Symmetric Teleparallel Geometries

TL;DR

This paper derives the general form of coframes and spin connections for symmetric teleparallel geometries with affine symmetries, aiding the understanding of $F(T)$ gravity and its special cases like Teleparallel Robertson Walker and de Sitter geometries.

Contribution

It provides the explicit forms of coframes and spin connections for teleparallel geometries with affine symmetries, including new solutions like Teleparallel Robertson Walker and de Sitter models.

Findings

Explicit coframe and spin connection forms for $G_6$ affine symmetric geometries.

Field equations for $F(T)$ gravity in these symmetric geometries.

Introduction of Teleparallel de Sitter geometry with additional symmetry.

Abstract

In teleparallel gravity and, in particular, in $F(T)$ teleparallel gravity, there is a challenge in determining an appropriate (co-)frame and its corresponding spin connection to describe the geometry. Very often, the "proper" frame, the frame in which all inertial effects are absent, is not the simplest (e.g, diagonal) (co-)frame. The determination of the frame and its corresponding spin connection for $F(T)$ teleparallel gravity theories when there exist affine symmetries is of much interest. In this paper we present the general form of the coframe and its corresponding spin connection for teleparallel geometries which are invariant under a $G_6$ group of affine symmetries. The proper coframe and the corresponding $F(T)$ field equations are also shown for these Teleparallel Robertson Walker (TRW) geometries. Further, with the addition of an additional affine symmetry, it is possible to define a Teleparallel de Sitter (TdS) geometry.