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Tetrads in General Relativity

XII. Einstein-Cartan Theory

Noether Identities

   In the usual procedure Noether identities are obtained by implementing symmetries, in the present case diffeomorphisms and local Lorentz transformations, in the action. The result would then imply the Bianchi identities (12.13,14) and the dynamical equations (12,13,14). However, since these dynamical equations have already been established, the shorter alternative is to derive the Noether identities laws therefrom. By some algebraic manipulations one then obtains the differential Noether identities of the Einstein-Cartan theory:

12.16

\[{\text{D}}{{\mathbf{G}}_a} = {\mathbf{T}}_a^b \wedge {{\mathbf{G}}_b} - \frac{1}{2}{\bs{\mathcal{R}}}_a^{bc} \wedge {{\mathbf{S}}_{bc}}\]

12.17

\[{\text{D}}{{\mathbf{S}}_{ab}} = {{\mathbf{e}}_b} \wedge {{\mathbf{G}}_a} - {{\mathbf{e}}_a} \wedge {{\mathbf{G}}_b}\]

   These equations look much like balance equations for energy-momentum and angular momentum; this will be further explored in Conservation Laws. For vanishing torsion, GRT is recovered as shown in (12.15). In this limit equation (12.17) enforces the symmetry of the Einstein tensor which is conserved on account of (12.16). As an aside it may be is mentioned that in the limit of zero torsion and curvature the above Noether equations reduce to the conservation laws of energy-momentum and angular momentum in Minkowski spacetime.

12.18

Notes

  1. The 1-form fields appearing in (12.16) are defined as the contractions
    \[{\mathbf{T}}_b^a: = {{\mathbf{e}}_b} \cdot {{\mathbf{T}}^a} = T_{bc}^a{{\mathbf{e}}^c} \quad {\bs{\mathcal{R}}}_c^{ab} : = {{\mathbf{e}}_c} \cdot {{\bs{\mathcal{R}}}^{ab}} = \mathcal{R}_{cd}^{ab}{{\mathbf{e}}^d}\]
  2. Since ${{\mathbf{G}}_a}$ is a 3-form, $({{\mathbf{T}}^b} \wedge {{\mathbf{G}}_b})$ is a 5-form, hence vanishes identically. By contracting with ${{\mathbf{e}}_a}$ and working out the inner product with the algebraic rule (E.1), one finds that the right-hand sides of (12.13) and (12.16) are equivalent.
  3. To arrive at (12.17), one may use (E.2d) and the auxiliary formula
    \[{{\mathbf{e}}^k} \wedge {{\mathbf{G}}_a} = - \frac{1}{2}\left( {{{\bs{\eta }}_{ab}} \wedge {{\bs{\mathcal{R}}}^{bk}} + {{\bs{\eta }}_{ca}} \wedge {{\bs{\mathcal{R}}}^{kc}} + \delta _a^k{{\bs{\eta }}_{bc}} \wedge {{\bs{\mathcal{R}}}^{bc}}} \right)\]