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XII. Einstein-Cartan Theory
By construction the action (12.5) of the Einstein-Cartan theory is
invariant under local Lorentz transformations (LLTs). The
action is also invariant (up to a neglected boundary term) under manifold
diffeomorphisms (MDs), i.e. smooth, invertible mappings
${\mathcal{M}_4} \to {\mathcal{M}_4}$. The latter symmetry derives
from the general covariance under a change of coordinates. These two
transformations comprise the set of fundamental symmetries
of general relativity. They may be seen as gauge symmetries
of the theory, where it should be noted that diffeomorphism
invariance of ECT is closely related to the symmetry group of local
spacetime translations $T(4)$; see Riemann-Cartan
Geometry.
The two fundamental gauge symmetries are continuous symmetries. So,
by Noether's second theorem one expects two
constraint equations. In both GRT and ECT they take the form of two
sets of (contracted) Bianchi identities intrinsic to the
geometrical structure of the GR and RC spacetimes. In ECT these
identities are ${\text{D}}{{\mathbf{T}}^a} =
{{\bs{\mathcal{R}}}^a}_b \wedge {{\mathbf{e}}^b}$,
${\text{D}}{{\bs{\mathcal{R}}}^a}_b = 0$; see (10.9),(10.13) for
their Riemannian equivalents. They may be used to evaluate the
exterior coderivatives of the Einstein and torsion 3-forms
${{\mathbf{G}}_a}$ (12.7) and ${\mathbf{S}}_{ab}$ (12.11):
\[{\text{D}}{{\mathbf{G}}_a} =
\frac{1}{2}{\varepsilon _{abcd}}{{\mathbf{T}}^b} \wedge
{{\bs{\mathcal{R}}}^{cd}} - \Lambda {{\mathbf{T}}^b} \wedge
{{\bs{\eta }}_{ab}}\]
\[{\text{D}}{{\mathbf{S}}_{ab}}
= {{\bs{\eta }}_{ac}} \wedge {{\bs{\mathcal{R}}}_b}^c - {{\bs{\eta
}}_{bc}} \wedge {{\bs{\mathcal{R}}}_a}^c\]
In the derivation of the last equation identity (E.2d) has been
used.
Equation (12.13) is the generalization to RC space, of the twice
contracted Bianchi identity (10.22) which is crucial in standard
GRT. It is interpreted to be a consequence of diffeomorphism
invariance. The second equation derives from Lorentz symmetry
and relates to the conservation of angular momentum.
GRT
- Without torsion (12.13) reduces to the GRT Bianchi identity
${\text{D}}{{\mathbf{G}}_a} = 0$.
- Proof: the dual of equation (12.8c) gives the Einstein
3-form
\[{{\mathbf{G}}_a} = \left( {{G^b}_a - \Lambda \delta _a^b}
\right){{\bs{\eta }}_b}\]
expanded with respect to the basis 3-form ${{\bs{\eta }}_b}$
defined in (C.3.b).
- Since ${\text{D}}{{\bs{\eta }}_b} = 0$ (no torsion), one
obtains with (E.2a):
\[-
*{\text{D}}{{\mathbf{G}}_a} = {D_b}\left( {{G^b}_a - \Lambda
\delta _a^b}\right)=0\]
which is the tetrad form of the covariant conservation equation
(10.24) of the (symmetric) Einstein tensor (10.23).