\( \newcommand{\bs}{\boldsymbol} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \newcommand{\third}{{\scriptstyle \frac{1}{3}}} \newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter] \newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)

Tetrads in General Relativity

XII. Einstein Cartan Theory

Einstein-Cartan Action

   In the Einstein-Cartan theory the Cartan curvature tensor (12.2) 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), with the Ricci scalar replaced by the Riemann–Cartan curvature scalar $\mathcal{R}$, into the form of a directed integral. This may be accomplished by inserting the product of the unit volume element and its inverse (A.1):

12.3

\[{S_{{\text{EC}}}} := \frac{1}{{2\kappa }}\int {{d^4}x} \sqrt {\left| g \right|} {\operatorname{I} _4} \operatorname{I}^{4} \mathcal{R} = \frac{1}{{2\kappa }}\int {d{x_4}} \operatorname{I} ^{4}\mathcal{R}\]

The volume element $d{x_4}: = {d^4}x\sqrt {\left| g \right|} {\operatorname{I} _4}$ is the signed invariant volume measure implying a choice of orientation for $\mathcal{M}$; see Integration Measure.

   Subsequently, the definition of the Riemann-Cartan curvature scalar in the integrand may be reworked into

12.4

\[\mathcal{R}: = \mathcal{R}_{cd}^{cd} = \frac{1}{2}\delta _{cd}^{mn}\mathcal{R}_{mn}^{cd} \]

The pre-factor is the generalized Kronecker-delta symbol (B.1) for $p=2$. The first identity (B.4), together with the tetrad form (A.2) of the inverse unit volume and formula (10.5) for the tetrad curvature 2-form, then allows the Einstein-Cartan action, including the cosmological constant, to be cast in the tetradic Einstein-Cartan form:

12.5

\[{S_{{\text{EC}}}}[{\mathbf{e}}, {\boldsymbol{\omega }}] = \frac{1}{{4\kappa }}\int {dx_4}\, {\varepsilon _{abcd}}{{\mathbf{e}}^a} \wedge {{\mathbf{e}}^b} \wedge \left( {{{{\bs{\mathcal{R}}}}^{cd}}[{\boldsymbol{\omega }}] + \frac{\Lambda }{6}{{\mathbf{e}}^c} \wedge {{\mathbf{e}}^d}} \right)\]

The Cartan curvature ${{\bs{\mathcal{R}}}^{a}}_{b}$ is the functional of ${{\boldsymbol{\omega }}^{a}}_{b}$ as given in (12.2).

   Without embellishments, Einstein-Cartan theory is a viable modification of Einstein-Hilbert gravitation, the frame field and the connection having the dynamical interpretation of $\text{T}(4)$ and ${\text{SO}}(1,3)$ gauge potentials. Moreover, the theory is completely consistent with all observational tests of gravity so far. For these reasons the Einstein-Cartan formulation is sometimes preferred over the canonical Einstein-Hilbert formulation e.g. in explorations of quantum gravity. Significant departures from the Einstein theory are expected only for matter densities well beyond the nuclear threshold.