Teleparallel gravity theories employ a tetrad and a Lorentz spin connection as independent variables in their covariant formulation. In order to solve their field equations, it is helpful to search for solutions which exhibit certain amounts of symmetry, such as spherical or cosmological symmetry. In this article we present how to apply the notion of spacetime symmetries known from Cartan geometry to teleparallel geometries. We explicitly derive the most general tetrads and spin connections which are compatible with axial, spherical, cosmological, and maximal symmetry. For homogeneous and isotropic spacetime symmetry we find that the tetrads and spin connection found by the symmetry constraints are universal solutions to the antisymmetric part of the field equations of any teleparallel theory of gravity. In other words, for cosmological symmetry we find what has become known as “good tetrads” in the context of $f(T)$ gravity.

• Received 30 May 2019

DOI:https://doi.org/10.1103/PhysRevD.100.084002