6.3. General Bianchi identities
Let us now check the Bianchi identities for general covariant derivatives with torsion. Note that these identities are not directly encoded in xTensor`, but can be easily computed.
Define a covariant derivative with torsion. Note the special symmetry properties of the defined tensors:
In[462]:=
DefCovD[CD[-a], {";", "▽"}, Torsion→True]
** DefCovD: Defining covariant derivative CD[-a] .
** DefTensor: Defining torsion tensor TorsionCD[a, -b, -c] .
** DefTensor: Defining non-symmetric Christoffel tensor ChristoffelCD[a, -b, -c] .
** DefTensor: Defining Riemann tensor RiemannCD[-a, -b, -c, d] . Antisymmetric only in the first pair.
** DefTensor: Defining non-symmetric Ricci tensor RicciCD[-a, -b] .
** DefCovD: Contractions of Riemann automatically replaced by Ricci.
This is the general form of the first Bianchi identity:
In[463]:=
RiemannCD[-a, -b, -c, d] + CD[-a][ TorsionCD[d, -b, -c] ] - TorsionCD[e, -a, -b] TorsionCD[d, -c, -e]
Out[463]=
R[▽] _abc ^( d) - T[▽] _ ( ce)^d T[▽] _ ( ab)^e + ▽_a^ T[▽] _ ( bc)^d
In[464]:=
6Antisymmetrize[%, {-a, -b, -c}]
Out[464]=
The computation can be performed by transforming the derivative CD and its associated tensors into PD and Christoffel tensors:
In[465]:=
%//ChangeCovD
Out[465]=
In[466]:=
%//RiemannToChristoffel
Out[466]=
In[467]:=
%//TorsionToChristoffel
Out[467]=
In[468]:=
%//Expand
Out[468]=
This is the general form of the second Bianchi identity:
In[469]:=
CD[-a][ RiemannCD[-b, -c, -d, e] ] - TorsionCD[f, -a, -b] RiemannCD[-c, -f, -d, e]
Out[469]=
-R[▽] _cfd ^( e) T[▽] _ ( ab)^f + ▽_a^ R[▽] _bcd ^( e)
In[470]:=
6Antisymmetrize[%, {-a, -b, -c}]
Out[470]=
The computation proceeds along the same lines:
In[471]:=
%//ChangeCovD
Out[471]=
In[472]:=
%//RiemannToChristoffel
Out[472]=
In[473]:=
%//TorsionToChristoffel
Out[473]=
In[474]:=
%//Expand
Out[474]=
In[475]:=
%//ToCanonical
Out[475]=
Clean up:
In[476]:=
UndefCovD[CD]
** UndefTensor: Undefined non-symmetric Christoffel tensor ChristoffelCD
** UndefTensor: Undefined non-symmetric Ricci tensor RicciCD
** UndefTensor: Undefined Riemann tensor RiemannCD
** UndefTensor: Undefined torsion tensor TorsionCD
** UndefCovD: Undefined covariant derivative CD