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Tetrads in General Relativity

IV. Spin Connection

First Cartan Equation

   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 (4.15) allows the replacement of the Levi-Civita connection by the spin (Lorentz) connection:

4.24

\[{{\mathbf{T}}^a}: = {e_\kappa }^a{{\mathbf{T}}^\kappa } =  \left( {{\partial _\mu }{e_\nu }^a + {\Sigma _\mu }{{^a}_b}{e_\nu }^b} \right){{\mathbf{g}}^\mu } \wedge {{\mathbf{g}}^\nu }\]

The term with the spin connection at the right-hand side can be rewritten in terms of two different wedge 2-forms

4.25

\[-{\omega _{[bc]}}^a{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^c} =  {{\bs{\omega }}^a}_b \wedge {{\mathbf{e}}^b}\]

build out of anti-symmetrized Ricci coefficients (4.20), left, and the Cartan connection (4.22), right.

  The Cartan 2-form results in the first Cartan structure equation for the torsion 2-form

4.26

\[  {{\mathbf{T}}^a} = {\text{d}}{{\mathbf{e}}^a} + {{\bs{\omega }}^a}_b \wedge {{\mathbf{e}}^b} : = {\text{D}}{{\mathbf{e}}^a}  \]

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 ${{\bs{\omega }}^a}_b$. This derivative transforms covariantly under local Lorentz transformations on account of (4.10). 

   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

4.27

\[ \nabla \wedge {{\mathbf{e}}^a} + {{\bs{\omega }}^a}_b \wedge {{\mathbf{e}}^b} = 0\]