A frame based approach to computing symmetries with non-trivial isotropy groups

Abstract

A frame approach to determining the most general solution admitting a desired symmetry group has previously been examined in Riemannian and teleparallel geometries with some success. In teleparallel geometries, one must determine the general form of the frame and spin connection to generate a general solution admitting the desired symmetry group. Current approaches often rely on the use of the proper frame, where the spin connection is zero. However, this leads to particular theoretical and practical problems. In this paper, we introduce an entirely general approach to determining the most general Riemann–Cartan geometries that admit a given symmetry group and apply these results to teleparallel geometries. To illustrate the approach, we determine the most general geometries, with the minimal number of arbitrary functions, for particular choices of symmetry groups with dimension one, three, six, and seven. In addition, we rigorously show how the teleparallel analog of the Robertson–Walker, de Sitter, and Einstein static spacetimes can be determined.