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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 a Riemann–Cartan geometry. The main 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 $\left\{{{{\mathbf{e}}^a},{{\boldsymbol{\omega }}^a}_b} \right\}$ of frame field and 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 $D_\mu$ that ensures 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, from equations (4.24) and (6.3), found to be torsion and curvature as defined in GRT:

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

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

   As already shown, the tensor equations for torsion and curvature are equivalent to, respectively, the First Cartan Equation (4.25) and the Second Cartan Equation (10.6). Thus, gauging the Poincaré group leads directly to the Riemann-Cartan geometry of spacetime. Notable is the correspondence between torsion and translation symmetry.