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Tetrads in General Relativity

XII. Einstein-Cartan Theory

ECKS Theory

   By the addition of a matter action ${S_{{\text{EC}}}}[{\mathbf{e}},{\boldsymbol{\omega }}] \to {S_{{\text{EC}}}}[{\mathbf{e}},{\boldsymbol{\omega }}] - {S_\text{M}}[{\mathbf{e}},{\boldsymbol{\omega }},\psi ]$ to the EC-action (12.5), both EC field equations (12.7) and (12.11) acquire a source term depending on $\{ {{\mathbf{e}}^a},{{\boldsymbol{\omega }}^{ab}}\} $ as well as on one or more matter fields $\psi$:

12.13

\[{{\mathbf{G}}_a} =  \kappa \frac{{\delta {S_{\text{M}}} [{\mathbf{e}},{\boldsymbol{\omega }},\psi ]}}{{\delta {{\mathbf{e}}^a}}} := \kappa {{\boldsymbol{\tau }}_a}\]

12.14

\[{{\mathbf{S}}_{ab}} = 2\kappa \frac{{\delta {S_{\text{M}}} [{\mathbf{e}},{\boldsymbol{\omega }},\psi ]}}{{\delta {{\boldsymbol{\omega }}^{ab}}}} : = \kappa {{\boldsymbol{\sigma }}_{ab}}\]

These equations, first proposed by Tom Kibble (1961) and Dennis Sciama (1962), are the field equations of the ECKS-theory of gravity. This is the special case of a gauge theory which has the curvature of the Riemann-Cartan spacetime as gravitational action. Sciama and Kibble localized the Poincaré group ${\text{P}}(1,3)$ of spacetime symmetries and in this way established that gravity can consistently described as a gauge theory.

   In Equation of Motion it has been demonstratedthat the 3-form ${{\mathbf{G}}_a}$ is equivalent to the Einstein tensor including cosmological constant. With the help of (12.8a), equation (12.13) may then be cast in the form of the (generalized) Einstein equation

12.15

\[{{\mathbf{G}}_a} = \left( {{G_a}^b - \Lambda \delta _a^b} \right){{\boldsymbol{\eta }}_b}{\text{ = }} \kappa {\tau _a}^b{{\boldsymbol{\eta }}_b}\]

This equation shows that the 3-form ${{\boldsymbol{\tau }}_a}$ is the source of curvature and may be identified as the (canonical) energy-momentum density of matter. The energy-momentum tensor defined through ${{\boldsymbol{\tau }}_a}: = {\tau _a}^b{{\boldsymbol{\eta }}_b}$, is asymmetric like the Ricci tensor.

   By a similar reasoning, and the use of (12.12), the torsion equation (12.14) can be rewritten into

12.16

\[{{\mathbf{S}}_{ab}} = \left( {T_{ab}^c + 2\delta _{[a}^cT_{b]d}^d} \right){{\boldsymbol{\eta }}_c} = \kappa \sigma _{ab}^c{{\boldsymbol{\eta }}_c}\]

Analogously to curvature being sourced by the energy-momentum density of the matter sources, torsion is sourced by the spin density  ${{\boldsymbol{\sigma }}_{ab}} =: \sigma _{ab}^c{{\boldsymbol{\eta }}_c}$ of matter. The torsion equation can be solved to give the so-called Cartan equations

12.17

\[T_{ab}^c = \kappa \left( {\sigma _{ab}^c + \frac{1}{2}\delta _a^c\sigma _{bd}^d + \frac{1}{2}\delta _b^c\sigma _{da}^d} \right)\]

These equations are linear and algebraic which implies that, outside regions with spin densities, torsion vanishes, or stated differently, torsion does not propagate. If the matter is altogether spinless, the torsion vanishes identically and the equation of motion (12.15) reduces to the Einstein field equation of GRT.