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IV. Spin Connection
The tetrad-based Cartan formalism is largely equivalent to
the conventional tensor calculus of Riemannian geometry and easily
integrated in the GA formulation sharing the advantage of a compact
notation. The point of difference is that the Cartan theory is
formulated within the framework of a Riemann–Cartan geometry
in which the notion of torsion is contained in an essential
way. A Riemann-Cartan geometry with vanishing torsion is identical
to the Riemannian geometry of general relativity.
The close correspondence between the Catan- and GA-formalisms may be
illustrated by considering the torsion 2-form (1.24). The tetrad
postulate (4.14) allows the replacement of the Levi-Civita
connection by the spin connection.
\[{{\mathbf{T}}^a}: =
{e_\kappa }^a{{\mathbf{T}}^\kappa } = \left( {{\partial _\mu
}{e_\nu }^a + {\Sigma _\mu }{{^a}_b}{e_\nu }^b}
\right){{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }\]
The term with the spin connection at the right-hand side can be
rewritten in terms of two different wedge 2-forms
\[{\omega
_{[bc]}}^a{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} = -
{{\mathbf{\omega }}^a}_b \wedge {{\mathbf{e}}^b}\]
build out of anti-symmetrized Ricci coefficients (4.19), left, and
the Cartan connection (4.21), right.
The
Cartan 2-form results in the first Cartan structure equation
for the torsion 2-form
\[ {{\mathbf{T}}^a} =
{\text{d}}{{\mathbf{e}}^a} + {{\mathbf{\omega }}^a}_b \wedge
{{\mathbf{e}}^b} : = {\text{D}}{{\mathbf{e}}^a} \]
The operator ${\text{d}}$ is the exterior derivative as defined in
(1.7). ${\text{D}}$ is the Cartan exterior coderivative
associated with the Cartan connection ${{\mathbf{\omega }}^a}_b$.
This derivative transforms covariantly under local Lorentz
transformations on account of (4.9).
In Einsteins GRT torsion vanishes. In that case the exterior
derivative ${\text{d}}$ is completely equivalent to the torsion-free
curl: ${\text{d}} = \nabla \wedge $. The first Cartan equation then
reduces to the equality
\[ \nabla \wedge
{{\mathbf{e}}^a} + {{\mathbf{\omega }}^a}_b \wedge {{\mathbf{e}}^b}
= 0\]
which is the Cartan equivalent of the Levi-Civita torsion
constraint (1.26).