Teleparallel Geometry with a Single Affine Symmetry

TL;DR

This paper develops an algorithm to analyze teleparallel geometries with a single affine symmetry, focusing on the constraints imposed by such symmetries in the context of teleparallel gravity theories.

Contribution

It introduces a systematic method for studying geometries with affine symmetries using a locally Lorentz covariant approach and complex null gauge.

Findings

Constraints on geometry from affine symmetry are fully characterized.

Severe restrictions on the geometry arise in $f(T)$ teleparallel gravity.

An algorithm for constructing symmetric teleparallel geometries is proposed.

Abstract

In teleparallel geometries, symmetries are represented by affine frame symmetries which constrain both the (co)frame basis and the spin-connection (which are the primary geometric objects). In this paper we shall study teleparallel geometries with a single affine symmetry, utilizing the locally Lorentz covariant approach and adopting a complex null gauge. We first introduce an algorithm to study geometries with an affine frame symmetry, which consists of choosing coordinates adapted to the symmetry, constructing a canonical frame, and solving the equations describing the symmetry. %for the spin connection (to determine the six real Lorentz functions. All of the constraints on the geometry are determined in the case of a single affine symmetry, but there are additional constraints arising from the field equations for a given theory of teleparallel gravity. In particular, we find that in $f(T)$ teleparallel gravity there will be severe constraints on the geometry arising from the antisymmetric part of the field equations.