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Tetrads in General Relativity

XIII. ECKS Theory

Einstein Equation

   Because of the linear structure of equations (13.1,2), the pair of Noether identities (12.16,17) immediately implies the conservation laws of energy-momentum and angular momentum in the ECKS theory:

13.6

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

13.7

\[ \operatorname{D} {{\bs{\sigma }}_{ab}} = 2 {{\mathbf{e}}_{[b}} \wedge {{\bs{\tau }}_{a]}}\]

   Under the assumption that the torsion tensor is completely anti-symmetric,  these equations read in tensor form, see  (12.22,23):

13.8

\[{D_b}{\tau ^b}_a = T_{ab}^c{\tau ^b}_c - \frac{1}{2}R_{ab}^{cd}\sigma _{cd}^b\]

13.9

\[{D_c}\sigma _{ab}^c = {\tau _{ba}} - {\tau _{ab}}\]

They are ten covariant Riemann-Cartan conservation laws for the canonical energy-momentum ${\tau^b}_a$ and spin angular momentum $\sigma _{ab}^c$ tensors.

   The divergence of the spin current (13.9) is directly related to the anti-symmetric part of the canonical energy momentum tensor. Analogously to (12.24), this allows a Belinfante-Rosenfeld (BR) construction of a symmetric energy-momentum tensor of matter

13.10

\[t_{ab}:=\mathcal{\tau}_{(ab)} ={\mathcal{\tau}}_{ab} + \frac{1}{2}{D_c}\sigma_{ab}^c\]

Hence, local conservation of angular momentum (13.9) leads to the Einstein equation of the ECKS theory

13.11

\[{G_{ab}}- \Lambda \delta_{ab} = \kappa \,{t_{ab}}\]