Tue Feb 10 12:26:32 2004 run on Linux
% Test of CANTENS.RED
%
%     Authors: H. Caprasse <hubert.caprasse@ulg.ac.be>
%
% Version and Date:  Version 1.1, 15 September 1998.
%----------------------------------------------------------------
off errcont;


% Default : 
onespace ?;


yes

wholespace_dim ?;


dim
  
global_sign ? ;


1
 
signature ?;


0
 
% answers to the 4 previous commands: yes, dim, 1, 0
wholespace_dim 4;


4
 
signature 1;


1

global_sign(-1);


-1

% answers to the three previous commands: 4, 1, (-1) 
% answer to the command below: {}
show_spaces();


{}

% Several spaces:
off onespace;


onespace ?;


no
 
% answer: no
show_spaces();


{}

define_spaces wholespace={6,signature=1,indexrange=0 .. 5};


t

% indexrange command is superfluous since 'wholespace':
show_spaces();


{{wholespace,6,signature=1,indexrange=0 .. 5}}

rem_spaces wholespace;


t

define_spaces wholespace={11,signature=1};


t
 
define_spaces mink={4,signature=1,indexrange=0 .. 3};


t
 
define_spaces eucl={6,euclidian,indexrange=4 .. 9};


t
 
show_spaces();


{{wholespace,11,signature=1},

 {mink,4,signature=1,indexrange=0 .. 3},

 {eucl,6,euclidian,indexrange=4 .. 9}}

%
% if input error or modifications necessary:
%
define_spaces eucl={7,euclidian,indexrange=4 .. 10};

*** Warning: eucl  cannot be (or is already) defined as space identifier 
t

%
% do:
%
rem_spaces eucl;


t

define_spaces eucl={7,euclidian,indexrange=4 .. 10};


t

show_spaces();


{{wholespace,11,signature=1},

 {mink,4,signature=1,indexrange=0 .. 3},

 {eucl,7,euclidian,indexrange=4 .. 10}}

% done
%
define_spaces eucl1={1,euclidian,indexrange=11 .. 11};


t
 
show_spaces();


{{wholespace,11,signature=1},

 {mink,4,signature=1,indexrange=0 .. 3},

 {eucl,7,euclidian,indexrange=4 .. 10},

 {eucl1,1,euclidian,indexrange=11 .. 11}}

rem_spaces wholespace,mink,eucl,eucl1;


t

show_spaces();


{}

%
% Indices can be made to belong to a subspace or replaced 
% in the whole space:
define_spaces eucl={3,euclidean};


t

show_spaces();


{{eucl,3,euclidean}}

mk_ids_belong_space({a1,a2},eucl);


t

% a1,a2 belong to the subspace eucl.
mk_ids_belong_anyspace a1,a2;


t

% replaced in the whole space.
rem_spaces eucl;


t

%%  
%% GENERIC TENSORS:
on onespace;


wholespace_dim dim;


dim

tensor te;


t
 
te(3,a,-4,b,-c,7);


  3 a   b   7
te
      4   c 
 
te(3,a,{x,y},-4,b,-c,7);


   3 a   b   7
te            (x,y)
       4   c 

te(3,a,-4,b,{u,v},-c,7);


   3 a   b   7
te            (u,v)
       4   c 

te({x,y});


te(x,y)

 
make_variables x,y;


t

te(x,y);


te(x,y)


te(x,y,a);


   a
te  (x,y)
   
 
remove_variables x;


t

te(x,y,a);


   x a
te    (y)
   

remove_variables y;


t

%
% implicit dependence:
%
operator op2;


depend op1,op2(x);


df(op1,op2(x));


df(op1,op2(x))

% the next response is 0:
df(op1,op2(y));


0

clear op2;


% case of a tensor: 
operator op1;


depend te,op1(x);

 
df(te(a,-b),op1(x));


     a 
df(te   ,op1(x))
       b

% next the outcome is 0:
df(te(a,-b),op1(y));


0

%
tensor x;


t

depend te,x;


% outcome is NOT 0:
df(te(a,-b),x(c));


     a    c
df(te   ,x )
       b

%
% Substitutions:
sub(a=-c,te(a,b));


    b
te
  c 

sub(a=-1,te(a,b));


    b
te
  1 
 
% the following operation is wrong:
sub(a=-0,te(a,b));


  0 b
te
  
% should be made as following to be correct:
sub(a=-!0,te(a,b));


    b
te
  0 
 
% dummy indices recognition
dummy_indices();


{}

te(a,b,-c,-a);


  a b 
te
      c a

dummy_indices();


{a}

te(a,b,-c,-a);


  a b 
te
      c a

dummy_indices();


{a}

% hereunder an error message correctly occurs:
on errcont;


te(a,b,-c,a);


***** ((c) (a b a)) are inconsistent lists of indices 

off errcont;


sub(c=b,te(a,b,-c,-a));


  a b 
te
      b a

dummy_indices();


{b,a}

% dummy indices suppression:
on errcont;


te(d,-d,d);


***** ((d) (d d)) are inconsistent lists of indices 

off errcont;


dummy_indices();


{d,b,a}

rem_dummy_indices d;


t

te(d,d);


  d d
te

dummy_indices();


{b,a}

rem_dummy_indices a,b;


t

onespace ?;


yes

% case of space of integer dimension:
wholespace_dim 4;


4

signature 0;


0

% 7 out of range
on errcont;


te(3,a,-b,7);


***** numeric indices out of range 

off errcont;


te(3,a,-b,3);


  3 a   3
te
      b 

te(4,a,-b,4);


  4 a   4
te
      b 

% an 'out-of-range' error is issued:
on errcont;


sub(a=5,te(3,a,-b,3));


***** numeric indices out of range 

off errcont;


signature 1;


1

% now indices should run from 0 to 3 => error: 
on errcont;


te(4,a,-b,4);


***** numeric indices out of range 

off errcont;


% correct:
te(0,a,-b,3);


  0 a   3
te
      b 

%
off onespace;


define_spaces wholespace={4,euclidean};


t

% We MUST say that te BELONG TO A SPACE, here to wholespace:
make_tensor_belong_space(te,wholespace);


wholespace

on errcont;


te(a,5,-b);


***** numeric indices out of range 

off errcont;


te(a,4,-b);


  a 4 
te
      b

rem_spaces wholespace;


t

define_spaces wholespace={5,signature=1};


t

define_spaces eucl={1,signature=0};


t

show_spaces();


{{wholespace,5,signature=1},

 {eucl,1,signature=0}}

make_tensor_belong_space(te,eucl);


eucl

te(1);


  1
te

% hereunder, an error message is issued: 
on errcont;


te(2);


***** numeric indices out of range 

off errcont;


% hereunder, an error message should be issued, it is not 
% because no indexrange has been declared:
te(0);


  0
te

rem_spaces eucl;


t

define_spaces eucl={1,signature=0,indexrange=1 .. 1};


t

% NOW an error message is issued:
on errcont;


te(0);


***** numeric indices do not belong to (sub)-space 

off errcont;


te(1);


  1
te

% again an error message:
on errcont;


te(2);


***** numeric indices do not belong to (sub)-space 

off errcont;


%
rem_dummy_indices a,b,c,d;


t

% symmetry properties:
%
symmetric te;


te(a,-b,c,d);


  a c d 
te
        b

remsym te;


antisymmetric te;


te(a,b,-c,d);


     a b d 
 - te
           c

remsym te;


% mixed symmetries:
tensor r;


t

% 
symtree(r,{!+,{!-,1,2},{!-,3,4}});


ra:=r(b,a,c,d)$

 
canonical ra;


    a b c d
 - r
 
ra:=r(c,d,a,b)$


canonical ra;


 a b c d
r

% here canonical is short-cutted
ra:=r(b,b,c,a);


ra := 0

%
% symmetrization:
on onespace;


symmetrize(r(a,b,c,d),r,permutations,perm_sign);


 a b c d    a b d c    a c b d    a c d b    a d b c    a d c b    b a c d
r        - r        - r        + r        + r        - r        - r

    b a d c    b c a d    b c d a    b d a c    b d c a    c a b d    c a d b
 + r        + r        - r        - r        + r        + r        - r

    c b a d    c b d a    c d a b    c d b a    d a b c    d a c b    d b a c
 - r        + r        + r        - r        - r        + r        + r

    d b c a    d c a b    d c b a
 - r        - r        + r

canonical ws;


    a b c d    a c b d    a d b c
8*(r        - r        + r       )

off onespace;


symmetrize({a,b,c,d},r,cyclicpermlist);


 a b c d    b c d a    c d a b    d a b c
r        + r        + r        + r

canonical ws;


    a b c d    a d b c
2*(r        - r       )

rem_tensor r;


t

% Declared bloc-diagonal tensor:
rem_spaces wholespace,eucl;


t

define_spaces wholespace={7,signature=1};


t

define_spaces mink={4,signature=1,indexrange=0 .. 3};


t

define_spaces eucl={3,euclidian,indexrange=4 .. 6};


t

show_spaces();


{{wholespace,7,signature=1},

 {mink,4,signature=1,indexrange=0 .. 3},

 {eucl,3,euclidian,indexrange=4 .. 6}}

make_tensor_belong_space(te,eucl);


eucl

make_bloc_diagonal te;


t

mk_ids_belong_space({a,b,c},eucl);


t

te(a,b,z);


  a b z
te

mk_ids_belong_space({m1,m2},mink);


t

te(a,b,m1);


0

te(a,b,m2);


0

mk_ids_belong_anyspace a,b,c,m1,m2;


t

te(a,b,m2);


  a b m2
te

% how to ASSIGN a particular component ?
% take the simplest context: 
rem_spaces wholespace,mink,eucl;


t

on onespace;


te({x,y},a,-0)==x*y*te(a,-0);


  a 
te   *x*y
    0
 
te({x,y},a,-0);


  a 
te   *x*y
    0
 
te({x,y},a,0);


   a 0
te    (x,y)
   
 
% hereunder an error message  is issued because already assigned:
on errcont;


te({x,y},a,-0)==x*y*te(a,-0);


        a 
***** te   *x*y invalid as setvalue kernel
          0

off errcont;


% clear value:
rem_value_tens te({x,y},a,-0);


t

te({x,y},a,-0);


   a 
te    (x,y)
     0
 
te({x,y},a,-0)==(x+y)*te(a,-0);


  a 
te   *(x + y)
    0

% A small illustration
te(1)==sin th * cos phi;


cos(phi)*sin(th)
 
te(-1)==sin th * cos phi;


cos(phi)*sin(th)

te(2)==sin th * sin phi;


sin(phi)*sin(th)

te(-2)==sin th * sin phi;


sin(phi)*sin(th)

te(3)==cos th ;


cos(th)

te(-3)==cos th ;


cos(th)

for i:=1:3 sum te(i)*te(-i);


        2        2          2           2        2
cos(phi) *sin(th)  + cos(th)  + sin(phi) *sin(th)

rem_value_tens te;


t

te(2);


  2
te

let te({x,y},-0)=x*y;


te({x,y},-0);


x*y

te({x,y},0);


   0
te  (x,y)
   

te({x,u},-0);


te  (x,u)
   0

for all x,a let te({x},a,-b)=x*te(a,-b);


te({u},1,-b);


  1 
te   *u
    b

te({u},c,-b);


  c 
te   *u
    b

te({u},b,-b);


  b 
te   *u
    b

te({u},a,-a);


   a 
te    (u)
     a

for all x,a clear te({x},a,-b);


te({u},c,-b);


   c 
te    (u)
     b

% rule for indices only
for all a,b let te({x},a,-b)=x*te(a,-b);


te({x},c,-b);


  c 
te   *x
    b

te({x},a,-a);


  a 
te   *x
    a

% A BUG still exists for -0 i.e. rule does NOT apply:
te({x},a,-0);


   a 
te    (x)
     0

% the cure is to use -!0 in this case
te({x},0,-!0);


  0 
te   *x
    0

%
% local rules: 
%
rul:={te(~a) => sin a};


          ~a
rul := {te   => sin(a)}

te(1) where rul;


sin(1)

% 
rul1:={te(~a,{~x,~y}) => x*y*sin(a)};


           ~a 
rul1 := {te   (~x,~y) => x*y*sin(a)}

 
%
te(a,{x,y}) where rul1;


sin(a)*x*y

te({x,y},a) where rul1;


sin(a)*x*y

%
rul2:={te(-~a,{~x,~y}) => x*y*sin(-a)};


rul2 := {te   (~x,~y) => x*y*sin( - a)}
           ~a 

% 
te(-a,{x,y}) where rul2;


 - sin(a)*x*y
 
te({x,y},-a) where rul2;


 - sin(a)*x*y

%% CANONICAL 
%
% 1. Coherence of tensorial indices.
%
tensor te,tf;

*** Warning:  te redefined as generic tensor  
t
 
dummy_indices();


{a,b}

make_tensor_belong_anyspace te;


t

on errcont;


bb:=te(a,b)*te(-b)*te(b);


            a b   b
bb := te *te   *te
        b

% hereunder an error message is issued:
canonical bb;


***** ((b) (a b b)) are inconsistent lists of indices 

off errcont;


bb:=te(a,b)*te(-b);


            a b
bb := te *te
        b

% notice how it is rewritten by canonical:
canonical bb;


  a     b
te   *te
    b

% 
dummy_indices();


{a,b}

aa:=te(d,-c)*tf(d,-c);


        d     d 
aa := te   *tf
          c     c

% if a and c are FREE no error message:
canonical aa;


  d     d 
te   *tf
    c     c

% do NOT introduce powers for NON-INVARIANT tensors:
aa:=te(d,-c)*te(d,-c);


         d   2
aa := (te   )
           c

% Powers are taken away
canonical aa;


  d 
te
    c

% A trace CANNOT be squared because powers are removed by 'canonical':
cc:=te(a,-a)^2$


canonical cc;


  a 
te
    a

%
% Correct writing of the previous squared:
cc:=te(a,-a)*te(b,-b)$


canonical cc;


  a     b 
te   *te
    a     b

% all terms must have the same variance:
on errcont;


aa:=te(a,c)+x^2;


        a c    2
aa := te    + x

canonical aa;


***** scalar added with tensor(s) 

aa:=te(a,b)+tf(a,c);


        a b     a c
aa := te    + tf

canonical aa;


***** mismatch in free indices :  ((a c) (a b)) 

off errcont;


dummy_indices();


{a,b}

rem_dummy_indices a,b,c;


t

dummy_indices();


{}

% a dummy VARIABLE is NOT a dummy INDEX
dummy_names b;


t

dummy_indices();


{}

% so, no error message in the following: 
canonical(te(b,c)*tf(b,c));


  b c   b c
te   *tf

% it is an incorrect input for a variable.
% correct input is:
canonical(te({b},c)*tf({b},c));


   c       c
te  (b)*tf  (b)
           

clear_dummy_names;


t

% contravariant indices are placed before covariant ones if possible.
% i.e. Riemanian spaces by default: 
pp:=te(a,-a)+te(-a,a)+1;


          a     a 
pp := te    + te    + 1
        a         a
 
canonical pp;


    a 
2*te    + 1
      a

pp:=te(a,-c)+te(-b,b,a,-c);


          b a       a 
pp := te        + te
        b     c       c

canonical pp;


  a       b   a 
te    + te
    c       b   c

pp:=te(r,a,-f,d,-a,f)+te(r,-b,-c,d,b,c);


        r     d b c     r a   d   f
pp := te            + te
          b c               f   a 

canonical pp;


    r a b d 
2*te
            a b

% here, a case where a normal form cannot be obtained:
tensor nt;


t

a1:=nt(-a,d)*nt(-c,a);


          d     a
a1 := nt   *nt
        a     c 

a2:=nt(-c,-a)*nt(a,d);


              a d
a2 := nt   *nt
        c a
 
% obviously, a1-a2 =0, but ....
canonical(a1-a2);


       d     a           a d
 - nt   *nt    + nt   *nt
     a     c       c a

% does give the same expression with the sign changed.
% zero is either:
canonical a1 -a2;


0

% or
a1 -canonical a2;


0

% below the result is a2:
canonical a1;


        a d
nt   *nt
  c a

% below result is a1 again: 
canonical ws;


    d     a
nt   *nt
  a     c 

%  the above manipulations are NOT DONE if space is AFFINE
off onespace;


define_spaces aff={dd,affine};


t

make_tensor_belong_space(te,aff);


aff
 
% dummy indices MUST be declared to belong 
% to a well defined space. here to 'aff':
mk_ids_belong_space({a,b},aff);


t

canonical(te(-a,a));


    a
te
  a 

canonical(te(-a,a)+te(b,-b));


    a     a 
te    + te
  a         a

canonical(te(-a,c));


    c
te
  a 

% put back  the system in the previous status: 
make_tensor_belong_anyspace te;


t

mk_ids_belong_anyspace a,b;


t

rem_spaces aff;


t

on onespace;


%
% 2. Summations with DELTA tensor.
%
make_partic_tens(delta,delta);


t

aa:=delta(a,-b)*delta(b,-c)*delta(c,-a) + 1;


           a       b       c 
aa := delta  *delta  *delta   + 1
           b       c       a

% below, answer is dim+1:
canonical aa;


dim + 1

aa:=delta(a,-b)*delta(b,-c)*delta(c,-d)*te(d,e)$


canonical aa;


  a e
te

% 3. Summations with DELTA and ETA tensors.
make_partic_tens(eta,eta);


t

signature 1;


1

aa:=eta(a,b)*eta(-b,-c);


                a b
aa := eta   *eta
         b c

canonical aa;


     a 
delta
     c

aa:=eta(a,b)*eta(-b,-c)*eta(c,d);


                a b    c d
aa := eta   *eta   *eta
         b c

canonical aa;


   a d
eta

aa:=eta(a,b)*eta(-b,-c)*eta(d,c)*te(d,-a) +te(d,d);


                a b    c d   d       d d
aa := eta   *eta   *eta   *te    + te
         b c                   a

canonical aa;


    d d
2*te

aa:=delta(a,-b)*eta(b,c);


           a     b c
aa := delta  *eta
           b

canonical aa;


   a c
eta

aa:=delta(a,-b)*delta(d,-a)*eta(-c,-d)*eta(b,c);


           a       d            b c
aa := delta  *delta  *eta   *eta
           b       a     c d

% below the answer is dim:
canonical aa;


dim

aa:=delta(a,-b)*delta(d,-a)*eta(-d,-e)*te(f,g,e);


           a       d           f g e
aa := delta  *delta  *eta   *te
           b       a     d e

canonical aa;


  f g 
te
      b

% Summations with the addition of the METRIC tensor:
make_partic_tens(g,metric);


t

g(1,2,{x})==1/4*sin x;


 sin(x)
--------
   4

g({x},1,2);


 sin(x)
--------
   4

aa:=g(a,b)*g(-a,-c);


            a b
aa := g   *g
       a c
 
canonical aa;


     b 
delta
     c

aa:=g(a,b)*g(c,d)*eta(-c,-b);


              a b  c d
aa := eta   *g   *g
         b c

% answer is g(a,d):
canonical aa;


 a d
g

tensor te;

*** Warning:  te redefined as generic tensor  
t

aa:=g(a,b)*g(c,d)*eta(-c,-e)*eta(e,f)*te(-f,g);


                e f  a b  c d     g
aa := eta   *eta   *g   *g   *te
         c e                    f 

canonical aa;


 a b   d g
g   *te

% Summations with the addition of the EPSILON tensor.
dummy_indices();


{c,f,b,a}

rem_dummy_indices a,b,c,f;


t

dummy_indices();


{}

wholespace_dim ?;


dim

signature ?;


1

% define the generalized delta function:
make_partic_tens(gd,del);


t

make_partic_tens(epsilon,epsilon);


t

aa:=epsilon(a,b)*epsilon(-c,-d);


                        a b
aa := epsilon   *epsilon
             c d

% Minus sign reflects the chosen signature.
canonical aa;


     a b
 - gd
     c d 

aa:=epsilon(a,b)*epsilon(-a,-b);


                        a b
aa := epsilon   *epsilon
             a b

canonical aa;


dim*( - dim + 1)

aa:=epsilon(a,b,c,d)*epsilon(-a,-b,-c,-e);


                            a b c d
aa := epsilon       *epsilon
             a b c e

canonical aa;


     d         3        2
delta  *( - dim  + 6*dim  - 11*dim + 6)
     e

on exdelt;


% extract delta function down to the bottom:
aa:=epsilon(a,b,c)*epsilon(-b,-d,-e);


                          a b c
aa := epsilon     *epsilon
             b d e

canonical aa;


     a       c               a       c         a       c 
delta  *delta  *dim - 2*delta  *delta   - delta  *delta  *dim
     d       e               d       e         e       d

          a       c 
 + 2*delta  *delta
          e       d

off exdelt;


% below expressed in terms of 'gd' tensor.
canonical aa;


  a c
gd    *(dim - 2)
  d e 

rem_dummy_indices a;


t

aa:=epsilon(- b,-c)*eta(a,b)*eta(a,c);


                    a b    a c
aa := epsilon   *eta   *eta
             b c

% answer below is zero:
canonical aa;


0

aa:=epsilon(a,b,c)*te(-a)*te(-b);


             a b c
aa := epsilon     *te *te
                     a   b

% below the result is again zero.
canonical aa;


0

%
tensor tf,tg;

*** Warning:  tf redefined as generic tensor  
t

aa:=epsilon(a,b,c)*te(-a)*tf(-b)*tg(-c)+epsilon(d,e,f)*te(-d)*tf(-e)*tg(-f);


             a b c                      d e f
aa := epsilon     *te *tf *tg  + epsilon     *te *tf *tg
                     a   b   c                  d   e   f

% below the result is twice the first term.
canonical aa;


         a b c
2*epsilon     *te *tf *tg
                 a   b   c

aa:=epsilon(a,b,c)*te(-a)*tf(-c)*tg(-b)+epsilon(d,e,f)*te(-d)*tf(-e)*tg(-f);


             a b c                      d e f
aa := epsilon     *te *tf *tg  + epsilon     *te *tf *tg
                     a   c   b                  d   e   f

% below the result is zero.
canonical aa;


0

% An illustration when working inside several spaces. 
rem_dummy_indices a,b,c,d,e,f;


t

off onespace;


define_spaces wholespace={dim,signature=1};


t

define_spaces sub4={4,signature=1};


t

define_spaces subd={dim-4,signature=0};


t

show_spaces();


{{wholespace,dim,signature=1},

 {sub4,4,signature=1},

 {subd,dim - 4,signature=0}}

make_partic_tens(epsilon,epsilon);

*** Warning: epsilon redefined as particular tensor 
t

make_tensor_belong_space(epsilon,sub4);


sub4

make_partic_tens(kappa,epsilon);

*** Warning: kappa MUST belong to a space 
t

make_tensor_belong_space(kappa,subd);


subd

show_epsilons();


{{kappa,subd},{epsilon,sub4}}

mk_ids_belong_space({i,j,k,l,m,n,r,s},sub4);


t

mk_ids_belong_space({a,b,c,d,e,f},subd);


t

off exdelt;


aa:=kappa(a,b,c)*kappa(-d,-e,-f)*epsilon(i,j,k,l)*epsilon(-k,-l,-i,-j);


                            i j k l                 a b c
aa := epsilon       *epsilon       *kappa     *kappa
             i j k l                     d e f

canonical aa;


        a b c
 - 24*gd
        d e f 

aa:=kappa(a,b,c)*kappa(-d,-e,-f)*epsilon(i,j,k,l)*epsilon(-m,-n,-r,-s);


                            i j k l                 a b c
aa := epsilon       *epsilon       *kappa     *kappa
             m n r s                     d e f

canonical aa;


     a b c    i j k l
 - gd      *gd
     d e f    m n r s 

end;


Time for test: 40 ms
