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XII. Einstein-Cartan Theory
The field equations in the Einstein-Cartan theory are obtained from
the tetradic action (12.5) by performing independent stationary
variations with respect to the gravitational gauge fields,
i.e. the frame field $\{{{\mathbf{e}}^a}\}$ and the Lorentz
connection $\{ {{\boldsymbol{\omega }}^{ab}}\} $. The variation with
respect to the frame field yields:
\[ {\delta
_{\mathbf{e}}}{S_{{\text{EC}}}} [{\mathbf{e}},{\boldsymbol{\omega
}}] = \frac{1}{{2\kappa }}\int {d{x_4}}\, {\varepsilon
_{abcd}}\delta {{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge
\left( {{{{\bs{\mathcal{R}}}}^{cd}} + \frac{\Lambda
}{3}{{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}} \right)\]
The condition that the action vanishes for arbitrary variations
$\delta {{\mathbf{e}}^a}$ gives the equation of motion
\[{{\mathbf{G}}_a}:=\frac{1}{2}{\varepsilon_{abcd}}{{\mathbf{e}}^b}
\wedge \left( {{{{\bs{\mathcal{R}}}}^{cd}} + \frac{\Lambda
}{3}{{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}} \right) = 0\]
which is the Einstein (vacuum) field equation in the RC
tetrad formalism, including the cosmological constant.
Einstein Tensor
- Expand the curvature 2-form $\bs{\mathcal{R}}^{cd}$ in (12.7)
to create the Einstein 3-form
\[{{\mathbf{G}}_a} = \frac{1}{4}{\varepsilon
_{abcd}}{{\mathbf{e}}^b} \wedge {{\mathbf{e}}^m} \wedge
{{\mathbf{e}}^n}\left( {\mathcal{R}_{mn}^{cd} + \frac{\Lambda
}{3}\delta _{mn}^{cd}} \right)\]
- Multiply from the left with the pseudoscalar
${\operatorname{I} _4}$. Then use (C.2) and (B.4) for $p=3$ to
get the 1-form:
\[*\mathbf{G}_a = - \frac{1}{4}\delta _{acd}^{bmn}\left(
{\mathcal{R}_{mn}^{cd} + \frac{\Lambda }{3}\delta _{mn}^{cd}}
\right){{\mathbf{e}}_b}\]
- Expand the generalized Kronecker delta's with (B.1) to obtain
the tensor form of the EC vacuum field equations:
\[ * {{\mathbf{G}}_a} = \left( {{\mathcal{G}^b}_a
- \Lambda \delta _a^b} \right){{\mathbf{e}}_b} = 0 \]
The Einstein tensor ${\mathcal{G}}_{ba}
= {\mathcal{G}}_{(ab)} - {\mathcal{R}}_{[ab]}$ is asymmetric
due to the asymmetry of the Ricci tensor when torsion is
present.