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XII. Einstein-Cartan Theory
Noether's (first) theorem states that symmetries of the action are
associated with conserved quantities. In the usual procedure
the conservation laws are obtained by implementing the symmetries
(in the present case diffeomorphisms and local Lorentz
transformations) in the action. The result would then imply the
Bianchi identities (12.13,14). However, since these Bianchi
identities have already been established, the shorter alternative is
to derive the conservation laws therefrom. By some algebraic
manipulations one then obtains the differential Noether
identities of the Einstein-Cartan theory:
\[{\text{D}}{{\mathbf{G}}_a} =
{\mathbf{T}}_a^b \wedge {{\mathbf{G}}_b} -
\frac{1}{2}{\bs{\mathcal{R}}}_a^{bc} \wedge {{\mathbf{S}}_{bc}}\]
\[{\text{D}}{{\mathbf{S}}_{ab}}
= {{\mathbf{e}}_b} \wedge {{\mathbf{G}}_a} - {{\mathbf{e}}_a} \wedge
{{\mathbf{G}}_b}\]
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) forces 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.
Einstein 3-form
- The 1-form fields appearing in (12.16) are defined as the
contractions
\[{\mathbf{T}}_b^a: = {{\mathbf{e}}_b} \cdot {{\mathbf{T}}^a}
= T_{bc}^a{{\mathbf{e}}^c} \quad {\bs{\mathcal{R}}}_c^{ab} : =
{{\mathbf{e}}_c} \cdot {{\bs{\mathcal{R}}}^{ab}} =
\mathcal{R}_{cd}^{ab}{{\mathbf{e}}^d}\]
- 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.
Torsion 2-form
- To arrive at (12.17), one may use (E.2d) and the auxiliary
formula
\[{{\mathbf{e}}^k} \wedge {{\mathbf{G}}_a} = -
\frac{1}{2}\left( {{{\bs{\eta }}_{ab}} \wedge
{{\bs{\mathcal{R}}}^{bk}} + {{\bs{\eta }}_{ca}} \wedge
{{\bs{\mathcal{R}}}^{kc}} + \delta _a^k{{\bs{\eta }}_{bc}}
\wedge {{\bs{\mathcal{R}}}^{bc}}} \right)\]