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Tetrads in General Relativity

XIII. ECKS Theory

Matter Action

   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 $\psi$:

13.1

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

13.2

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

These equations, independently 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. 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 demonstrated that the Einstein 3-form ${{\mathbf{G}}_a}$ is equivalent to an asymmetric Einstein tensor including cosmological constant. Using representation (12.15b), one may then cast equation (13.1) in the form of the (generalized) Einstein equation

13.3

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

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

   By a similar reasoning, and the use of (12.12), the torsion equation (13.2) can be rewritten into

13.4

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

Analogously to curvature being sourced by the energy-momentum density of the matter sources, torsion is sourced by the spin angular momentum 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

13.5

\[T_{ab}^c = \kappa \left( {\sigma _{ab}^c + \delta _{[a}^c\sigma _{b]d}^d} \right)\]

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 spinless altogether, torsion vanishes identically and the equation of motion (13.3) reduces to the Einstein field equation of GRT.