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Tetrads in General Relativity

XII. Einstein-Cartan Theory

BR Symmetrization

   The Noether identities (12.16,17) may be brought in component form by substituting the representations, see (12.8,12),

12.20

\[{{\mathbf{G}}_a} = \left( {{\mathcal{G}^b}_a - \Lambda \delta _a^b} \right){{\bs{\eta }}_b}{\quad}{{\mathbf{S}}_{{ab}}} = S_{ab}^c{{\bs{\eta }}_c}\]

With the help of of formulae (A.2),(B.4) and C3.b), one finds that the exterior derivative ${\text{D}} = {{\mathbf{e}}^c} \wedge {D_c}$ of ${{\bs{\eta }}_a}$ is proportional to the trace of the torsion tensor:

12.21

\[{\text{D}}{{\bs{\eta }}_a} = {{\mathbf{T}}^b} \wedge {{\bs{\eta }}_{ab}} = - T_{ab}^b{\operatorname{I} ^4}\]

This trace vanishes if the torsion tensor is completely anti-symmetric which is e.g. the case for the Dirac field; see ECD Gravity.

   These manipulations result in ten differential Noether (Bianchi) identities involving the Einstein tensor ${\mathcal{G}^b}_a$ and modified torsion tensor $S_{ab}^c$:

12.22

\[ - * {\text{D}}{{\mathbf{G}}_a} = \mathop {{D_b}}\limits^* {\mathcal{G}^b}_a = T_{ab}^c{\mathcal{G}^b}_c - \frac{1}{2}\mathcal{R}_{ab}^{cd}S_{cd}^b\]

12.23

\[ - * {\text{D}}{{\mathbf{S}}_{ab}} = \mathop {{D_c}}\limits^* S_{ab}^c = {\mathcal{G}_{ba}}-{\mathcal{G}_{ab}} = - 2{\mathcal{R}_{[ab]}}\]

with derivative operator $\mathop {{D_a}}\limits^* : = {D_a} + T_{ab}^b$.

   It is seen that the divergence of modified torsion tensor is directly related to the anti-symmetric part of the Einstein tensor ${\mathcal{G}_{ab}}$ or, what amounts to the same, the Ricci tensor:

12.24

\[G_{ab}:= \mathcal{G}_{(ab)} =\mathcal{G}_{ab} + {\frac{1}{2}}\mathop {{D_c}}\limits^* S_{ab}^c \]