_{abc} -\end{equation} - -\begin{center} -\begin{tabular}{|c|c|c|} -\hline object & exists if & and has $n$ components \\ -\hline -\vv$Q_{abc}$ & & $\frac{d^2(d-1)}{2}$ \\[1mm] -\hline\vv$\stackrel{\rm c}{Q}_{abc}$ & $d\geq3$ & $\frac{d(d^2-4)}{3}$ \\ -\vv$\stackrel{\rm t}{Q}_{abc}$ & & $d$ \\ -\vv$\stackrel{\rm a}{Q}_{abc}$ & $d\geq3$ & $\frac{d(d-1)(d-2)}{6}$ \\[1mm] -\hline -\end{tabular} -\end{center} - -The corresponding union of three objects {\tt Torsion 2-forms} is -\object{Traceless Torsion 2-form THQC'a} -{\stackrel{\rm c}{\Theta}\!{}^a=\frac{1}{2} - \stackrel{\rm c}{Q}\!{}^a{}_{bc}\,S^{bc}} -\object{Torsion Trace 2-form THQT'a} -{\stackrel{\rm t}{\Theta}\!{}^a=\frac{1}{2} - \stackrel{\rm t}{Q}\!{}^a{}_{bc}\,S^{bc}} -\object{Antisymmetric Torsion 2-form THQA'a} -{\stackrel{\rm a}{\Theta}\!{}^a=\frac{1}{2} - \stackrel{\rm a}{Q}\!{}^a{}_{bc}\,S^{bc}} - -And the auxiliary quantities -\object{Torsion Trace QT'a}{Q^a} -\object{Torsion Trace 1-form QQ}{Q=-\partial_a\ipr\Theta^a} -\object{Antisymmetric Torsion 3-form QQA}{\stackrel{\rm a}{Q}=\theta_a\wedge\Theta^a} - -The torsion trace $Q^a=Q^m{}_{am}$ can be obtained {\tt From torsion -trace 1-form} -\begin{equation} -Q^a = \partial^a\ipr Q -\end{equation} - -The {\tt Standard way} for the irreducible torsion 2-forms is -\begin{equation} -\stackrel{\rm t}{\Theta}\!{}^a = -\frac{1}{(d-1)}\theta^a\wedge Q -\end{equation} -\begin{equation} -\stackrel{\rm t}{\Theta}\!{}^a = \frac{1}{3}\partial^a\ipr\stackrel{\rm a}{Q} -\end{equation} -\begin{equation} -\stackrel{\rm c}{\Theta}\!{}^a = \Theta^a --\stackrel{\rm t}{\Theta}\!{}^a --\stackrel{\rm a}{\Theta}\!{}^a -\end{equation} - -The rest of this section is valid in dimension 4 only. - -In this case one can introduce the torsion pseudo trace -\object{Torsion Pseudo Trace QP'a}{ -P^a = \stackrel{*}{Q}\!{}^{ma}{}_{m}, -\ \stackrel{*}{Q}\!{}^a{}_{bc} = \frac{1}{2}{\cal E}_{bc}{}^{pq} -Q^a{}_{pq}} -which can be computed {\tt From antisymmetric torsion 3-form} -\begin{equation} -P^a = \partial^a\ipr\,*\!\stackrel{\rm a}{Q} -\end{equation} - -Finally let us consider the spinorial representation of the -torsion. -Below the upper sign corresponds to the -\seethis{See \pref{spinors}\ or \ref{spinors1}.} -signature ${\scriptstyle(-,+,+,+)}$ and lower one to the -signature ${\scriptstyle(+,-,-,-)}$. - -First we introduce the spinorial analog of the torsion tensor -\begin{equation} -Q_{abc}\tsst Q_{A\dot{A} BC}\epsilon_{\dot{B}\dot{C}} -+Q_{A\dot{A}\dot{B}\dot{C}}\epsilon_{BC} -\end{equation} -where -\begin{equation} -Q_{A\dot{A} BC}=-i*(\Theta_{A\dot{A}}\wedge S_{BC}),\qquad -Q_{A\dot{A}\dot{B}\dot{C}}=i*(\Theta_{A\dot{A}}\wedge S_{\dot{B}\dot{C}}) -\end{equation} -These spinors are reducible but the -\object{Traceless Torsion Spinor QC.ABC.D\cc}{C_{ABC\dot D}} -\[ -\stackrel{\rm c}{Q}_{abc}\tsst C_{ABC\dot A}\epsilon_{\dot{B}\dot{C}} -+Q_{\dot{A}\dot{B}\dot{C}A}\epsilon_{BC},\quad -C_{\dot{A}\dot{B}\dot{C} A}=\overline{C_{ABC\dot{A}}} -\] -is irreducible (symmetric in $\scriptstyle ABC$). And it can be -computed {\tt From torsion} by the relation -\begin{equation} -C_{ABC\dot A} = Q_{(A|\dot{A}|BC)} -\end{equation} - -The torsion trace can be calculated {\tt From torsion using spinors} -\begin{equation} -Q^a\tsst Q^{A\dot{A}},\quad -Q_{A\dot{B}}=\mp(Q^M{}_{\dot{B}MA}+Q_A{}^{\dot M}{}_{\dot M\dot{B}}) -\end{equation} - -And similarly the torsion pseudo-trace can be found -{\tt From torsion using spinors} -\begin{equation} -P^a\tsst P^{A\dot{A}},\quad -P_{A\dot{B}}=\mp i(Q^M{}_{\dot{B}MA}-Q_A{}^{\dot M}{}_{\dot M\dot{B}}) -\end{equation} - -Finally we introduce the {\tt Undotted trace 2-forms} -which are selfdual parts of the irreducible torsion 2-forms -\object{Undotted Traceless Torsion 2-form THQCU'a} -{\stackrel{\rm c}{\vartheta}\!{}^a} -\object{Undotted Torsion Trace 2-form THQTU'a} -{\stackrel{\rm t}{\vartheta}\!{}^a} -\object{Undotted Antisymmetric Torsion 2-form THQAU'a} -{\stackrel{\rm a}{\vartheta}\!{}^a} \seethis{See \pref{thetau}.} -These quantities will be used in the gravitational equations. - -This complex 2-forms can be obtained by the equations -({\tt Standard way}): -\begin{eqnarray} -\stackrel{\rm c}{\vartheta}\!{}^a &\tsst& \stackrel{\rm c}{\vartheta}\!{}^{A\dot A} -=C^A_{\ \ BC}{}^{\dot{A}}S^{BC}\\[1mm] -\stackrel{\rm t}{\vartheta}\!{}^a &\tsst& \stackrel{\rm t}{\vartheta}\!{}^{A\dot A} -=\mp\frac13 Q_{M}^{\ \ \ \dot{A}}S^{AM}\\[1mm] -\stackrel{\rm a}{\vartheta}\!{}^a &\tsst& \stackrel{\rm a}{\vartheta}\!{}^{A\dot A} -=\pm\frac{i}3 P_{M}^{\ \ \ \dot{A}}S^{AM} -\end{eqnarray} - - - -\section{Nonmetricity Decomposition} - -In general the nonmetricity tensor -\begin{equation} -N_{abc}=N_{(ab)c},\qquad N_{ab}=N_{abc}\theta^c -\end{equation} -consist of 4 irreducible pieces -\begin{equation} -N_{abcd} = -\stackrel{\rm c}{N}_{abc} -+\stackrel{\rm a}{N}_{abc} -+\stackrel{\rm t}{N}_{abc} -+\stackrel{\rm w}{N}_{abc} -\end{equation} - -\begin{center} -\begin{tabular}{|c|c|c|} -\hline object & exists if & and has $n$ components \\ -\hline -\vv$N_{abc}$ & & $\frac{d^2(d+1)}{2}$ \\[1mm] -\hline\vv$\stackrel{\rm c}{N}_{abc}$ & & $\frac{d(d-1)(d+4)}{6}$ \\ -\vv$\stackrel{\rm a}{N}_{abc}$ & $d\geq3$ & $\frac{d(d^2-4)}{3}$ \\ -\vv$\stackrel{\rm t}{N}_{abc}$ & & $d$ \\ -\vv$\stackrel{\rm w}{N}_{abc}$ & & $d$ \\[1mm] -\hline -\end{tabular} -\end{center} - -The corresponding union of objects {\tt Nonmetricity 1-forms} -consist of -\object{Symmetric Nonmetricity 1-form NC.a.b} -{\stackrel{\rm c}{N}_{ab}=\stackrel{\rm c}{N}_{abc}\theta^c} -\object{Antisymmetric Nonmetricity 1-form NA.a.b} -{\stackrel{\rm a}{N}_{ab}=\stackrel{\rm a}{N}_{abc}\theta^c} -\object{Nonmetricity Trace 1-form NT.a.b} -{\stackrel{\rm t}{N}_{ab}=\stackrel{\rm t}{N}_{abc}\theta^c} -\object{Weyl Nonmetricity 1-form NW.a.b} -{\stackrel{\rm w}{N}_{ab}=\stackrel{\rm w}{N}_{abc}\theta^c} - -We have also two auxiliary 1-forms -\object{Weyl Vector NNW}{\stackrel{\rm w}{N}} -\object{Nonmetricity Trace NNT}{\stackrel{\rm t}{N}} - -They are computed according to the following formulas -\begin{equation} -\stackrel{\rm w}{N} = N^a{}_a -\end{equation} -\begin{equation} -\stackrel{\rm t}{N} = \theta^a\,\partial^b\ipr N_{ab} -- \frac{1}{d} \stackrel{\rm w}{N} -\end{equation} -\begin{equation} -\stackrel{\rm w}{N}_{ab} = \frac{1}{d}g_{ab}\stackrel{\rm w}{N} -\end{equation} -\begin{equation} -\stackrel{\rm t}{N}_{ab}=\frac{d}{(d-1)(d+2)}\left[ -\theta_b\partial_a\ipr\stackrel{\rm t}{N} -+\theta_a\partial_b\ipr\stackrel{\rm t}{N} --\frac{2}{d} g_{ab} \stackrel{\rm t}{N}\right] -\end{equation} -\begin{equation} -\stackrel{\rm a}{N}_{ab}=\frac{1}{3}\left[ -\partial_a\ipr(\theta^m\wedge\stackrel{0}{N}_{bm}) -+\partial_b\ipr(\theta^m\wedge\stackrel{0}{N}_{am})\right] -\end{equation} -where -\[ -\stackrel{\rm 0}{N}_{ab}= -N_{abc} --\stackrel{\rm t}{N}_{abc} --\stackrel{\rm w}{N}_{abc} -\] -And finally -\begin{equation} -\stackrel{\rm c}{N}_{ab}= -N_{abc} --\stackrel{\rm a}{N}_{abc} --\stackrel{\rm t}{N}_{abc} --\stackrel{\rm w}{N}_{abc} -\end{equation} - -\section{Newman-Penrose Formalism} - -The method of spinorial differential forms described in the -previous sections are essentially equivalent to the well -known Newman-Penrose formalism but for the sake of convenience -\grg\ has complete set of macro objects which allows to -write the Newman-Penrose equations in -traditional notation. All these objects refer (up to some sign -and 1/2 factors) to other \grg\ built-in objects. - -In this section upper sign corresponds to the -signature ${\scriptstyle(-,+,+,+)}$ and lower one to the -signature ${\scriptstyle(+,-,-,-)}$. -\seethis{See \pref{spinors}.} -The frame must be null as explained in section \ref{spinors}. - -For the Newman-Penrose formalism we use notation and conventions -of the book \emph{Exact Solutions of the Einstein Field Equations} -by D. Kramer, H. Stephani, M. MacCallum and E. Herlt, ed. -E. Schmutzer (Berlin, 1980). We denote this book as ESEFE. - -We chose the relationships between NP null tetrad and \grg\ null -frame as follows -\begin{equation} -l^\mu=h^\mu_0,\quad -k^\mu=h^\mu_1,\quad -\overline{m}\!{}^\mu=h^\mu_2,\quad -m^\mu=h^\mu_3 -\end{equation} - -The NP vector operators are just the components of the -vector frame $\partial_a$ -\begin{eqnarray} -\mbox{\tt DD}&=& D =\partial_1 \\ -\mbox{\tt DT}&=& \Delta=\partial_0 \\ -\mbox{\tt du}&=& \delta=\partial_3 \\ -\mbox{\tt dd}&=& \overline\delta=\partial_2 -\end{eqnarray} - -The spin coefficient are the components of the connection -1-form -\object{SPCOEF.AB.c}{ \omega_{AB\,c}=\partial_c\ipr\omega_{AB}} -or in the NP notation -\begin{eqnarray} -\mbox{\tt alphanp }&=& \alpha =\pm\omega_{(1)2} \\ -\mbox{\tt betanp }&=& \beta =\pm\omega_{(1)3} \\ -\mbox{\tt gammanp }&=& \gamma =\pm\omega_{(1)0} \\ -\mbox{\tt epsilonnp }&=& \epsilon =\pm\omega_{(1)1} \\ -\mbox{\tt kappanp }&=& \kappa =\pm\omega_{(0)1} \\ -\mbox{\tt rhonp }&=& \rho =\pm\omega_{(0)2} \\ -\mbox{\tt sigmanp }&=& \sigma =\pm\omega_{(0)3} \\ -\mbox{\tt taunp }&=& \tau =\pm\omega_{(0)0} \\ -\mbox{\tt munp }&=& \mu =\pm\omega_{(2)3} \\ -\mbox{\tt nunp }&=& \nu =\pm\omega_{(2)0} \\ -\mbox{\tt lambdanp }&=& \lambda =\pm\omega_{(2)2} \\ -\mbox{\tt pinp }&=& \pi =\pm\omega_{(2)1} \\ -\end{eqnarray} -where the first index of the -quantity $\omega_{(AB)c}$ is included inn parentheses to remind -that it is summed spinorial index. - -Finally for the curvature we have -\object{PHINP.AB.CD\cc }{ -\Phi_{AB\dot{C}\dot{D}} = \pm\frac{1}{2}C_{AB\dot C\dot D} } -\object{PSINP.ABCD }{\Psi_{ABCD}=C_{ABCD}} -the conventions for the scalar curvature $R$ in ESEFE and -in \grg\ are the same. - -For the signature ${\scriptstyle(-,+,+,+)}$ the Newman-Penrose equations for -the quantities introduced above can be found in section 7.1 of ESEFE. -For other signature ${\scriptstyle(+,-,-,-)}$ one must alter the sign of -$\Psi_{ABCD}$, $\Phi_{AB\dot{C}\dot{D}}$ and $R$ in Eqs. (7.28)--(7.45). - -\section{Electromagnetic Field} - -Formulas in this section are valid only in spaces -with the signature ${\scriptstyle(-,+,\dots,+)}$ and -${\scriptstyle(+,-,\dots,-)}$. -The sign factor $\sigma$ in the expressions below is -$\sigma=-{\rm diag}_0$ ($+1$ for the first signature and $-1$ -for the second). - -Let us introduce the -\object{EM Potential A}{A=A_\mu dx^\mu} -and the -\object{Current 1-form J}{J=j_\mu dx^\mu} - -The EM strength tensor -$F_{\alpha\beta}=\partial_\alpha A_\beta-\partial_\beta A_\alpha$ -\object{EM Tensor FT.a.b}{F_{ab}= -\partial_b\ipr\partial_a\ipr F} -where $F$ is the -\object{EM 2-form FF}{F} -which can be found {\tt From EM potential} -\begin{equation} -F=dA -\end{equation} -or {\tt From EM tensor} -\begin{equation} -F = \frac{1}{2}F_{ab}\,S^{ab} -\end{equation} - -The EM action $d$-form -\object{EM Action EMACT}{L_{\rm EM}= --\frac{1}{8\pi}\,F\wedge *F} - -The {\tt Maxwell Equations} -\object{First Maxwell Equation MWFq}{d*F=-4\pi\sigma\,(-1)^{d}\,*J} -\object{Second Maxwell Equation MWSq}{dF=0} - -The current must satisfy the -\object{Continuity Equation COq}{d*J=0} - -The -\object{EM Energy-Momentum Tensor TEM.a.b}{T_{ab}^{\rm EM}} -is given by the equation -\begin{equation} -T^{\rm EM}_{ab} = \frac{\sigma}{4\pi} -F_{am}F_b{}^m +s\sigma\,g_{ab}\,*L_{\rm EM} -\end{equation} - -The rest of the section is valid in the dimension 4 only. - -In 4 dimensions the tensor $F_{ab}$ and its dual -$\stackrel{*}{F}_{ab}=\frac{1}{2}{\cal E}_{ab}{}^{mn}F_{mn}$ -are expressed via usual 3-dimensional vectors $\vec E$ and -$\vec H$ -\begin{eqnarray} -F_{ab}&=&-\sigma\left(\begin{array}{rrr} -E_1&E_2&E_3\\ -&-H_3&H_2\\ -&&-H_1\end{array}\right)\\[1.5mm] -\stackrel{*}{F}_{ab}&=&\sigma\left(\begin{array}{rrr} -H_1&H_2&H_3\\ -&E_3&-E_2\\ -&&E_1\end{array}\right) -\end{eqnarray} -Similarly for the current we have -\begin{equation} -J=\sigma(-\rho dt + \vec j\,d\vec x) -\end{equation} - -The {\tt EM scalars} -\object{First EM Scalar SCF}{I_1=\frac12F_{ab}F^{ab} -={\vec H}^2-{\vec E}^2} -\object{Second EM Scalar SCS}{I_2=\frac12\stackrel{*}{F}_{ab}F^{ab} -=2\vec E\cdot\vec H} -can be obtained as follows by {\tt Standard way} -\begin{equation} -I_1 = -*(F\wedge*F) -\end{equation} -\begin{equation} -I_2 = *(F\wedge F) -\end{equation} - -The -\object{Complex EM 2-form FFU}{\Phi} -can be found {\tt From EM 2-form} -\begin{equation} -\Phi=F-i*F -\end{equation} -or {\tt From EM Spinor} -\begin{equation} -\Phi = 2\Phi_{AB}\,S^{AB} -\end{equation} - -The 2-form $\Phi$ must obey the -\object{Selfduality Equation SDq.AB\cc}{\Phi\wedge S_{\dot A\dot B}} -and gives rise to the -\object{Complex Maxwell Equation MWUq}{d\Phi=-4i\sigma\pi\,*J} - -The EM 2-form $F$ can be restored {\tt From Complex EM 2-form} -\begin{equation} -F=\frac{1}{2}(\Phi+\overline\Phi) -\end{equation} - -The symmetric -\object{Undotted EM Spinor FIU.AB}{\Phi_{AB}} -is the spinorial analog of the tensor $F_{ab}$ -\begin{equation} - F_{ab} \tsst \epsilon_{AB} \Phi_{\dot A\dot B} -+ \epsilon_{\dot A\dot B} \Phi_{AB} -\end{equation} -It can be obtained either {\tt From complex EM 2-form} -\begin{equation} -\Phi_{AB} = -\frac{i}{2}*(\Phi\wedge S_{AB}) -\end{equation} -of {\tt From EM 2-form} -\begin{equation} -\Phi_{AB} = -i*(F\wedge S_{AB}) -\end{equation} - -The -\object{Complex EM Scalar SCU}{\iota=I_1-iI_2} -can be found {\tt From EM Spinor} -\begin{equation} -\iota = 2\Phi_{AB}\Phi^{AB} -\end{equation} -or {\tt From Complex EM 2-form} -\begin{equation} -\iota = -\frac{i}{2} *(\Phi\wedge\Phi) -\end{equation} - -Finally we have the -\object{EM Energy-Momentum Spinor TEMS.AB.CD\cc} -{T^{\rm EM}_{AB\dot A\dot B}=\frac{1}{2\pi}\Phi_{AB}\Phi_{\dot A\dot B}} - - -\section{Dirac Field} - -In this section upper sign corresponds to the -signature ${\scriptstyle(-,+,+,+)}$ and lower one to the -signature ${\scriptstyle(+,-,-,-)}$. - -The four component Dirac spinor consists of two 1-index spinors -\begin{equation} -\psi=\left(\begin{array}{c}\phi^A\\ \chi_{\dot A}\end{array}\right),\ \ -\overline\psi=\left(\chi_A\ \ \phi^{\dot A}\right) -\end{equation} -Thus we have the {\tt Dirac spinor} as the union of two objects -\object{Phi Spinor PHI.A}{\phi_A} -\object{Chi Spinor CHI.B}{\chi_B} - -The gamma-matrices are expressed via sigma-matrices as follows -\begin{equation} -\gamma^m=\sqrt2\left(\begin{array}{cc} -0&\sigma^{mA\dot B}\\ \sigma^m\!{}_{B\dot A}&0\end{array}\right) -\end{equation} - -Dirac field action 4-form -\begin{eqnarray} -&&\mbox{\tt Dirac Action 4-form DACT}=L_{\rm D}=\nonumber\\[1mm] -&&\quad=\left[\frac{i}2(\overline\psi\gamma^a -(\nabla_a+ieA_a)\psi-(\nabla_a-ieA_a)\overline\psi\gamma^a\psi) --m_{\rm D}\overline\psi\psi\right]\upsilon -\end{eqnarray} - -The {\tt Standard way} to compute this quantity is -\begin{eqnarray} -L_{\rm D} &=& -\frac{i}{\sqrt2}\left[ -\phi_{\dot A}\theta^{A\dot A}\!\wedge*(D+ieA)\phi_A-{\rm c.c.} --\chi_{\dot A} \theta^{A\dot A}\!\wedge*(D-ieA)\chi_A -{\rm c.c.}\right]- -\nonumber\\[1mm]&&\qquad\qquad\quad --m_{\rm D}\left(\phi^A\chi_A+{\rm c.c.}\right)\upsilon -\end{eqnarray} - -The {\tt Dirac equation} is -\object{Phi Dirac Equation DPq.A\cc}{ -i\sqrt2\partial_{B\dot A}\ipr(D+ieA-\frac12Q)\phi^B-m_{\rm D}\chi_{\dot A}=0} -\object{Chi Dirac Equation DCq.A\cc}{ -i\sqrt2\partial_{B\dot A}\ipr(D-ieA-\frac12Q)\chi^B-m_{\rm D}\phi_{\dot A}=0} -where $Q$ is the torsion trace 1-form. Notice that terms with the -electromagnetic field $eA$ are included in equations iff -the value of $A$ is defined. The unit charge $e$ is given by the -constant \comm{ECONST}. - -The current 1-form can be computed {\tt From Dirac Spinor} -\begin{equation} -J=\mp\sqrt2e(\phi_A\phi_{\dot A}+\chi_A\chi_{\dot A})\theta^{A\dot A} -\end{equation} - -The symmetrized -\object{Dirac Energy-Momentum Tensor TDI.a.b}{T^{\rm D}_{ab}} -can be obtained as follows -\begin{eqnarray} -T^{\rm D}_{ab}&=& -*(\theta_{(a}\wedge T^{\rm D}_{b)})\nonumber\\[1mm] -T^{\rm D}_a&=&\mp\frac{i}{\sqrt2}\Big[ -*\theta^{A\dot A}\partial_a\ipr(D+ieA)\phi_A\phi_{\dot A} --{\rm c.c.}\nonumber\\ -&&\qquad-*\theta^{A\dot A}\partial_a\ipr(D-ieA)\chi_A\chi_{\dot A} --{\rm c.c.}\Big] -\pm\partial_a\ipr L_{\rm D} -\end{eqnarray} - -The -\object{Undotted Dirac Spin 3-Form SPDIU.AB}{s^{\rm D}_{AB}} -\begin{equation} -s^{\rm D}_{AB}=\frac{i}{2\sqrt2} -\left(*\theta_{(A|\dot A}\phi_{B)}\phi^{\dot A} --*\theta_{(A|\dot A}\chi_{B)}\chi^{\dot A}\right) -\end{equation} - -The Dirac field mass $m_{\rm D}$ is given by the constant -\comm{DMASS}. - - -\section{Scalar Field} - -Formulas in this section are valid in any dimension -with the signature ${\scriptstyle(-,+,\dots,+)}$ and -${\scriptstyle(+,-,\dots,-)}$. -The sign factor $\sigma$ is $\sigma=-{\rm diag}_0$ -($+1$ for the first signature and $-1$ for the second). - -The scalar field -\object{Scalar Field FI}{\phi} - -The minimal scalar field action $d$-form -\object{Minimal Scalar Action SACTMIN}{ -L_{\rm Smin}= --\frac{1}{2}\left[\sigma(\partial_\alpha\phi)^2+ -m_{\rm s}^2 \phi^2\right]\upsilon} - -The nonminimal scalar field action -\object{Scalar Action SACT}{ -L_{\rm S}= --\frac{1}{2}\left[\sigma(\partial_\alpha\phi)^2+ -(m_{\rm s}^2+a_0R) \phi^2\right]\upsilon} - -The scalar field equation -\object{Scalar Equation SCq} -{s\sigma(-1)^d*d*d\phi-(m_{\rm s}^2+a_0R)\phi=0} -which gives -\[ --\sigma\rim{\nabla}{}^\pi\rim{\nabla}_\pi\phi-(m_{\rm s}^2+a_0R)\phi=0 -\] - -The minimal energy-momentum tensor is -\begin{eqnarray} -&&\mbox{\tt Minimal Scalar Energy-Momentum Tensor TSCLMIN.a.b} -=T^{\rm Smin}_{ab}= \nonumber\\ -&&\qquad\qquad=\partial_a\phi\partial_b\phi+s\sigma\,g_{ab} -*L_{\rm Smin} -\end{eqnarray} -The nonminimal part of the scalar field energy-momentum -\seethis{See pages \pageref{graveq}\ and \pageref{metreq}.} -tensor can be taken into account in the left-hand side -of gravitational equations. - -The scalar field mass $m_{\rm s}$ are given by the -constant {\tt SMASS}. The nonminimal interaction -terms are included iff the switch \comm{NONMIN} \swind{NONMIN} -is turned on and the value of nonminimal interaction constant -$a_0$ is determined by the object -\object{A-Constants ACONST.i2}{a_i} -The default value of $a_0$ is the constant \comm{AC0}. - -\section{Yang-Mills Field} - -Formulas in this section are valid in any dimension -with the signature ${\scriptstyle(-,+,\dots,+)}$ and -${\scriptstyle(+,-,\dots,-)}$. -The sign factor $\sigma$ in the expressions below is -$\sigma=-{\rm diag}_0$ ($+1$ for the first signature and $-1$ -for the second). The indices $\scriptstyle i,j,k,l,m,n$ -are the internal space Yang-Mills indices and we a -assume that the internal Yang-Mills metric is $\delta_{ij}$. - -The Yang-Mills potential 1-form -\object{YM Potential AYM.i9}{A^i=A^i_\mu dx^\mu} - -The structural constants -\object{Structural Constants SCONST.i9.j9.k9}{c^i{}_{jk}=c^i{}_{[jk]}} - -The Yang-Mills strength 2-form -\object{YM 2-form FFYM.i9}{F^i} -and strength tensor -\object{YM Tensor FTYM.i9.a.b}{F^i{}_{ab}} - -The $F^i$ can be computed {\tt From YM potential} -\begin{equation} -F^i = dA^i + \frac12 c^i{}_{jk} \, A^j\wedge A^k -\end{equation} -or {\tt From YM tensor} -\begin{equation} -F^i = \frac12 F^i{}_{ab}\, S^{ab} -\end{equation} - -The {\tt Standard way} to find Yang-Mills strength tensor is -\begin{equation} -F^i{}_{ab}=\partial_b\ipr\partial_a\ipr F^i -\end{equation} - -The Yang-Mills action $d$-form -\object{YM Action YMACT}{L_{\rm YM}= --\frac{1}{8\pi}F^i\wedge*F_i} - -The {\tt YM Equations} -\object{First YM Equation YMFq.i9}{d*F^i + c^i{}_{jk} \, A^j\wedge *F^k=0} -\object{Second YM Equation YMSq.i9}{dF^i + c^i{}_{jk} \, A^j\wedge F^k=0} - -The energy-momentum tensor -\object{YM Energy-Momentum Tensor TYM.a.b} -{\frac{\sigma}{4\pi}F^i{}_{am}F^i{}_b{}^m + s\sigma\,g_{ab}\, -*L_{\rm YM}} - - -\section{Geodesics} - -The geodesic equation -\object{Geodesic Equation GEOq\^m}{ -\frac{d^2x^\mu}{dt^2}+\{^\mu_{\pi\tau}\} -\frac{dx^\pi}{dt}\frac{dx^\tau}{dt}=0} -Here the parameter $t$ must be declared by the -\seethis{See page \pageref{affpar}.} -\cmdind{Affine Parameter} -{\tt Affine parameter} declaration. - -\section{Null Congruence and Optical Scalars} - -Let us consider the congruence defined by the vector field -$k^\alpha$ -\object{Congruence KV}{k=k^\mu\partial_\mu} - -This congruence is null iff -\object{Null Congruence Condition NCo}{k\cdot k=0} -holds. - -The congruence is geodesic iff the condition -\object{Geodesics Congruence Condition GCo'a}{k^\mu\rim{\nabla}_\mu k^a=0} -is fulfilled. - -For the null geodesic congruence one can calculate the -{\tt Optical scalars} -\object{Congruence Expansion thetaO}{\theta= -\frac{1}{2}\rim{\nabla}{}^\pi k_\pi} -\object{Congruence Squared Rotation omegaSQO}{\omega^2= -\frac{1}{2}(\rim{\nabla}_{[\alpha}k_{\beta]})^2} -\object{Congruence Squared Shear sigmaSQO}{\sigma\overline\sigma= -\frac{1}{2}\left[ (\rim{\nabla}_{(\alpha}k_{\beta)})^2 --2\theta^2\right]} - -\section{Timelike Congruences and Kinematics} - -Let us consider the congruence determined by the velocity -vector $u^\alpha$ -\object{Velocity UU'a}{u^a} -\object{Velocity Vector UV}{u=u^a\partial_a} - -The velocity vector must be normalized and the quantity -\object{Velocity Square USQ}{u^2=u\cdot u} -must be constant but nonzero. - -If the frame metric coincides with its default -diagonal value \seethis{See \pref{defaultmetric}.} -$g_{ab}={\rm diag}(-1,\dots)$ -then {\tt By default} we have for the velocity -\begin{equation} -u^a=(1,0,\dots,0) -\end{equation} -which means that the congruence is comoving in the given frame. - -In general case the velocity can be obtained -{\tt From velocity vector} -\begin{equation} -u^a=u\ipr \theta^a -\end{equation} - -We introduce the auxiliary object -\object{Projector PR'a.b}{P^a{}_b= -\delta^a_b-\frac{1}{u^2}u^an_b} - -The following four quantities called {\tt Kinematics} -comprise the complete set of the congruence characteristics -\object{Acceleration accU'a}{A^a=\rim{\nabla}_uu^a} -\object{Vorticity omegaU.a.b}{\omega_{ab}= -P^m{}_aP^n{}_b \rim{\nabla}_{[m}u_{n]}} -\object{Volume Expansion thetaU}{\Theta=\rim{\nabla}_au^a} -\object{Shear sigmaU.a.b}{ -P^m{}_aP^n{}_b \rim{\nabla}_{(m}u_{n)}- -\frac{1}{(d-1)}P_{ab}\Theta} - - -\section{Ideal And Spin Fluid} - - -The ideal fluid is characterized by the -\object{Pressure PRES}{p} -and -\object{Energy Density ENER}{\varepsilon} - -The ideal fluid energy-momentum tensor is -\begin{eqnarray} -&&\mbox{\tt Ideal Fluid Energy-Momentum Tensor TIFL.a.b}= -T^{\rm IF}_{ab} = \nonumber\\ -&&\qquad\qquad=(\varepsilon+p)u_a u_b - u^2p g_{ab} -\end{eqnarray} - -The rest of the section requires the nonmetricity be zero -(\comm{NONMETR} is off). - -In addition spin-fluid is characterized by -\object{Spin Density SPFLT.a.b }{S^{\rm SF}_{ab}=S^{\rm SF}_{[ab]}} -or equivalently by -\object{Spin Density 2-form SPFL }{S^{\rm SF}} - -The spin 2-form can be obtained {\tt From spin density} -\begin{equation} -S^{\rm SF}=\frac{1}{2}S^{\rm SF}_{ab} \theta^a\wedge\theta^a -\end{equation} -and $s_{ab}$ is determined {\tt From spin density 2-form} -\begin{equation} -S^{\rm SF}_{ab}= \partial_b\ipr\partial_a\ipr S^{\rm SF} -\end{equation} - -The spin density must satisfy the Frenkel condition -\object{Frenkel Condition FCo}{u\ipr S^{\rm SF}=0} - -The spin fluid energy-momentum tensor is -\begin{eqnarray} -&&\mbox{\tt Spin Fluid Energy-Momentum Tensor TSFL.a.b}=T^{\rm SF}_{ab}= -\nonumber\\ -&&\qquad\qquad=(\varepsilon+p)u_a u_b - u^2p g_{ab}+\Delta_{(ab)} -\end{eqnarray} -where -\begin{equation} -\Delta_{ab}=-2(g^{cd}+u^{-2}\,u^cu^d) \nabla_c S^{\rm SF}_{(ab)d} -\end{equation} -\begin{equation} -s^{\rm SF}_{abc}=u_a\,S^{\rm SF}_{bc} -\end{equation} -if torsion is zero (\comm{TORSION} off) and -\begin{equation} -\Delta_{ab}=2u^{-2}\,u_au^d\,\nabla_u S^{\rm SF}_{bd} -\end{equation} -if torsion is nonzero (\comm{TORSION} on). - -Notice that the energy-momentum \seethis{See \pref{tsym}.} -tensor $T^{\rm SF}_{ab}$ is symmetrized. - -Finally yet another representation for the spin -is the undotted spin 3-form -\object{Undotted Fluid Spin 3-form SPFLU.AB }{s^{\rm SF}_{AB}} -which is given by the standard spinor $\tsst$ tensor correspondence rules -\begin{equation} - s^{\rm SF}_{mab}\,*\theta^m \tsst \epsilon_{AB} s^{\rm SF}_{\dot A\dot B} -+ \epsilon_{\dot A\dot B}s^{\rm SF}_{AB} -\end{equation} -according to Eq. (\ref{asys}). \seethis{See \pref{asys}.} -This quantity is used in the right-hand side of gravitational equations. - -\section{Total Energy-Momentum And Spin} -\label{totalc} - -\enlargethispage{4mm} - - -The total energy-momentum tensor -\object{Total Energy-Momentum Tensor TENMOM.a.b}{T_{ab}} -and the total undotted spin 3-form \seethis{See pages \pageref{graveq}\ and \pageref{metreq}.} -\object{Total Undotted Spin 3-form SPINU.AB}{s_{AB}} -play the role of sources in the right-hand side of the -gravitational equations. - -The expression for these quantities read -\begin{equation} -T_{ab} = -T^{\rm D}_{ab}+ -T^{\rm EM}_{ab}+ -T^{\rm YM}_{ab}+ -T^{\rm Smin}_{ab}+ -T^{\rm IF}_{ab}+ -T^{\rm SF}_{ab} \label{b1} -\end{equation} -\begin{equation} -s_{AB} = s_{AB}^{\rm D} + s_{AB}^{\rm SF} \label{b2} -\end{equation} -When $T_{ab}$ and -$s_{AB}$ are calculated \grg\ does not tries to find value -of all objects in the right-hand side of Eqs. (\ref{b1}), (\ref{b2}) -instead it adds only the quantities whose value are currently -defined. In particular if none of above tensors and spinors are -defined then $T_{ab}=s_{AB}=0$. - -Notice that $T_{ab}$ and all tensors in the right-hand side -of Eq. (\ref{b1}) are symmetric. -\seethis{See \pref{tsym}.} -They are the symmetric parts of the canonical energy-momentum tensors. - -In addition we introduce the -\object{Total Energy-Momentum Trace TENMOMT}{T=T^a{}_a} -and the spinor -\object{Total Energy-Momentum Spinor TENMOMS.AB.CD\cc}{T_{AB\dot C\dot D}} -is a spinorial equivalent of the traceless part of $T_{ab}$ -\begin{equation} -T_{ab}-\frac{1}{4}g_{ab}T \tsst T_{AB\dot A\dot B} -\end{equation} - - -\section{Einstein Equations} - -The Einstein equation -\object{Einstein Equation EEq.a.b} -{R_{ab}-\frac{1}{2}g_{ab}R +\Lambda R =8\pi G\, T_{ab}} - -And the {\tt Spinor Einstein equations} -\object{Traceless Einstein Equation CEEq.AB.CD\cc}{ -C_{AB\dot C\dot D} = 8\pi G\, T_{AB\dot C\dot D}} -\object{Trace of Einstein Equation TEEq} -{R-4\Lambda = -8\pi G\, T} - -The cosmological constant is included in these equations -iff the switch \comm{CCONST} is turned on \swind{CCONST} -and its value is given by the constant \comm{CCONST}. -The gravitational constant $G$ is given by the constant \comm{GCONST}. - - -\section{Gravitational Equations in Space With Torsion} - -Equations in this section are valid in dimension $d=4$ -with the signature ${\scriptstyle(-,+,+,+)}$ and -${\scriptstyle(+,-,-,-)}$ only. -The $\sigma=1$ for the first signature and $\sigma=-1$ -for the second. The nonmetricity must be zero and the -switch \comm{NONMETR} turned off. - -Let us consider the action -\begin{equation} -S=\int\left[\frac{\sigma}{16\pi G}L_{\rm g} -+L_{\rm m}\right] -\end{equation} -where -\object{Action LACT}{L_{\rm g}=\upsilon\,{\cal L}_{\rm g}} -is the gravitational action 4-form and -\begin{equation} -L_{\rm m} = \upsilon\,{\cal L}_{\rm m} -\end{equation} -is the matter action 4-form. - -Let us define the following variational derivatives -\begin{equation} -Z^\mu{}_{a} = \frac{1}{\sqrt{-g}} -\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta h^a_\mu} -,\qquad -t^\mu{}_{a} = \frac{\sigma}{\sqrt{-g}} -\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta h^a_\mu} -\end{equation} -\begin{equation} -V^\mu{}_{ab} = \frac{1}{\sqrt{-g}} -\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta \omega^{ab}{}_\mu} -,\qquad -s^\mu{}_{ab} = \frac{\sigma}{\sqrt{-g}} -\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta \omega^{ab}{}_\mu} -\end{equation} -Then the gravitational equations reads -\begin{eqnarray} -Z^\mu{}_a &=& -16\pi G\,t^\mu{}_a \label{zma} \\[2mm] -V^\mu{}_{ab} &=& -16\pi G\,s^\mu{}_{ab} \label{vab} -\end{eqnarray} -Here the first equation is an analog of Einstein equation -and has the canonical nonsymmetric energy-momentum -tensor $t^\mu{}_a$ as a source. The source in the second -equation is the spin tensor $s^\mu{}_{ab}$. - -Now we rewrite these equation in other equivalent form. -First let us define the following 3-forms -\begin{equation} -Z_a = Z^m{}_a\,*\theta_m,\qquad t_a = t^m{}_a\,*\theta_m -\end{equation} -\begin{equation} -V_{ab} = V^m{}_{ab}\,*\theta_m,\qquad s_{ab} = s^m{}_{ab}\,*\theta_m -\end{equation} -Notice that Eq. (\ref{zma}) is not symmetric but \label{tsym} -the antisymmetric part of this equation is expressed via second -Eq. (\ref{vab}) due to Bianchi identity. Therefore only the -symmetric part of Eq. (\ref{zma}) is essential. -Eq. (\ref{vab}) is -antisymmetric and we can consider its spinorial analog -using the standard relations -\begin{eqnarray} -V_{ab} &\tsst& V_{A\dot AB\dot B}= -\epsilon_{AB} V_{\dot A\dot B} + \epsilon_{\dot A\dot B}V_{AB} \\ -s_{ab} &\tsst& s_{A\dot AB\dot B}= -\epsilon_{AB} s_{\dot A\dot B} + \epsilon_{\dot A\dot B}s_{AB} -\end{eqnarray} \seethis{See \pref{asys}.} - -Finally we define the {\tt Gravitational equations} in the form \label{graveq} -\object{Metric Equation METRq.a.b}{-\frac12Z_{(ab)}=8\pi G\,T_{ab}} -\object{Torsion Equation TORSq.AB}{V_{AB}=-16\pi G\,s_{AB}} -where the currents in the right-hand side of equations are -\seethis{See \pref{totalc}.} -\object{Total Energy-Momentum Tensor TENMOM.a.b}{T_{ab}=t_{(ab)}} -\object{Total Undotted Spin 3-form SPINU.AB}{s_{AB}} - -Now let us consider the equations which are used in \grg\ to -compute the left-hand side of the gravitational equations -$Z_{(ab)}$ and $V_{AB}$. We have to emphasize that we use -\seethis{See \pref{spinors}.} -spinors and all restrictions imposed by the spinorial formalism -must be fulfilled. - -We consider the Lagrangian which is an arbitrary algebraic function -of the curvature and torsion tensors -\begin{equation} -{\cal L}_{\rm g} = {\cal L}_{\rm g}(R_{abcd},Q_{abc}) -\end{equation} -No derivatives of the torsion or curvature are permitted. -For such a Lagrangian we define so called curvature and torsion -momentums -\begin{equation} -\widetilde{R}{}^{abcd} = -2\frac{\partial{\cal L}_{\rm g}(R,Q)}{\partial R_{abcd}},\qquad -\widetilde{Q}{}^{abc} = -2\frac{\partial{\cal L}_{\rm g}(R,Q)}{\partial Q_{abc}},\qquad -\end{equation} - -The corresponding objects are -\object{Undotted Curvature Momentum POMEGAU.AB}{\widetilde{\Omega}_{AB}} -\object{Torsion Momentum PTHETA'a}{\widetilde{\Theta}{}^a} -where -\begin{eqnarray} -\widetilde{\Omega}_{ab} &=& \frac12 \widetilde{R}_{abcd}\,S^{cd} \\[1mm] -\widetilde{\Theta}{}^a &=& \frac12 \widetilde{Q}{}^a{}_{cd}\,S^{cd} -\end{eqnarray} -and -\begin{equation} -\widetilde{\Omega}_{ab} \tsst \widetilde{\Omega}_{A\dot AB\dot B}= -\epsilon_{AB} \widetilde{\Omega}_{\dot A\dot B} -+ \epsilon_{\dot A\dot B}\widetilde{\Omega}_{AB} -\end{equation} - -If value of three objects $L_{\rm g}$ ({\tt Action}), -$\widetilde{\Omega}_{AB}$ ({\tt Undotted curvature momentum}) -and $\widetilde{\Theta}{}^a$ are specified then the -{\tt Gravitational equations} can be calculated using equations -({\tt Standard way}) -\begin{eqnarray} -Z_{(ab)} &=& *(\theta_{(a}\wedge Z_{b)}),\nonumber\\[1mm] -Z_a &=& D\widetilde{\Theta}_a - + (\partial_a\ipr\Theta^b)\wedge\widetilde{\Theta}_b - +2(\partial_a\ipr\Omega^{MN})\wedge\widetilde{\Omega}_{MN} -\nonumber\\ -&& + {\rm c.c.}-\partial_a L_{\rm g} -\end{eqnarray} -\begin{eqnarray} -&&V_{AB} = -D\widetilde{\Omega}_{AB} - \widetilde{\Theta}_{AB},\nonumber\\[1mm] -&& -\theta_{[a}\wedge\widetilde{\Theta}_{b]} \tsst -\epsilon_{AB} \widetilde{\Theta}_{\dot A\dot B} -+ \epsilon_{\dot A\dot B}\widetilde{\Theta}_{AB} -\end{eqnarray} - -Since gravitational equations are computed in the -spinorial formalism with the standard null frame -\seethis{See pages \pageref{spinors}\ and \pageref{spinors1}.} -the metric equation is complex and components $\scriptstyle02$, -$\scriptstyle12$, $\scriptstyle22$ are conjugated to $\scriptstyle03$. -$\scriptstyle13$, $\scriptstyle33$. Since these components are not independent -For the sake of efficiency by default \grg\ computes only -the $\scriptstyle00$, $\scriptstyle01$, $\scriptstyle02$, -$\scriptstyle11$, $\scriptstyle12$, $\scriptstyle22$ and $\scriptstyle23$ -components of $Z_{(ab)}$ only. -If you want to have all components the switch \comm{FULL} must be -turned on. \swind{FULL} - -These equations allows one to compute field equations for -gravity theory with an arbitrary Lagrangian. -But the value of three quantities $L_{\rm g}$, -$\widetilde{\Omega}_{AB}$ and $\widetilde{\Theta}{}^a$ -must be specified by the user. In addition \grg\ has built-in -formulas for the most general quadratic in torsion and curvature -Lagrangian. The {\tt Standard way} for $L_{\rm g}$, -$\widetilde{\Omega}_{AB}$ and $\widetilde{\Theta}{}^a$ is \label{thetau} -\begin{eqnarray} -\widetilde{\Theta}{}^a &=& -i\mu_1 (\stackrel{\scriptscriptstyle\rm c}{\vartheta}{}^a -{\rm c.c.}) -+i\mu_2 (\stackrel{\scriptscriptstyle\rm t}{\vartheta}{}^a -{\rm c.c.}) -+i\mu_3 (\stackrel{\scriptscriptstyle\rm a}{\vartheta}\!{}^a -{\rm c.c.}), \\[2mm] -\widetilde{\Omega}_{AB} &=& -i(\lambda_0-\sigma\,8\pi G\, a_0\phi^2)\, S_{AB} \nonumber\\&& -+i\lambda_1 \OO{w}_{AB} --i\lambda_2 \OO{c}_{AB} -+i\lambda_3 \OO{r}_{AB} \nonumber\\&& -+i\lambda_4 \OO{a}_{AB} --i\lambda_5 \OO{b}_{AB} -+i\lambda_6 \OO{d}_{AB} , \\[2mm] -L_{\rm g} &=& (-2\Lambda +\frac{1}{2}\lambda_0R --\sigma\,4\pi G a_0 \phi^2 R) \upsilon -+ \Omega^{AB}\wedge\widetilde{\Omega}_{AB} + {\rm c.c.} \nonumber\\&& -+ \frac{1}{2} \Theta^a\wedge\widetilde{\Theta}_a -\end{eqnarray} - -The cosmological term $\Lambda$ is included into -equations iff the switch \comm{CCONST} is turned on \swinda{CCONST} -and the value of $\Lambda$ is given by the constant \comm{CCONST}. -The term with the scalar field $\phi$ is included into -equations iff the switch \comm{NONMIN} is on. \swinda{NONMIN} -The gravitational constant $G$ is given by the constant \comm{GCONST}. -The parameters of the quadratic Lagrangian are given by the objects -\object{L-Constants LCONST.i6}{\lambda_i} -\object{M-Constants MCONST.i3}{\mu_i} -\object{A-Constants ACONST.i2}{a_i} -The default value of these objects ({\tt Standard way}) is -\begin{eqnarray} -\lambda_i &=& (\mbox{\tt LC0},\mbox{\tt LC1},\mbox{\tt LC2},\mbox{\tt LC3},\mbox{\tt LC4},\mbox{\tt LC5},\mbox{\tt LC6}), \\ -\mu_i &=& (0,\mbox{\tt MC1},\mbox{\tt MC2},\mbox{\tt MC32}), \\ -a_i &=& (\mbox{\tt AC0},0,0) -\end{eqnarray} - -\section{Gravitational Equations in Riemann Space} - -Equations in this section are valid in dimension $d=4$ -with the signature ${\scriptstyle(-,+,+,+)}$ and -${\scriptstyle(+,-,-,-)}$ only. -The $\sigma=1$ for the first signature and $\sigma=-1$ -for the second. The nonmetricity and torsion must be zero and the -switches \comm{NONMETR} and \comm{TORSION} must be turned off. - -Let us consider the action -\begin{equation} -S=\int\left[\frac{\sigma}{16\pi G}L_{\rm g} -+L_{\rm m}\right] -\end{equation} -where -\object{Action LACT}{L_{\rm g}=\upsilon\,{\cal L}_{\rm g}} -is the gravitational action 4-form and -\begin{equation} -L_{\rm m} = \upsilon\,{\cal L}_{\rm m} -\end{equation} -is the matter action 4-form. - -Let us define the following variational derivatives -\begin{equation} -Z^\mu{}_{a} = \frac{1}{\sqrt{-g}} -\frac{\delta\sqrt{-g}{\cal L}_{\rm g}}{\delta h^a_\mu} -,\qquad -T^\mu{}_{a} = \frac{\sigma}{\sqrt{-g}} -\frac{\delta\sqrt{-g}{\cal L}_{\rm m}}{\delta h^a_\mu} -\end{equation} -Then the {\tt Metric equation} is \label{metreq} -\object{Metric Equation METRq.a.b}{-\frac12Z_{ab}=8\pi G\,T_{ab}} -Notice that $Z_{ab}$ and $T_{ab}$ are automatically symmetric. - -Let us define 3-form -\begin{equation} -Z_a = Z^m{}_a\,*\theta_m,\qquad t_a = t^m{}_a\,*\theta_m -\end{equation} - -Now we consider the equations which are used in \grg\ to -compute the left-hand side of the metric equation -$Z_{ab}$. We have to emphasize that we use -spinors and all restrictions imposed by the spinorial formalism -\seethis{See pages \pageref{spinors}\ or \pageref{spinors1}.} -must be fulfilled. - -We consider the Lagrangian which is an arbitrary algebraic function -of the curvature tensor -\begin{equation} -{\cal L}_{\rm g} = {\cal L}_{\rm g}(R_{abcd}) -\end{equation} -No derivatives of the curvature are permitted. -For such a Lagrangian we define so called curvature momentum -\begin{equation} -\widetilde{R}{}^{abcd} = -2\frac{\partial{\cal L}_{\rm g}(R)}{\partial R_{abcd}} -\end{equation} - -The corresponding \grg\ built-in object is -\object{Undotted Curvature Momentum POMEGAU.AB}{\widetilde{\Omega}_{AB}} -where -\begin{eqnarray} -\widetilde{\Omega}_{ab} &=& \frac12 \widetilde{R}_{abcd}\,S^{cd} \\[1mm] -\end{eqnarray} -and -\begin{equation} -\widetilde{\Omega}_{ab} \tsst \widetilde{\Omega}_{A\dot AB\dot B}= -\epsilon_{AB} \widetilde{\Omega}_{\dot A\dot B} -+ \epsilon_{\dot A\dot B}\widetilde{\Omega}_{AB} -\end{equation} - -If value of the objects $L_{\rm g}$ ({\tt Action}) and -$\widetilde{\Omega}_{AB}$ ({\tt Undotted curvature momentum}) is specified -then the {\tt Metric equation} can be calculated using equations -({\tt Standard way}) -\begin{eqnarray} -Z_{ab} &=& *(\theta_{(a}\wedge Z_{b)}),\nonumber\\[1mm] -Z_a &=& D [ -2\partial_m\ipr D\widetilde{\Omega}_a{}^{m} --{\frac{1}{2}}\theta_a\!\wedge -(\partial_m\ipr\partial_n\ipr D\widetilde{\Omega}{}^{mn})] -\nonumber\\&& - +2(\partial_a\ipr\Omega^{MN})\wedge\widetilde{\Omega}_{MN} - + {\rm c.c.}-\partial_a L_{\rm g} -\end{eqnarray} - -Since gravitational equations are computed in the -spinorial formalism with the standard null frame -\seethis{See \pref{spinors}\ or \pref{spinors1}.} -the metric equation is complex and components $\scriptstyle02$, -$\scriptstyle12$, $\scriptstyle22$ are conjugated to $\scriptstyle03$, -$\scriptstyle13$, $\scriptstyle33$. -For the sake of efficiency by default \grg\ computes only -the components $\scriptstyle00$, $\scriptstyle01$, $\scriptstyle02$, -$\scriptstyle11$, $\scriptstyle12$, $\scriptstyle22$ and $\scriptstyle23$ -only. If you want to have all components the switch \comm{FULL} must be -turned on. \swinda{FULL} - -These equations allows one to compute field equations for -gravity theory with an arbitrary Lagrangian. -But the value of three quantities $L_{\rm g}$ and -$\widetilde{\Omega}_{AB}$ must be specified by user. -In addition \grg\ has built-in -formulas for the most general quadratic in the curvature -Lagrangian. The {\tt Standard way} for $L_{\rm g}$ and -$\widetilde{\Omega}_{AB}$ is -\begin{eqnarray} -\widetilde{\Omega}_{AB} &=& -i(\lambda_0-\sigma8\pi G\, a_0\phi^2)\, S_{AB} \nonumber\\&& -+i\lambda_1 \OO{w}_{AB} --i\lambda_2 \OO{c}_{AB} -+i\lambda_3 \OO{r}_{AB}, \\[2mm] -L_{\rm g} &=& (-2\Lambda +{\frac{1}{2}}\lambda_0R --\sigma4\pi G a_0 \phi^2 R) \upsilon -+ \Omega^{AB}\wedge\widetilde{\Omega}_{AB} + {\rm c.c.} -\end{eqnarray} - -The cosmological term is included into -equations iff the switch \comm{CCONST} is on \swinda{CCONST} -and the value of $\Lambda$ is given by the constant \comm{CCONST}. -The term with the scalar field $\phi$ is included into -equations iff the switch \comm{NONMIN} is on. \swinda{NONMIN} -The gravitational constant $G$ is given by the constant \comm{GCONST}. -The parameters of the quadratic lagrangian are given by the object -\object{L-Constants LCONST.i6}{\lambda_i} -\object{A-Constants ACONST.i2}{a_i} -The default value of these objects ({\tt Standard way}) is -\begin{eqnarray} -\lambda_i &=& (\mbox{\tt LC0},\mbox{\tt LC1},\mbox{\tt LC2},\mbox{\tt LC3},\mbox{\tt LC4},\mbox{\tt LC5},\mbox{\tt LC6}), \\ -a_i &=& (\mbox{\tt AC0},0,0) -\end{eqnarray} - - - -\appendix - -\chapter{\grg\ Switches}\vspace*{-6mm} -\index{Switches} - -\tabcolsep=1.5mm - -\begin{tabular}{|c|c|l|c|} -\hline -Switch & Default &\qquad Description & See \\ - & State & & page\\ -\hline -\tt AEVAL & Off & Use {\tt AEVAL} instead of {\tt REVAL}. &\pageref{AEVAL}\\ -\tt WRS & On & Re-simplify object before printing. &\pageref{WRS}\\ -\tt WMATR & Off & Write 2-index objects in matrix form. &\pageref{WMATR}\\ -\tt TORSION & Off & Torsion. &\pageref{TORSION}\\ -\tt NONMETR & Off & Nonmetricity. &\pageref{NONMETR}\\ -\tt UNLCORD & On & Save coordinates in {\tt Unload}. &\pageref{UNLCORD}\\ -\tt AUTO & On & Automatic object calculation in expressions. &\pageref{AUTO}\\ -\tt TRACE & On & Trace the calculation process. &\pageref{TRACE}\\ -\tt SHOWCOMMANDS & Off & Show compound command expansion. &\pageref{SHOWCOMMANDS}\\ -\tt EXPANDSYM & Off & Enable {\tt Sy Asy Cy} in expressions &\pageref{EXPANDSYM}\\ -\tt DFPCOMMUTE & On & Commutativity of {\tt DFP} derivatives. &\pageref{DFPCOMMUTE}\\ -\tt NONMIN & Off & Nonminimal interaction for scalar field. &\pageref{NONMIN}\\ -\tt NOFREEVARS & Off & Prohibit free variables in {\tt Print}. &\pageref{NOFREEVARS}\\ -\tt CCONST & Off & Include cosmological constant in equations. &\pageref{CCONST}\\ -\tt FULL & Off & Number of components in {\tt Metric Equation}. &\pageref{FULL}\\ -\tt LATEX & Off & \LaTeX\ output mode. &\pageref{LATEX}\\ -\tt GRG & Off & \grg\ output mode. &\pageref{GRG}\\ -\tt REDUCE & Off & \reduce\ output mode. &\pageref{REDUCE}\\ -\tt MAPLE & Off & {\sc Maple} output mode. &\pageref{MAPLE}\\ -\tt MATH & Off & {\sc Mathematica} output mode. &\pageref{MATH}\\ -\tt MACSYMA & Off & {\sc Macsyma} output mode. &\pageref{MACSYMA}\\ -\tt DFINDEXED & Off & Print {\tt DF} in index notation. &\pageref{DFINDEXED}\\ -\tt BATCH & Off & Batch mode. &\pageref{BATCH}\\ -\tt HOLONOMIC & On & Keep frame holonomic. &\pageref{HOLONOMIC}\\ -\tt SHOWEXPR & Off & Print expressions during algebraic &\pageref{SHOWEXPR}\\ -\tt & & classification. &\\ -\hline -\end{tabular} - -\chapter{Macro Objects} -\index{Macro Objects} - -Macro objects can be used in expression, in {\tt Write} and -{\tt Show} commands but not in the {\tt Find} command. -The notation for indices is the same as in the {\tt New Object} -declaration (see page \pageref{indices}). - -\begin{center} - -\section{Dimension and Signature} - -\begin{tabular}{|l|l|} -\hline -\tt dim & Dimension $d$ \\ -\hline -\tt sdiag.idim & {\tt sdiag(\parm{n})} is the $n$'th element of the \\ - & signature diag($-1,+1$\dots) \\ -\hline -\tt sign & Product of the signature specification \\ -\tt sgnt & elements $\prod_{n=0}^{d-1}\mbox{\tt sdiag(}n\mbox{\tt)}$ \\[1mm] -\hline -\tt mpsgn & {\tt sdiag(0)} \\ -\tt pmsgn & {\tt -sdiag(0)} \\ -\hline -\end{tabular} - -\section{Metric and Frame} - -\begin{tabular}{|l|l|} -\hline -\tt x\^m & $m$'th coordinate \\ -\tt X\^m & \\ -\hline -\tt h'a\_m & Frame coefficients \\ -\tt hi.a\^m & \\ -\hline -\tt g\_m\_n & Holonomic metric \\ -\tt gi\^m\^n & \\ -\hline -\end{tabular} - -\section{Delta and Epsilon Symbols} - -\begin{tabular}{|l|l|} -\hline -\tt del'a.b & Delta symbols \\ -\tt delh\^m\_n & \\ -\hline -\tt eps.a.b.c.d & Totally antisymmetric symbols \\ -\tt epsi'a'b'c'd & (number of indices depend on $d$) \\ -\tt epsh\_m\_n\_p\_q & \\ -\tt epsih\^m\^n\^p\^q & \\ -\hline -\end{tabular} - -\section{Spinors} - -\begin{tabular}{|l|l|} -\hline -\tt DEL'A.B & Delta symbol \\ -\hline -\tt EPS.A.B & Spinorial metric \\ -\tt EPSI'A'B & \\ -\hline -\tt sigma'a.A.B\cc & Sigma matrices \\ -\tt sigmai.a'A'B\cc & \\ -\hline -\tt cci.i3 & Frame index conjugation in standard null frame \\ - & {\tt cci(0)=0}\ {\tt cci(1)=1}\ {\tt cci(2)=3}\ {\tt cci(3)=2} \\ -\hline -\end{tabular} - -\section{Connection Coefficients} - -\begin{tabular}{|l|l|} -\hline -\tt CHR\^m\_n\_p & Christoffel symbols $\{{}^\mu_{\nu\pi}\}$ \\ -\tt CHRF\_m\_n\_p & and $[{}_{\mu},_{\nu\pi}]$ \\ -\tt CHRT\_m & Christoffel symbol trace $\{{}^\pi_{\pi\mu}\}$ \\ -\hline -\tt SPCOEF.AB.c & Spin coefficients $\omega_{AB\,c}$ \\ -\hline -\end{tabular} - -\section{NP Formalism} - -\begin{tabular}{|l|c|} -\hline -\tt PHINP.AB.CD~ & $\Phi_{AB\dot{c}\dot{D}}$ \\ -\tt PSINP.ABCD & $\Psi_{ABCD}$ \\ -\hline -\tt alphanp & $\alpha$ \\ -\tt betanp & $\beta$ \\ -\tt gammanp & $\gamma$ \\ -\tt epsilonnp & $\epsilon$ \\ -\tt kappanp & $\kappa$ \\ -\tt rhonp & $\rho$ \\ -\tt sigmanp & $\sigma$ \\ -\tt taunp & $\tau$ \\ -\tt munp & $\mu$ \\ -\tt nunp & $\nu$ \\ -\tt lambdanp & $\lambda$ \\ -\tt pinp & $\pi$ \\ -\hline -\tt DD & $D$ \\ -\tt DT & $\Delta$ \\ -\tt du & $\delta$ \\ -\tt dd & $\overline\delta$ \\ -\hline -\end{tabular} - -\end{center} - -\chapter{Objects} - -Here we present the complete list of built-in objects -with names and identifiers. -The notation for indices is the same as in the -{\tt New Object} declaration (see page \pageref{indices}). -Some names (group names) refer to a set of objects. -For example the group name {\tt Spinorial S - forms} below -denotes {\tt SU.AB} and {\tt SD.AB\cc} - -\begin{center} - - -\section{Metric, Frame, Basis, Volume \dots} -\begin{tabular}{|l|l|}\hline -\tt Frame &\tt T'a\\ -\tt Vector Frame &\tt D.a\\ -\hline -\tt Metric &\tt G.a.b\\ -\tt Inverse Metric &\tt GI'a'b\\ -\tt Det of Metric &\tt detG\\ -\tt Det of Holonomic Metric &\tt detg\\ -\tt Sqrt Det of Metric &\tt sdetG\\ -\hline -\tt Volume &\tt VOL\\ -\hline -\tt Basis &\tt b'idim \\ -\tt Vector Basis &\tt e.idim \\ -\hline -\tt S-forms &\tt S'a'b\\ -\hline -\multicolumn{2}{|c|}{\tt Spinorial S-forms} \\ -\tt Undotted S-forms &\tt SU.AB\\ -\tt Dotted S-forms &\tt SD.AB\cc\\ -\hline\end{tabular} - -\section{Rotation Matrices} -\begin{tabular}{|l|l|}\hline -\tt Frame Transformation &\tt L'a.b \\ -\tt Spinorial Transformation &\tt LS.A'B \\ -\hline\end{tabular} - -\section{Connection and related objects} -\begin{tabular}{|l|l|}\hline -\tt Frame Connection &\tt omega'a.b\\ -\tt Holonomic Connection &\tt GAMMA\^m\_n\\ -\hline -\multicolumn{2}{|c|}{\tt Spinorial Connection}\\ -\tt Undotted Connection &\tt omegau.AB\\ -\tt Dotted Connection &\tt omegad.AB\cc\\ -\hline -\tt Riemann Frame Connection &\tt romega'a.b\\ -\tt Riemann Holonomic Connection &\tt RGAMMA\^m\_n\\ -\hline -\multicolumn{2}{|c|}{\tt Riemann Spinorial Connection}\\ -\tt Riemann Undotted Connection &\tt romegau.AB\\ -\tt Riemann Dotted Connection &\tt romegad.AB\cc\\ -\hline -\tt Connection Defect &\tt K'a.b\\ -\hline\end{tabular} - -\section{Torsion} -\begin{tabular}{|l|l|}\hline -\tt Torsion &\tt THETA'a\\ -\tt Contorsion &\tt KQ'a.b\\ -\tt Torsion Trace 1-form &\tt QQ\\ -\tt Antisymmetric Torsion 3-form &\tt QQA\\ -\hline -\multicolumn{2}{|c|}{\tt Spinorial Contorsion}\\ -\tt Undotted Contorsion &\tt KU.AB\\ -\tt Dotted Contorsion &\tt KD.AB\cc\\ -\hline -\multicolumn{2}{|c|}{\tt Torsion Spinors }\\ -\multicolumn{2}{|c|}{\tt Torsion Components }\\ -\tt Torsion Trace &\tt QT'a\\ -\tt Torsion Pseudo Trace &\tt QP'a\\ -\tt Traceless Torsion Spinor &\tt QC.ABC.D\cc\\ -\hline -\multicolumn{2}{|c|}{\tt Torsion 2-forms}\\ -\tt Traceless Torsion 2-form &\tt THQC'a\\ -\tt Torsion Trace 2-form &\tt THQT'a\\ -\tt Antisymmetric Torsion 2-form &\tt THQA'a\\ -\hline -\multicolumn{2}{|c|}{\tt Undotted Torsion 2-forms}\\ -\tt Undotted Torsion Trace 2-form &\tt THQTU'a\\ -\tt Undotted Antisymmetric Torsion 2-form &\tt THQAU'a\\ -\tt Undotted Traceless Torsion 2-form &\tt THQCU'a\\ -\hline\end{tabular} - - -\section{Curvature} - -\label{curspincoll} -\begin{tabular}{|l|l|}\hline -\tt Curvature &\tt OMEGA'a.b\\ -\hline -\multicolumn{2}{|c|}{\tt Spinorial Curvature}\\ -\tt Undotted Curvature &\tt OMEGAU.AB\\ -\tt Dotted Curvature &\tt OMEGAD.AB\cc\\ -\hline -\tt Riemann Tensor &\tt RIM'a.b.c.d\\ -\tt Ricci Tensor &\tt RIC.a.b\\ -\tt A-Ricci Tensor &\tt RICA.a.b\\ -\tt S-Ricci Tensor &\tt RICS.a.b\\ -\tt Homothetic Curvature &\tt OMEGAH\\ -\tt Einstein Tensor &\tt GT.a.b\\ -\hline -\multicolumn{2}{|c|}{\tt Curvature Spinors}\\ -\multicolumn{2}{|c|}{\tt Curvature Components}\\ -\tt Weyl Spinor &\tt RW.ABCD\\ -\tt Traceless Ricci Spinor &\tt RC.AB.CD\cc\\ -\tt Scalar Curvature &\tt RR\\ -\tt Ricanti Spinor &\tt RA.AB\\ -\tt Traceless Deviation Spinor &\tt RB.AB.CD\cc\\ -\tt Scalar Deviation &\tt RD\\ -\hline -\multicolumn{2}{|c|}{\tt Undotted Curvature 2-forms}\\ -\tt Undotted Weyl 2-form &\tt OMWU.AB \\ -\tt Undotted Traceless Ricci 2-form &\tt OMCU.AB \\ -\tt Undotted Scalar Curvature 2-form &\tt OMRU.AB \\ -\tt Undotted Ricanti 2-form &\tt OMAU.AB \\ -\tt Undotted Traceless Deviation 2-form &\tt OMBU.AB \\ -\tt Undotted Scalar Deviation 2-form &\tt OMDU.AB \\ -\hline -\multicolumn{2}{|c|}{\tt Curvature 2-forms}\\ -\tt Weyl 2-form &\tt OMW.a.b \\ -\tt Traceless Ricci 2-form &\tt OMC.a.b \\ -\tt Scalar Curvature 2-form &\tt OMR.a.b \\ -\tt Ricanti 2-form &\tt OMA.a.b \\ -\tt Traceless Deviation 2-form &\tt OMB.a.b \\ -\tt Antisymmetric Curvature 2-form &\tt OMD.a.b \\ -\tt Homothetic Curvature 2-form &\tt OSH.a.b \\ -\tt Antisymmetric S-Ricci 2-form &\tt OSA.a.b \\ -\tt Traceless S-Ricci 2-form &\tt OSC.a.b \\ -\tt Antisymmetric S-Curvature 2-form &\tt OSV.a.b \\ -\tt Symmetric S-Curvature 2-form &\tt OSU.a.b \\ -\hline -\end{tabular} - - -\section{Nonmetricity} -\begin{tabular}{|l|l|}\hline -\tt Nonmetricity &\tt N.a.b\\ -\tt Nonmetricity Defect &\tt KN'a.b\\ -\tt Weyl Vector &\tt NNW\\ -\tt Nonmetricity Trace &\tt NNT\\ -\hline -\multicolumn{2}{|c|}{\tt Nonmetricity 1-forms}\\ -\tt Symmetric Nonmetricity 1-form &\tt NC.a.b\\ -\tt Antisymmetric Nonmetricity 1-form &\tt NA.a.b\\ -\tt Nonmetricity Trace 1-form &\tt NT.a.b\\ -\tt Weyl Nonmetricity 1-form &\tt NW.a.b\\ -\hline\end{tabular} - - -\section{EM field} -\begin{tabular}{|l|l|}\hline -\tt EM Potential &\tt A\\ -\tt Current 1-form &\tt J\\ -\tt EM Action &\tt EMACT\\ -\tt EM 2-form &\tt FF\\ -\tt EM Tensor &\tt FT.a.b\\ -\hline -\multicolumn{2}{|c|}{\tt Maxwell Equations}\\ -\tt First Maxwell Equation &\tt MWFq\\ -\tt Second Maxwell Equation &\tt MWSq\\ -\hline -\tt Continuity Equation &\tt COq\\ -\tt EM Energy-Momentum Tensor &\tt TEM.a.b\\ -\hline -\multicolumn{2}{|c|}{\tt EM Scalars}\\ -\tt First EM Scalar &\tt SCF\\ -\tt Second EM Scalar &\tt SCS\\ -\hline -\tt Selfduality Equation &\tt SDq.AB\cc\\ -\tt Complex EM 2-form &\tt FFU\\ -\tt Complex Maxwell Equation &\tt MWUq\\ -\tt Undotted EM Spinor &\tt FIU.AB\\ -\tt Complex EM Scalar &\tt SCU\\ -\tt EM Energy-Momentum Spinor &\tt TEMS.AB.CD\cc\\ -\hline\end{tabular} - -\section{Scalar field} -\begin{tabular}{|l|l|}\hline -\tt Scalar Equation &\tt SCq\\ -\tt Scalar Field &\tt FI\\ -\tt Scalar Action &\tt SACT\\ -\tt Minimal Scalar Action &\tt SACTMIN\\ -\tt Minimal Scalar Energy-Momentum Tensor &\tt TSCLMIN.a.b\\ -\hline\end{tabular} - - -\section{YM field} -\begin{tabular}{|l|l|}\hline -\tt YM Potential &\tt AYM.i9\\ -\tt Structural Constants &\tt SCONST.i9.j9.k9\\ -\tt YM Action &\tt YMACT\\ -\tt YM 2-form &\tt FFYM.i9\\ -\tt YM Tensor &\tt FTYM.i9.a.b\\ -\hline -\multicolumn{2}{|c|}{\tt YM Equations}\\ -\tt First YM Equation &\tt YMFq.i9\\ -\tt Second YM Equation &\tt YMSq.i9\\ -\hline -\tt YM Energy-Momentum Tensor &\tt TYM.a.b\\ -\hline\end{tabular} - -\section{Dirac field} -\begin{tabular}{|l|l|}\hline -\multicolumn{2}{|c|}{\tt Dirac Spinor}\\ -\tt Phi Spinor &\tt PHI.A\\ -\tt Chi Spinor &\tt CHI.B\\ -\hline -\tt Dirac Action 4-form &\tt DACT\\ -\tt Undotted Dirac Spin 3-Form &\tt SPDIU.AB\\ -\tt Dirac Energy-Momentum Tensor &\tt TDI.a.b\\ -\hline -\multicolumn{2}{|c|}{\tt Dirac Equation}\\ -\tt Phi Dirac Equation &\tt DPq.A\cc\\ -\tt Chi Dirac Equation &\tt DCq.A\cc\\ -\hline\end{tabular} - -\section{Geodesics} -\begin{tabular}{|l|l|}\hline -\tt Geodesic Equation &\tt GEOq\^m\\ -\hline\end{tabular} - -\section{Null Congruence} -\begin{tabular}{|l|l|}\hline -\tt Congruence &\tt KV\\ -\tt Null Congruence Condition &\tt NCo\\ -\tt Geodesics Congruence Condition&\tt GCo'a\\ -\hline -\multicolumn{2}{|c|}{\tt Optical Scalars}\\ -\tt Congruence Expansion &\tt thetaO\\ -\tt Congruence Squared Rotation &\tt omegaSQO\\ -\tt Congruence Squared Shear &\tt sigmaSQO\\ -\hline\end{tabular} - -\section{Kinematics} -\begin{tabular}{|l|l|}\hline -\tt Velocity Vector &\tt UV\\ -\tt Velocity &\tt UU'a\\ -\tt Velocity Square &\tt USQ\\ -\tt Projector &\tt PR'a.b\\ -\hline -\multicolumn{2}{|c|}{\tt Kinematics}\\ -\tt Acceleration &\tt accU'a\\ -\tt Vorticity &\tt omegaU.a.b\\ -\tt Volume Expansion &\tt thetaU\\ -\tt Shear &\tt sigmaU.a.b\\ -\hline\end{tabular} - -\section{Ideal and Spin Fluid} -\begin{tabular}{|l|l|}\hline -\tt Pressure &\tt PRES\\ -\tt Energy Density &\tt ENER\\ -\tt Ideal Fluid Energy-Momentum Tensor &\tt TIFL.a.b\\ -\hline -\tt Spin Fluid Energy-Momentum Tensor &\tt TSFL.a.b \\ -\tt Spin Density &\tt SPFLT.a.b \\ -\tt Spin Density 2-form &\tt SPFL \\ -\tt Undotted Fluid Spin 3-form &\tt SPFLU.AB \\ -\tt Frenkel Condition &\tt FCo \\ -\hline\end{tabular} - -\section{Total Energy-Momentum and Spin} -\begin{tabular}{|l|l|}\hline -\tt Total Energy-Momentum Tensor &\tt TENMOM.a.b\\ -\tt Total Energy-Momentum Spinor &\tt TENMOMS.AB.CD\cc\\ -\tt Total Energy-Momentum Trace &\tt TENMOMT\\ -\tt Total Undotted Spin 3-form &\tt SPINU.AB\\ -\hline\end{tabular} - -\section{Einstein Equations} -\begin{tabular}{|l|l|}\hline -\tt Einstein Equation &\tt EEq.a.b\\ -\hline -\multicolumn{2}{|c|}{\tt Spinor Einstein Equations}\\ -\tt Traceless Einstein Equation &\tt CEEq.AB.CD\cc\\ -\tt Trace of Einstein Equation &\tt TEEq\\ -\hline\end{tabular} - -\section{Constants} -\begin{tabular}{|l|l|}\hline -\tt A-Constants &\tt ACONST.i2\\ -\tt L-Constants &\tt LCONST.i6\\ -\tt M-Constants &\tt MCONST.i3\\ -\hline\end{tabular} - -\section{Gravitational Equations} -\begin{tabular}{|l|l|}\hline -\tt Action &\tt LACT\\ -\tt Undotted Curvature Momentum &\tt POMEGAU.AB\\ -\tt Torsion Momentum &\tt PTHETA'a\\ -\hline -\multicolumn{2}{|c|}{\tt Gravitational Equations}\\ -\tt Metric Equation &\tt METRq.a.b\\ -\tt Torsion Equation &\tt TORSq.AB\\ -\hline\end{tabular} - -\end{center} - - -\chapter{Standard Synonymy} -\index{Synonymy} - -Below we present the default synonymy as it is defined in the -global configuration file. See section \ref{tuning} to find out -how to change the default synonymy or define a new one. - -\begin{verbatim} - Affine Aff - Anholonomic Nonholonomic AMode ABasis - Antisymmetric Asy - Change Transform - Classify Class - Components Comp - Connection Con - Constants Const Constant - Coordinates Cord - Curvature Cur - Dimension Dim - Dotted Do - Equation Equations Eq - Erase Delete Del - Evaluate Eval Simplify - Find F Calculate Calc - Form Forms - Functions Fun Function - Generic Gen - Gravitational Gravity Gravitation Grav - Holonomic HMode HBasis - Inverse Inv - Load Restore - Next N - Normalize Normal - Object Obj - Output Out - Parameter Par - Rotation Rot - Scalar Scal - Show ? - Signature Sig - Solutions Solution Sol - Spinor Spin Spinorial Sp - standardlisp lisp - Switch Sw - Symmetries Sym Symmetric - Tensor Tensors Tens - Torsion Tors - Transformation Trans - Undotted Un - Unload Save - Vector Vec - Write W - Zero Nullify -\end{verbatim} - - -\makeatletter -\if@openright\cleardoublepage\else\clearpage\fi -\makeatother -\thispagestyle{empty} -\def\indexname{INDEX} -\printindex - -\end{document} - -%======== End of grg32.tex ==============================================% -