Using Gauge Covariant Lie Derivatives in Poincaré Gauge and Metric Teleparallel Theories of Gravity
A procedure to determine the initial ansatz for the co-frame and spin connection characterizing a Riemann-Cartan geometry respecting a given group of continuous symmetries is illustrated. Assuming invariance of the metric under a particular group of symmetries, this gauge covariant Lie derivative when applied to the metric and associated orthonormal co-frame determines the values of an antisymmetric compensating matrix. The gauge covariant Lie derivative of the spin connection depends on this matrix, which when solved yields the spin connection compatible with the chosen co-frame. The procedure is straightforward and can be employed for any Riemann-Cartan geometry having symmetries including those with a non-trivial isotropy subgroup. Here we illustrate the procedure with numerous examples, including, spherically symmetric, plane symmetric, locally rotationally symmetric Bianchi type III, Gödel, de Sitter and anti-de Sitter geometries. Further, we have also solved the zero curvature constraint to obtain the resulting spin connection for the corresponding metric teleparallel geometry having this same symmetry group. We complete this investigation by including the Lorentz transformation that yields the proper frame for some of these metric teleparallel geometries. This investigation provides a kinematical framework useful for the further study of Poincaré Gauge and metric teleparallel theories of gravity.