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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 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 symmetries are continuous symmetries. So, by
Noether's theorem, one expects two conservation laws. 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). They
may be used to evaluate the exterior coderivative of the 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. Here, 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 Bianchi identity
${\text{D}}{{\mathbf{G}}_a} = 0$.
- The dual of equation (12.8c) gives the Einstein 3-form
\[{{\mathbf{G}}_a} = \left( {G_a^b - \Lambda \delta _a^b}
\right){{\bs{\eta }}_b}\]
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):
\[-
*{\text{D}}{{\mathbf{G}}_a} = {D_b}\left( {G_a^b - \Lambda
\delta _a^b}\right)=0\]
which is the tetrad form of the covariant conservation equation
(10.24) of the (symmetric) Einstein tensor (10.23).