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

Tetrads in General Relativity

XII. Einstein-Cartan Theory

Riemann-Cartan Geometry

   The Einstein–Cartan theory of gravity (ECT) is a modification of GRT, allowing spacetime to have torsion in addition to curvature. This theory, also known as the Einstein–Cartan–Kibble–Sciama (ECKS) theory, has a long history with major contributions by Einstein, Cartan, Utimaya, Kibble and Sciama. Élie Cartan initiated this development in 1922 by reformulating general relativity on the basis of a new mathematical construct, now known as Riemann–Cartan (RC) geometry. The main difference with Riemannian geometry is that in the latter the affine connection is derived from the metric, whereas the primary structure characterizing the RC spacetime manifold, commonly denoted ${\mathcal{U}_4}$, is the independent pair $\left\{{{{\mathbf{e}}^a},{{\boldsymbol{\omega }}^a}_b} \right\}$ of frame field and Maurer-Cartan connection .

   In modern view, ECT is the simplest version of a Poincaré Gauge Theory (PGT). This class of theories within the Riemann-Cartan geometry describes gravitational interactions as a gauge theory based on the Poincaré group $\text{P}(1,3)$, i.e., the symmetry group underlying special relativity consisting of translations $\text{T(4)}$ and Lorentz rotations ${\text{SO}}(1,3)$. In PGT, these translational and rotational symmetries are 'gauged' to become local, inducing two sets of gauge potentials $\left\{ {{e_\mu }^a,{\Sigma _\mu }^{ab}} \right\}$: the vierbein field and its inverse associated with translations; the spin (Lorentz) connection associated with Lorentz transformations. These fields are used to define a coderivative $D_\mu$ thereby ensuring that the theory remains invariant under local Poincaré transformations; see Gauge Principle and [Wikipedia: Poincaré gauge theory].

   The corresponding 'gauge field strengths' tensors are obtained by commutation with the coderivative and found to be torsion and curvature as defined in (4.24) and (6.3) equivalent to, respectively, the First Cartan Equation (4.25) and Second Cartan Equation (10.6):

12.1

\[\;\, T_{\mu \nu }^a = 2{D_{[\mu }}e_{\nu ]}^a \qquad \; {{\mathbf{T}}^a} = {\text{d}}{{\mathbf{e}}^a} + {{\bs{\omega }}^a}_b \wedge {{\mathbf{e}}^b} \]

12.2

\[\qquad R_{\mu \nu }^{ab} = 2{D_{[\mu }}\Sigma _{\nu ]}^{ab}\qquad {{\bs{\mathcal{R}}}^{a}}_{b} = {\text{d}}{{\bs{\omega }}^a}_b + {{\bs{\omega }}^a}_c \wedge {{\bs{\omega }}^c}_b\]

Thus, gauging the Poincaré group leads directly to the RC geometry of spacetime. Notable is the correspondence between torsion and translation symmetry.

   In the EC theory the curvature 2-form ${{\bs{\mathcal{R}}}^{a}}_{b}$ differs from the Riemann 2-form ${{\bs{R}}^{a}}_{b}$ through the torsion dependence of Maurer-Cartan connection. The curvature tensor still satisfies ${\mathcal{R}}_{abcd}={\mathcal{R}}_{[ab][cd]}$. However, in the RC spacetime the Ricci tensor ${{\mathcal{R}}^{a}}_{b}:={\mathcal{R}}^{ac}_{bc}$, although well-defined, is not necessarily symmetric because the Cartan curvature has not the symmetry (10.4c) of the Riemann. When torsion vanishes, the Cartan and Riemann curvature tensors coincide.