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Tetrads in General Relativity

IV. Spin Connection

First Cartan Equation

   The tetrad-based Cartan formalism is equivalent to conventional tensor calculus of Riemannian geometry with the advantage of hiding some of its profusion of indices. Consider, for example, the 2-form

4.23

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

build out of anti-symmetrized Ricci coefficients (4.20). This form has a remarkable relation with the torsion tensor 2-form (1.24). The tetrad postulate (4.14) allows the replacement of the Levi-Civita connection by the spin connection. The result is the first Cartan structure equation for the torsion 2-form

4.24

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

Here ${\text{d}}$ is the exterior derivative as defined in (1.7), and ${\text{D}}$ the Cartan exterior coderivative associated with the connection ${{\mathbf{\omega }}^a}_b$; see (4.21). 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

4.25

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

which is the Cartan equivalent of the Levi-Civita torsion constraint (1.26).

   A corollary of the equivalence ${\text{d}} = \nabla \wedge $ is that the multivector torsion-free curl is independent of the metric, since it is equivalent to the exterior derivative of differential forms (which is defined without reference to a metric). In particular one has

4.26

\[{{\text{d}}^2}{\mathbf{A}} = \nabla \wedge (\nabla \wedge {\mathbf{A}}) = 0\]

for any multivector field ${\mathbf{A}}$. This example shows that the theory of multivector fields includes the theory of differential forms as a part of the GA algebra.