# Modified teleparallel theories of gravity in symmetric spacetimes

## TL;DR

This paper develops a systematic method to find symmetric solutions in teleparallel gravity theories, deriving universal tetrads and spin connections compatible with various spacetime symmetries, including cosmological cases.

## Contribution

It applies Cartan geometry to teleparallel theories, deriving general symmetric tetrads and spin connections, and identifies universal solutions for cosmological symmetry in teleparallel gravity.

## Key findings

- Derived general symmetric tetrads and spin connections for various symmetries.
- Identified universal solutions for cosmological symmetry in teleparallel gravity.
- Connected symmetry solutions to known 'good tetrads' in $f(T)$ gravity.

## Abstract

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 anti-symmetric 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.