Cartan structural equations and Bianchi identity
Theoretical background
Let M be a manifold, and g a (semi)Riemannian metric defined on M. Then the line element for the metric g is ds2(g)=gμνdxμ⊗dxν. Nonetheless, the information about the metric structure of the manifold can be translated to the language of frames, ds2(g)=gμνdxμ⊗dxν=ηabeaμ(x)ebν(x)dxμ⊗dxν=ηabea⊗eb. Therefore, the vielbein 1-form, ea≡eaμ(x)dxμ, encodes the information of the metric tensor. In order to complete the structure, one needs information about the transport of geometrical objects lying on bundles based on M. That information is encoded on the spin connection 1-form, ωab. Using these quantities one finds the generalisation of the structure equations of Cartan, dea+ωab∧eb=Ta,dωac+ωab∧ωbc=Rac. The torsion 2-form, Ta, and the curvature 2-form, Rac, measure the impossibility of endowing M with an Euclidean structure.Manipulation of the structural equations
Definitions
{a,b,c,l,m,n}::Indices.
d{#}::ExteriorDerivative;.
d{#}::LaTeXForm("\mathrm{d}").
T{#}::LaTeXForm("\mathrm{T}").
R{#}::LaTeXForm("\mathrm{R}").
{e^{a}, \omega^{a}_{b}}::DifferentialForm(degree=1);
{T^{a}, R^{a}_{b}}::DifferentialForm(degree=2);
Attached property DifferentialForm to [ea,ωab].
Attached property DifferentialForm to [Ta,Rab].
Cartan structural equations
struc1 := d{e^{a}} + \omega^{a}_{b} ^ e^{b} - T^{a} = 0;
dea+ωab∧eb−Ta=0
struc2 := d{\omega^{a}_{b}} + \omega^{a}_{m} ^ \omega^{m}_{b} - R^{a}_{b} = 0;
dωab+ωam∧ωmb−Rab=0
In the following, we will also use the structural equations as definitions of the
exterior derivatives of the vielbein and spin connection 1-forms. Therefore, we shall
utilise the
isolate
algorithm---from the cdb.core.manip
library---to
define substitution rules.from cdb.core.manip import *
de:= @(struc1):
isolate(de, $d{e^{a}}$);
dea=−ωab∧eb+Ta
domega := @(struc2):
isolate(domega, $d{\omega^{a}_{b}}$);
dωab=−ωam∧ωmb+Rab
Bianchi identities
The bianchi identities are obtained by applying the exterior derivative to the structural equations.First Bianchi identity
Bianchi1 := d{ @(struc1) };
d(dea+ωab∧eb−Ta)=0
distribute(Bianchi1)
product_rule(_);
dωab∧eb−ωab∧deb−dTa=0
substitute(Bianchi1, de, repeat=True)
substitute(Bianchi1, domega, repeat=True)
distribute(_);
−ωam∧ωmb∧eb+Rab∧eb+ωab∧ωbc∧ec−ωab∧Tb−dTa=0
rename_dummies(Bianchi1);
Rab∧eb−ωab∧Tb−dTa=0
In the absence of torsion, the above expression is the well-known algebraic Bianchi identity
Rμνλρ+Rμλρν+Rμρνλ=0.
Second Bianchi identity
Bianchi2 := d{ @(struc2) };
d(dωab+ωam∧ωmb−Rab)=0
distribute(Bianchi2)
product_rule(_);
dωam∧ωmb−ωam∧dωmb−dRab=0
substitute(Bianchi2, domega, repeat=True)
distribute(_)
rename_dummies(_);
Rac∧ωcb−ωac∧Rcb−dRab=0
The above result, when written in tensorial components, is the well-known
differential Bianchi identity:
Rμνλρ;σ+Rμνσλ;ρ+Rμνρσ;λ=0.