Nächste: Graßmann-Algebra, Vorige: Begleitende Vielbeine, Nach oben: Funktionen und Variablen für ITENSOR [Inhalt][Index]
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.
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
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.)
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.)
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.)
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.)
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
Nächste: Graßmann-Algebra, Vorige: Begleitende Vielbeine, Nach oben: Funktionen und Variablen für ITENSOR [Inhalt][Index]