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%==========================================================================%
% GRG 3.2 Manual (C) 1988-97 Vadim V. Zhytnikov %
%==========================================================================%
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GRG
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Version 3.2
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Computer Algebra System for
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Differential Geometry,
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Gravitation and
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Field Theory
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Vadim V. Zhytnikov
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Moscow, 1992–1997
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Chung-Li, 1994
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\begin_layout Chapter
Introduction
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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
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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
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covariant properties
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is based on the computer algebra system
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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.
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knows the covariant properties of these objects, you can easily raise and lower indices, compute covariant and Lie derivatives, perform coordinate and frame transformations.
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works in any dimension and allows one to represent tensor quantities with respect to holonomic, orthogonal and even any other arbitrary frame.
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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
<lyxtabular version="3" rows="11" columns="2">
<features rotate="0" tabularvalignment="middle" tabularwidth="0pt">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
E I PI INFINITY
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" rightline="true" usebox="none">
\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
</cell>
</row>
<row>
<cell alignment="left" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
FAILED
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
ECONST
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Charge of the electron
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
DMASS
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Dirac field mass
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
SMASS
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Scalar field mass
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
GCONST
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Gravitational constant
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
CCONST
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Cosmological constants
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
LC0 LC1 LC2 LC3
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Parameters of the quadratic
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
LC4 LC5 LC6
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
gravitational Lagrangian
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
MC1 MC2 MC3
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
AC0
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Nonminimal interaction constant
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\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
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\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
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\begin_layout Plain Layout
\backslash
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\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
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\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
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\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
<lyxtabular version="3" rows="6" columns="2">
<features rotate="0" tabularvalignment="middle" tabularwidth="0pt">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family roman
total symmetry
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family roman
antisymmetry in first pair of indices and
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family roman
symmetry in second pair
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family roman
symmetry in pair permutation
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family roman
antisymmetry in first and second pair of indices
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family roman
and symmetry in pair permutation
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\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
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\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
<lyxtabular version="3" rows="22" columns="3">
<features rotate="0" tabularvalignment="middle" tabularwidth="0pt">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\family default
\series bold
Operation
\family typewriter
\series default
\shape default
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\family default
\series bold
Description
\family typewriter
\series default
\shape default
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\family default
\series bold
Grouping
\family typewriter
\series default
\shape default
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
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[
\begin_inset Formula $v_1$
\end_inset
,
\begin_inset Formula $v_2$
\end_inset
]
\family typewriter
\series default
\shape default
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Vector bracket
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
@
\family typewriter
\series default
\shape default
\begin_inset Formula $x$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Holonomic vector
\begin_inset Formula $\partial_x$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Exterior differential
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\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
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Dualization
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
<cell alignment="none" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Complex conjugation
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Exponention
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
<cell alignment="none" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Division
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\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
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Multiplication
\end_layout
\end_inset
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<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Vector acting on scalar
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\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
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\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
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
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\begin_inset Formula $v$
\end_inset
\begin_inset space \space{}
\end_inset
\begin_inset ERT
status collapsed
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\backslash
ip
\end_layout
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\begin_inset space \space{}
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\begin_inset Formula $\omega$
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\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
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Interior product
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
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\begin_inset Formula $\Updownarrow$
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\end_layout
\end_inset
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<row>
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\begin_inset Text
\begin_layout Standard
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\begin_inset Formula $v_1$
\end_inset
\begin_inset space \space{}
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.
\family typewriter
\series default
\shape default
\begin_inset space \space{}
\end_inset
\begin_inset Formula $v_2$
\end_inset
\end_layout
\end_inset
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
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Scalar product
\end_layout
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\begin_inset Formula $v$
\end_inset
\begin_inset ERT
status collapsed
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\backslash
ii
\end_layout
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(
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\series default
\shape default
\begin_inset Formula $\omega_1$
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
w
\end_layout
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(
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\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
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<row>
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\begin_inset Text
\begin_layout Standard
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\begin_inset Formula $v$
\end_inset
\begin_inset space \space{}
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.
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\begin_inset space \space{}
\end_inset
\begin_inset Formula $o$
\end_inset
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
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<cell alignment="none" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
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<row>
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\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
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<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
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<cell alignment="none" valignment="top" rightline="true" usebox="none">
\begin_inset Text
\end_inset
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<row>
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\begin_inset Text
\begin_layout Standard
\family typewriter
\begin_inset Formula $\omega_1$
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\begin_inset space \space{}
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\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
w
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\begin_inset space \space{}
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
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Exterior product
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<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
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<row>
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\begin_inset Text
\begin_layout Standard
\family typewriter
+
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\series default
\shape default
\begin_inset space \space{}
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\begin_inset Formula $e$
\end_inset
\end_layout
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<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
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Prefix plus
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<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
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<row>
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\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Prefix minus
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Addition
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\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
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Subtraction
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\end_inset
</cell>
</row>
</lyxtabular>
\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{
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\end_layout
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}
\end_layout
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and covariant derivative
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v
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See page
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If
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grg
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omega
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Dc(OMEGA(a,b),romega)
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romega
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Dfc(v,RIC(m̂,_n))
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EXPANDSYM
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See page
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LatexCommand label
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Here
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The expression
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\begin_inset 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
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See page
\begin_inset CommandInset ref
LatexCommand pageref
reference "solutions"
plural "false"
caps "false"
noprefix "false"
\end_inset
about solutions.
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The substitution can be deactivated by the command
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matches all powers which are greater than one indicated in the substitution:
\begin_inset ERT
status collapsed
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<- Const a; <- (a+1)8̂;
\begin_inset ERT
status collapsed
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\backslash
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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
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<- Let a3̂=1; <- (a+1)8̂;
\begin_inset ERT
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2 85*a + 86*a + 85
\begin_inset ERT
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<- Clear a3̂; <- Match a3̂=1; <- (a+1)8̂;
\begin_inset ERT
status collapsed
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8 7 6 5 4 2 a + 8*a + 28*a + 56*a + 70*a + 28*a + 8*a + 57
\begin_inset ERT
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\begin_layout Standard
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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
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assign
\begin_inset Quotes erd
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value to the user-defined and built-in constants; (iii) to define differentiation rules for functions.
\end_layout
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After some substitution is activated it applies to every evaluated expression but value of the objects calculated
\emph on
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\emph default
remain unchanged. The command
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here
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is the object name, or identifier, or the group object name. Let us consider a simple
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\begin_inset space \space{}
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task which calculates the volume 4-form of some metric
\begin_inset ERT
status collapsed
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<- 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
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2 2 VOL = (SIN(a) + COS(a) ) d t
\begin_inset ERT
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d x
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d z
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status collapsed
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We see that
\begin_inset ERT
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do not know the appropriate trigonometric rule. Thus we are going to apply substitution
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
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\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
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VOL = d t
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
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d x
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d z
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status collapsed
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The situation has been improved. But actually, the
\emph on
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\emph default
representation of VOL
\family typewriter
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remains unchanged. Write
\family typewriter
\series default
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by default re-simplifies expressions before printing.
\begin_inset ERT
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WRS
\begin_inset ERT
status collapsed
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}
\end_layout
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By turning switch WRS
\family typewriter
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\shape default
off we can prevent this re-simplification:
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
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\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
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\begin_inset space \space{}
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d x
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d z
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Now we can apply
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:
\begin_inset ERT
status collapsed
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\end_layout
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<- 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
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\begin_inset space \space{}
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d x
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d z
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status collapsed
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We see that the internal value of VOL
\family typewriter
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now has been replaced by re-simplified expression.
\end_layout
\begin_layout Standard
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Notice that the command
\begin_inset ERT
status collapsed
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Evaluate All;
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applies
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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
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\end_layout
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\begin_inset space \space{}
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lacks the notion of partial derivative of a function. The expression
\begin_inset ERT
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DF(f(x,y),x)
\begin_inset ERT
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\end_layout
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is treated by
\begin_inset ERT
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\begin_inset space \space{}
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as the
\begin_inset Quotes eld
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derivative of the expression
\begin_inset ERT
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f(x,y)
\begin_inset ERT
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with respect to the variable
\begin_inset ERT
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x
\begin_inset ERT
status collapsed
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\begin_inset Quotes erd
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rather than the
\begin_inset Quotes eld
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derivative of the function
\begin_inset ERT
status collapsed
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f
\begin_inset ERT
status collapsed
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\end_layout
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with respect to its first argument
\begin_inset Quotes erd
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. Due to this
\begin_inset ERT
status collapsed
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\end_layout
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\begin_inset space \space{}
\end_inset
cannot handle chain differentiation rule etc. This problem is fixed by the package
\begin_inset ERT
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dfpart
\begin_inset ERT
status collapsed
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\end_layout
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written by H.
\begin_inset space ~
\end_inset
Melenk. This package introduces notion of generic function and partial derivative
\begin_inset ERT
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DFP
\begin_inset ERT
status collapsed
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\end_layout
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. If
\begin_inset ERT
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dfpart
\begin_inset ERT
status collapsed
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is installed on your
\begin_inset ERT
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\begin_inset space \space{}
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system
\begin_inset ERT
status collapsed
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\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
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\end_layout
\end_inset
Generic functions must be always declared with the list of parameters (
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
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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
}
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\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
}
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\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
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command{
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\end_inset
Normalize
\begin_inset ERT
status collapsed
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parm{
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\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
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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
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command{
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\end_inset
Evaluate
\begin_inset ERT
status collapsed
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parm{
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\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
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\begin_layout Standard
\family typewriter
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
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cmdind{
\end_layout
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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
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command{
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Write
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status collapsed
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parm{
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object
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status collapsed
\begin_layout Plain Layout
}
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\end_inset
;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
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\end_inset
prints value of the
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status collapsed
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parm{
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object
\begin_inset ERT
status collapsed
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. Here
\begin_inset ERT
status collapsed
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parm{
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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
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parm{
\end_layout
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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
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command{
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Write
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status collapsed
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parm{
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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 ~
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>
\begin_inset space ~
\end_inset
\begin_inset ERT
status collapsed
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"
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\end_inset
\begin_inset ERT
status collapsed
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\backslash
parm{
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file
\begin_inset ERT
status collapsed
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}
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\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{
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\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
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\begin_inset ERT
status collapsed
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parm{
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file
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status collapsed
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}
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\end_inset
\begin_inset ERT
status collapsed
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"
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
. Notice that
\begin_inset ERT
status collapsed
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\backslash
comm{
\end_layout
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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
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command{
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\end_inset
Write to
\begin_inset ERT
status collapsed
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parm{
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file
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}
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status collapsed
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"
\end_layout
\end_inset
;
\begin_inset Newline newline
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Write >
\begin_inset ERT
status collapsed
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"
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status collapsed
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parm{
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file
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\end_inset
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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<b;
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family typewriter
prints off-diagonal components of the metric.
\end_layout
\begin_layout Standard
\family typewriter
Both word
\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
and
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
For
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
parts of the command can be omitted and it is possible just to enter an expression
\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
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
and it will be evaluated and printed. The expression can contain indefinite identifiers and by default
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
grg
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
treats them similarly to the variables in the
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
For
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
part of the
\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. The range of such parameters are determined by the short summation variable specification as explained on page
\begin_inset CommandInset ref
LatexCommand pageref
reference "siter"
plural "false"
caps "false"
noprefix "false"
\end_inset
. For example the following four commands are equivalent. they all print the components of the holonomic metric
\begin_inset Formula $g_{\alpha\beta}$
\end_inset
\begin_inset listings
lstparams "float"
inline false
status collapsed
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Standard
\family typewriter
Print g(a,b) for a,b; For a,b Print g(a,b); g(a,b) for a,b; g(a,b);
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family typewriter
Here the 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
,
\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
run from 0 to
\begin_inset Formula $d-1$
\end_inset
.
\end_layout
\begin_layout Standard
\family typewriter
Unfortunately such treatment of unknown variables may create some confusion since occasionally misprinted identifier may be recognizes by
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
grg
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
as an iteration variable. If switch
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
swind{
\end_layout
\end_inset
NOFREEVARS
\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
NOFREEVARS
\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
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
becomes more scrupulous and any unknown variable will cause the error.
\end_layout
\begin_layout Subsection
\family typewriter
Controlling the Output
\end_layout
\begin_layout Standard
\family typewriter
There are several switches and commands which allow one to change output form of expressions. One needs to stress that all these facilities have no influence on the
\emph on
internal form
\emph default
of expressions, they alter the
\emph on
printout only
\emph default
.
\end_layout
\begin_layout Standard
\family typewriter
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
enlargethispage{2mm}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family typewriter
Switches ALLFAC
\family typewriter
\series default
\shape default
and command Factor
\family typewriter
\series default
\shape default
control factoring of subexpressions. In the on default position ALLFAC
\family typewriter
\series default
\shape default
makes the system search for a common factor and print it outside the expression. The command
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
cmdind{
\end_layout
\end_inset
Factor
\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
Factor
\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
makes the system collect together terms with different powers of subexpressions
\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
. Command
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
cmdind{
\end_layout
\end_inset
RemFac
\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
RemFac
\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
removes the action of the previous Factor
\family typewriter
\series default
\shape default
command.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{slisting}
\end_layout
\end_inset
<- Constants a,b,c; <- a*(a+b+1)2̂;
\begin_inset Newpage newpage
\end_inset
2 2 a*(a + 2*a*b + 2*a + b + 2*b + 1)
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- Off ALLFAC; <- a*(a+b+1)2̂;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
3 2 2 2 a + 2*a *b + 2*a + a*b + 2*a*b + a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- Factor b; <- a*(a+b+1)2̂;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 2 3 2 b *a + b*(2*a + 2*a) + a + 2*a + a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- On ALLFAC; <- a*(a+b+1)2̂;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 2 b *a + 2*b*a*(a + 1) + a*(a + 2*a + 1)
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
end{slisting}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family typewriter
Normally
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
reduce
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
prints terms in some canonical order. The switch REVPRI
\family typewriter
\series default
\shape default
prints terms in reverse order and command
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
cmdind{
\end_layout
\end_inset
Order
\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
Order
\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
specifies the required order of subexpressions explicitly.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{slisting}
\end_layout
\end_inset
<- Constants a,b,c; <- (a+b*c)3̂;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
3 2 2 2 3 3 a + 3*a *b*c + 3*a*b *c + b *c
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- On REVPRI; <- (a+b*c)3̂;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
3 3 2 2 2 3 b *c + 3*a*b *c + 3*a *b*c + a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- Order c,a,b; <- (a+b*c)3̂;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
3 2 2 2 3 3 a + 3*c*a *b + 3*c *a*b + c *b
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- Off REVPRI; <- (a+b*c)3̂;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
3 3 2 2 2 3 c *b + 3*c *a*b + 3*c*a *b + a
\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
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
reduce
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
prints fractions in two-dimensional format but turning off switch RATPRI
\family typewriter
\series default
\shape default
prevents this facility. Switch DIV
\family typewriter
\series default
\shape default
in the on position makes the system divide each term of the numerator by the denominator and to print the denominator in the form of negative powers. Switch RAT
\family typewriter
\series default
\shape default
works in combination with the Factor
\family typewriter
\series default
\shape default
command. In the on position it makes the system divide each term collected by the Factor
\family typewriter
\series default
\shape default
in the numerator by the denominator.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{slisting}
\end_layout
\end_inset
<- Const a,b,c; <- (a+b+1)2̂/a;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 2 a + 2*a*b + 2*a + b + 2*b + 1 ——————————— a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- Off RATPRI; <- (a+b+1)2̂/a;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 2 (a + 2*a*b + 2*a + b + 2*b + 1)/a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- On DIV; <- (a+b+1)2̂/a;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
-1 2 -1 -1 a + a *b + 2*a *b + a + 2*b + 2
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- Factor b; <- (a+b+1)2̂/a;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 -1 -1 -1 b *a + 2*b*(a + 1) + a + a + 2
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- Off DIV; <- (a+b+1)2̂/a;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 2 (b + 2*b*(a + 1) + a + 2*a + 1)/a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- On RAT; <- (a+b+1)2̂/a;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 2 b /a + 2*b*(a + 1)/a + (a + 2*a + 1)/a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
<- On RATPRI; <- (a+b+1)2̂/a;
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
par
\end_layout
\end_inset
2 2 b a + 1 a + 2*a + 1 —- + 2*b*——- + ————– a a a
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
end{slisting}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family typewriter
One needs to realize that output form transformations may require a long time and memory expense. There is a special switch PRI
\family typewriter
\series default
\shape default
which allows one to minimize this expense. If PRI
\family typewriter
\series default
\shape default
is turned off then the system will print all expressions exactly in their internal form and output control does not work. This is the fastest way to print result of calculations.
\end_layout
\begin_layout Standard
\family typewriter
The command
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
cmdind{
\end_layout
\end_inset
Line Length
\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
Line Length
\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
sets the output line length to
\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
.
\end_layout
\begin_layout Subsection
\family typewriter
\SpecialChar LaTeX
\begin_inset space \space{}
\end_inset
and Graphics Output
\end_layout
\begin_layout Standard
\family typewriter
\begin_inset Index idx
status collapsed
\begin_layout Plain Layout
LaTeX@\SpecialChar LaTeX
\begin_inset space \space{}
\end_inset
output mode
\end_layout
\end_inset
\begin_inset Index idx
status collapsed
\begin_layout Plain Layout
Graphics output mode
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family typewriter
Some versions of
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
reduce
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
running under Windows, OS/2 or X-windows are equipped with the graphic shells which provide book-style output with Greek characters, integral signs etc.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
grg
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
is compatible with these systems.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
swind{
\end_layout
\end_inset
FANCY
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
This graphic regime is activated by switch
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
FANCY
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
\family typewriter
Graphic output mode internally uses some subset of the \SpecialChar LaTeX
\begin_inset space \space{}
\end_inset
language.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
swind{
\end_layout
\end_inset
LATEX
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
Switch
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
LATEX
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
makes
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
grg
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
to print the output in the \SpecialChar LaTeX
\begin_inset space \space{}
\end_inset
format. This output can be written into a file and later directly inserted in a document. Notice that turning off switch
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
LATEX
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
returns graphic output mode with switch
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
FANCY
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
on while turning off
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
FANCY
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
automatically turns off
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
LATEX
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
as well and returns usual character output mode.
\end_layout
\begin_layout Standard
\family typewriter
In graphic regime the derivatives are printed in
\begin_inset Formula $\partial f/\partial x$
\end_inset
notation.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
swind{
\end_layout
\end_inset
DFINDEXED
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
Switch
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
comm{
\end_layout
\end_inset
DFINDEXED
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
makes the system to print derivatives in the indexed notation
\begin_inset Formula $f_x$
\end_inset
.
\end_layout
\begin_layout Standard
\family typewriter
The following expressions is the scalar curvature of the Bondi metric obtained by
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
grg
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
and directly inserted in this manual
\begin_inset Formula \begin{eqnarray*}
R &= &
\bigl(4\,e^{2\,\beta\,+\,2\,\gamma}\,\cos(\theta)\,\frac{\partial\,U}{\partial\,r}\,r^2\,-\,8\,e^{4\,\beta}\,\cos(\theta)\,\frac{\partial\,\beta}{\partial\,\theta}\,-\,\\
&&4\,e^{2\,\beta\,+\,2\,\gamma}\,\cos(\theta)\,\frac{\partial\,\gamma}{\partial\,r}\,U\,r^2\,+\,12\,e^{4\,\beta}\,\cos(\theta)\,\frac{\partial\,\gamma}{\partial\,\theta}\,+\,\\
&&12\,e^{2\,\beta\,+\,2\,\gamma}\,\cos(\theta)\,U\,r\,+\,4\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial^2\,U}{\partial\,r\,\partial\,\theta}\,\sin(\theta)\,r^2\,+\,\\
&&e^{4\,\gamma}\,(\frac{\partial\,U}{\partial\,r})^2\,\sin(\theta)\,r^4\,+\,4\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial\,U}{\partial\,r}\,\frac{\partial\,\beta}{\partial\,\theta}\,\sin(\theta)\,r^2\,+\,\\
&&4\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial\,U}{\partial\,\theta}\,\frac{\partial\,\gamma}{\partial\,r}\,\sin(\theta)\,r^2\,+\,12\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial\,U}{\partial\,\theta}\,\sin(\theta)\,r\,-\,\\
&&4\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial^2\,V}{\partial\,r^2}\,\sin(\theta)\,r\,-\,8\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial\,V}{\partial\,r}\,\frac{\partial\,\beta}{\partial\,r}\,\sin(\theta)\,r\,-\,\\
&&8\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial\,V}{\partial\,r}\,\sin(\theta)\,+\,8\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial^2\,\beta}{\partial\,r\,\partial\,\theta}\,\sin(\theta)\,U\,r^2\,-\,\\
&&8\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial^2\,\beta}{\partial\,r^2}\,\sin(\theta)\,V\,r\,+\,8\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial\,\beta}{\partial\,r}\,\sin(\theta)\,V\,-\,\\
&&8\,e^{4\,\beta}\,\frac{\partial^2\,\beta}{\partial\,\theta^2}\,\sin(\theta)\,-\,12\,e^{4\,\beta}\,(\frac{\partial\,\beta}{\partial\,\theta})^2\,\sin(\theta)\,+\,16\,e^{4\,\beta}\,\frac{\partial\,\beta}{\partial\,\theta}\,\frac{\partial\,\gamma}{\partial\,\theta}\,\sin(\theta)\,-\,\\
&&8\,e^{2\,\beta\,+\,2\,\gamma}\,(\frac{\partial\,\gamma}{\partial\,r})^2\,\sin(\theta)\,V\,r\,+\,8\,e^{2\,\beta\,+\,2\,\gamma}\,\frac{\partial\,\gamma}{\partial\,r}\,\frac{\partial\,\gamma}{\partial\,\theta}\,\sin(\theta)\,U\,r^2\,+\,\\
&&4\,e^{4\,\beta}\,\frac{\partial^2\,\gamma}{\partial\,\theta^2}\,\sin(\theta)\,-\,8\,e^{4\,\beta}\,(\frac{\partial\,\gamma}{\partial\,\theta})^2\,\sin(\theta)\,+\,4\,e^{4\,\beta}\,\sin(\theta)\bigr)/\\
&&\bigl(2\,e^{4\,\beta\,+\,2\,\gamma}\,\sin(\theta)\,r^2\bigr)
\end{eqnarray*}
\end_inset
\end_layout
\begin_layout Subsection
\family typewriter
Exporting Data Into Other Systems
\end_layout
\begin_layout Standard
\family typewriter
\begin_inset Index idx
status collapsed
\begin_layout Plain Layout
Output modes
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\family typewriter
Capabilities of major modern computer algebra systems are approximately equivalent but not quite. One system is better in doing one things and other is better for other purposes. It may happen that tools which you need are available only in one particular systems.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
grg
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
provides quite unique facility to export the data into other computer algebra systems. Turning on one of the following switches establishes the
\emph on
output mode
\emph default
in which all expressions are printed in the
\emph on
input
\emph default
language of other CAS. This output can be saved into a file and later you can use this CAS to proceed you analysis of the data. At present
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
grg
\end_layout
\end_inset
\begin_inset space \space{}
\end_inset
supports five output modes which are controlled by the switches
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
swind{
\end_layout
\end_inset
MACSYMA
\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
MAPLE
\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
MATH
\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
REDUCE
\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
GRG
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="2">
<features rotate="0" tabularvalignment="middle" tabularwidth="0pt">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
<cell alignment="left" valignment="top" usebox="none">
\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
</cell>
</row>
</lyxtabular>
\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
<lyxtabular version="3" rows="4" columns="2">
<features rotate="0" tabularvalignment="middle" tabularwidth="0pt">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\begin_inset Formula $X_{ABCD}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Weyl spinor type
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\begin_inset Formula $X_{AB\dot{C}\dot{D}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Traceless Ricci spinor type
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\begin_inset Formula $X_{AB}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Electromagnetic stress spinor type
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
\begin_inset Formula $X_{A\dot{B}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Standard
\family typewriter
Vector in the spinorial representation
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\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
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setenv grg /xxx/yyy/
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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 $
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\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 $
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\begin_inset Formula $. Thus, in addition we print coordinate
components of the tensor $
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\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
$
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\begin_inset Formula $. The metric is automatically assumed to be
Lorentzian $
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\begin_inset Formula $.
First \grg\ computed the frame connection 1-form $
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Next the curvature 2-form $
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\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
$
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\begin_inset Formula $ for
holonomic (coordinate) indices; $
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a,b,c,d,m,n
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\begin_inset Formula $ for
anholonomic frame indices and $
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\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 $
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\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 $
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\begin_inset Formula $
and $
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$
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\begin_inset Formula $ by $
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\begin_inset Formula $
($
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\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 $
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\begin_inset Formula $ and metric $
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\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 $
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\begin_inset Formula $ and frame $
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\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 $
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\begin_inset Formula $ only. In this section the upper sign relates to the
signature $
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\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
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conserving alphabetical relationship between tensor indices in the
left-hand side and spinorial one in the right-hand side:
$
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\begin_inset Formula $.
There is one quite important special case. Any real
antisymmetric tensor $
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\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 $
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\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 $
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\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 $
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\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 $
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\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 $
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\begin_inset Formula $ and
$
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\begin_inset Formula $ is the {\tt Standard way} to find Riemann
frame connection $
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\begin_inset Formula $.
If torsion is nonzero then $
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\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 $
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\begin_inset Formula $ is given by Eq. (\ref{solstr}).
Similarly if nonmetricity is nonzero then $
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\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 $
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\begin_inset Formula $ is given by Eq. (\ref{solstr}).
Finally if both torsion and nonmetricity are
nonzero then $
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\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 $
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\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 $
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\begin_inset Formula $
is obtained {\tt From torsion} by (\ref{solstr})
with $
\end_inset
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\begin_inset Formula $, $
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\begin_inset Formula $.
The nonmetricity defect $
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\begin_inset Formula $
is obtained {\tt From nonmetricity} by (\ref{solstr})
with $
\end_inset
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\begin_inset Formula $, $
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\begin_inset Formula $.
Analogously (\ref{solstr}) with $
\end_inset
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\begin_inset Formula $, $
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\begin_inset Formula $
is the {\tt Standard way} to compute the connection defect $
\end_inset
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\begin_inset Formula $.
The torsion $
\end_inset
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\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
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\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
$
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\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
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{
\end_layout
\end_inset
\begin_inset ERT
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\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
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\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
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\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
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\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
$
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\begin_inset Formula $ while lower one to the signature
$
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\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 $
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\begin_inset Formula $ and $
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\begin_inset Formula $ can be used to compute
the {\tt Curvature spinors} ($
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\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 $
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\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 $
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\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 $
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\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 $
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\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 $
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\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
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\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 $
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\begin_inset Formula $ and lower one to the
signature $
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\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 $
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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 $
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\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
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\end_layout
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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 $
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(AB)c
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\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 $
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\begin_inset Formula $ the Newman-Penrose equations for
the quantities introduced above can be found in section 7.1 of ESEFE.
For other signature $
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\begin_inset Formula $ one must alter the sign of
$
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\begin_inset Formula $ and $
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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 $
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$
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\begin_inset Formula $.
The sign factor $
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\begin_inset Formula $ in the expressions below is
$
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=-
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\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
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A
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\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 $
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\begin_inset Formula $ and its dual
$
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*
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E
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\begin_inset Formula $
are expressed via usual 3-dimensional vectors $
\end_inset
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\begin_inset Formula $ and
$
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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
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\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
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ab
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\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 $
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signature $
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\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
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\end_layout
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D
\begin_inset ERT
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\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
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(-,+,…,+)
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\end_layout
\end_inset
\begin_inset Formula $ and
$
\end_inset
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{
\end_layout
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\begin_inset ERT
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\end_layout
\end_inset
(+,-,…,-)
\begin_inset ERT
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}
\end_layout
\end_inset
\begin_inset Formula $.
The sign factor $
\end_inset
\begin_inset ERT
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\backslash
sigma
\end_layout
\end_inset
\begin_inset Formula $ is $
\end_inset
\begin_inset ERT
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sigma
\end_layout
\end_inset
=-
\begin_inset ERT
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{
\end_layout
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\end_layout
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diag
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}
\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
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s
\begin_inset ERT
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}
\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
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\end_layout
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\end_layout
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(-,+,…,+)
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}
\end_layout
\end_inset
\begin_inset Formula $ and
$
\end_inset
\begin_inset ERT
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{
\end_layout
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\begin_inset ERT
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\end_layout
\end_inset
(+,-,…,-)
\begin_inset ERT
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\begin_layout Plain Layout
}
\end_layout
\end_inset
\begin_inset Formula $.
The sign factor $
\end_inset
\begin_inset ERT
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sigma
\end_layout
\end_inset
\begin_inset Formula $ in the expressions below is
$
\end_inset
\begin_inset ERT
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sigma
\end_layout
\end_inset
=-
\begin_inset ERT
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{
\end_layout
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\end_layout
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diag
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}
\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
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\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
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delta
\end_layout
\end_inset
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\end_layout
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ij
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\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
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\backslash
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\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
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\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
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{
\end_layout
\end_inset
ab
\begin_inset ERT
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}
\end_layout
\end_inset
=
\begin_inset ERT
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{
\end_layout
\end_inset
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\end_layout
\end_inset
diag
\begin_inset ERT
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\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
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ab
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\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
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SF
\begin_inset ERT
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\end_layout
\end_inset
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\end_layout
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ab
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\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
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\backslash
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\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
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ab
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\end_inset
\begin_inset Formula $ and
$
\end_inset
s
\begin_inset ERT
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{
\end_layout
\end_inset
AB
\begin_inset ERT
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\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
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\end_layout
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ab
\begin_inset ERT
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\end_inset
=s
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\end_layout
\end_inset
AB
\begin_inset ERT
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\end_layout
\end_inset
=0
\begin_inset Formula $.
Notice that $
\end_inset
T
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ab
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\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
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ab
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\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
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(-,+,+,+)
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\end_inset
\begin_inset Formula $ and
$
\end_inset
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\end_layout
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\end_layout
\end_inset
(+,-,-,-)
\begin_inset ERT
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\end_layout
\end_inset
\begin_inset Formula $ only.
The $
\end_inset
\begin_inset ERT
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\end_layout
\end_inset
=1
\begin_inset Formula $ for the first signature and $
\end_inset
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\backslash
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\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
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\end_layout
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\begin_inset ERT
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\end_inset
a
\begin_inset Formula $ as a source. The source in the second
equation is the spin tensor $
\end_inset
s
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\end_layout
\end_inset
\begin_inset ERT
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\end_layout
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ab
\begin_inset ERT
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\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
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\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 $
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$
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\begin_inset Formula $ ({\tt Undotted curvature momentum})
and $
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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
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\begin_inset Formula $ are conjugated to $
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$
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\begin_inset Formula $. Since these components are not independent
For the sake of efficiency by default \grg\ computes only
the $
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$
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\begin_inset Formula $
components of $
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\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
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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 $
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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
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\begin_inset Formula $ is included into
equations iff the switch \comm{CCONST} is turned on \swinda{CCONST}
and the value of $
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\begin_inset Formula $ is given by the constant \comm{CCONST}.
The term with the scalar field $
\end_inset
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\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 $
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$
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The $
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=1
\begin_inset Formula $ for the first signature and $
\end_inset
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=-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 $
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\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
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\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 $
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$
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\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
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$
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\begin_inset Formula $ are conjugated to $
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$
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\begin_inset Formula $.
For the sake of efficiency by default \grg\ computes only
the components $
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$
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\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
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$
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\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
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\end_inset
\end_layout
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\begin_layout Standard
\family typewriter
\begin_inset ERT
status collapsed
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}
\end_layout
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}
\end_layout
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\end_body
\end_document