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XI. Field Equations
The Einstein–Cartan theory of gravity (ECT) is a
modification of GRT, allowing spacetime to have torsion in
addition to curvature. The theory has a long history with
contributions by Einstein, Cartan, Utimaya, Sciama, Kibble and
others. Élie Cartan started this development in 1922 by
reformulating general relativity on the basis of a Riemann–Cartan
geometry. The chief difference with the Riemannian geometry is that
in the latter the affine connection is derived from the metric,
whereas in the Riemann-Cartan geometry the primary structure
characterizing the geometry of the spacetime manifold is the
independent pair of frame field and Cartan-connection
$\left\{ {{{\mathbf{e}}^a},{{\mathbf{\omega }}^a}_b} \right\}$.
In the Riemann-Cartan geometry the Cartan curvature tensor
(10.6) plays central role in the construction of the action. The
first step to derive this action is to bring the Einstein-Hilbert
action (11.6) into the form of a directed integral. This may be
accomplished by inserting the product of the unit pseudo scalar and
its inverse (11.12):
\[{S_{{\text{EH}}}} =
\frac{1}{{2\kappa }}\int {{d^4}x} \sqrt {\left| g \right|}
{\operatorname{I} _4} \operatorname{I} _4^{ - 1}R: =
\frac{1}{{2\kappa }}\int {d{x_4}} \operatorname{I} _4^{ - 1}R\]
where the pseudoscalar volume $d{x_4}: = {d^4}x\sqrt {\left| g
\right|} {\operatorname{I} _4}$ is a directed measure.
Subsequently, the expression for the Ricci scalar in the integrand
may be reworked into
\[R = {e^\mu }_a{e^\nu }_bR_{\mu
\nu }^{ab} = \frac{1}{2}\delta _{ab}^{cd}{e^\mu }_c{e^\nu }_dR_{\mu
\nu }^{ab}\]
The identity (11.19) and the tetrad form (11.12) of the inverse
unit pseudoscalar then allow the Einstein-Hilbert action to be cast
in the tetradic Einstein-Cartan form:
\[{S_{{\text{EC}}}}[{\mathbf{e}},{\mathbf{\omega
}}] = \frac{1}{{4\kappa }}\int {dx_4}\, {\varepsilon
_{abcd}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge \left(
{{{\mathbf{R}}^{cd}}[{\mathbf{\omega }}] + \frac{\Lambda
}{6}{{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}} \right)\]
Without embellishments, Einstein-Cartan theory is a robust
modification of Einstein-Hilbert gravitation, the connection having
the dynamical interpretation of a ${\text{SO}}(1,3)$ gauge field.
Moreover, the theory is completely consistent with all the
experimental tests of gravity. For these reasons the Einstein-Cartan
formulation is often preferred over the canonical Einstein-Hilbert
formulation in explorations of quantum gravity.