Teleparallel gravity from the principal bundle viewpoint

TL;DR

This paper explores formulating Teleparallel Gravity as a gauge theory using principal bundles, analyzing the role of absolute elements and gauge symmetries to understand its structure.

Contribution

It provides a principal bundle framework for TEGR, clarifying the gauge group structure and the role of absolute elements and connections.

Findings

Identifies the gauge group of TEGR as potentially a subgroup of diffeomorphisms.

Shows that treating the teleparallel connection as an absolute element affects gauge symmetry.

Recovers the diffeomorphism group as the gauge group under certain assumptions.

Abstract

We examine whether the Teleparallel Equivalent of General Relativity (TEGR) can be formulated as a gauge theory in the language of connections on principal bundles. We argue in favor of using either the affine bundle with the Poincar\'e group or, equivalently, the orthonormal frame bundle with the Lorentz group as the structure group. Following the framework of Trautman--where gauge symmetries are determined using the absolute elements--we set to identify the absolute elements and gauge symmetries of TEGR. The problem of a non-dynamical teleparallel connection raises the question of whether it should be treated as an absolute element. If so, the gauge group of TEGR is potentially some undetermined subgroup of the diffeomorphism group. On the other hand, if the connection is allowed to be non-dynamical but the only absolute element is taken to be the canonical 1-form of the frame bundle, we recover the whole diffeomorphism group as the gauge group of TEGR.