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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 Cartan- 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 Maurer-Cartan connection (4.22), right.

  The latter 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). The Cartan exterior coderivative ${\text{D:}} = {{\mathbf{g}}^\mu } \wedge {D_\mu }$ is the covariant curl, see (3.10), associated with the Maurer-Cartan connection ${{\bs{\omega }}^a}_b$. This derivative raises the degree of any $p$-form by one and transforms covariantly under local Lorentz transformations on account of (4.10).

   In Einstein's GRT torsion vanishes. In that context the first Cartan equation reduces to the constraining equation

4.27

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