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20.2.2.6 Torsion und Nichtmetrizität

Maxima can now take into account torsion and nonmetricity. When the flag itorsion_flag is set to true, the contribution of torsion is added to the connection coefficients. Similarly, when the flag inonmet_flag is true, nonmetricity components are included.

Variable: inm

The nonmetricity vector. Conformal nonmetricity is defined through the covariant derivative of the metric tensor. Normally zero, the metric tensor’s covariant derivative will evaluate to the following when inonmet_flag is set to true:

g     =- g  inm
 ij;k     ij   k
Variable: inmc1

Covariant permutation of the nonmetricity vector components. Defined as

           g   inm  - inm  g   - g   inm
            ab    c      a  bc    ac    b
inmc1    = ------------------------------
     abc                 2

(Substitute ifg in place of g if a frame metric is used.)

Variable: inmc2

Contravariant permutation of the nonmetricity vector components. Used in the connection coefficients if inonmet_flag is true. Defined as:

                      c         c         cd
          -inm  kdelta  - kdelta  inm  + g   inm  g
     c        a       b         a    b          d  ab
inmc2   = -------------------------------------------
     ab                        2

(Substitute ifg in place of g if a frame metric is used.)

Variable: ikt1

Covariant permutation of the torsion tensor (also known as contorsion). Defined as:

                  d           d       d
          -g   itr  - g    itr   - itr   g
            ad    cb    bd    ca      ab  cd
ikt1    = ----------------------------------
    abc                   2

(Substitute ifg in place of g if a frame metric is used.)

Variable: ikt2

Contravariant permutation of the torsion tensor (also known as contorsion). Defined as:

    c     cd
ikt2   = g   ikt1
    ab           abd

(Substitute ifg in place of g if a frame metric is used.)

Variable: itr

The torsion tensor. For a metric with torsion, repeated covariant differentiation on a scalar function will not commute, as demonstrated by the following example:

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) imetric:g;
(%o2)                                  g
(%i3) covdiff( covdiff( f( [], []), i), j)
                      - covdiff( covdiff( f( [], []), j), i)$
(%i4) ishow(%)$
                                   %4              %2
(%t4)                    f    ichr2    - f    ichr2
                          ,%4      j i    ,%2      i j
(%i5) canform(%);
(%o5)                                  0
(%i6) itorsion_flag:true;
(%o6)                                true
(%i7) covdiff( covdiff( f( [], []), i), j)
                      - covdiff( covdiff( f( [], []), j), i)$
(%i8) ishow(%)$
                           %8             %6
(%t8)             f    icc2    - f    icc2    - f     + f
                   ,%8     j i    ,%6     i j    ,j i    ,i j
(%i9) ishow(canform(%))$
                                   %1             %1
(%t9)                     f    icc2    - f    icc2
                           ,%1     j i    ,%1     i j
(%i10) ishow(canform(ev(%,icc2)))$
                                   %1             %1
(%t10)                    f    ikt2    - f    ikt2
                           ,%1     i j    ,%1     j i
(%i11) ishow(canform(ev(%,ikt2)))$
                      %2 %1                    %2 %1
(%t11)          f    g      ikt1       - f    g      ikt1
                 ,%2            i j %1    ,%2            j i %1
(%i12) ishow(factor(canform(rename(expand(ev(%,ikt1))))))$
                           %3 %2            %1       %1
                     f    g      g      (itr    - itr   )
                      ,%3         %2 %1     j i      i j
(%t12)               ------------------------------------
                                      2
(%i13) decsym(itr,2,1,[anti(all)],[]);
(%o13)                               done
(%i14) defcon(g,g,kdelta);
(%o14)                               done
(%i15) subst(g,nounify(g),%th(3))$
(%i16) ishow(canform(contract(%)))$
                                           %1
(%t16)                           - f    itr
                                    ,%1    i j

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