Gauge invariance in teleparallel gravity theories: A solution to the background structure problem

TL;DR

This paper addresses the challenge of defining a background structure in tetrad theories of gravity by proposing a gauge-invariant approach that unifies teleparallel gravity and general relativity as gauge theories over specific groups.

Contribution

It introduces a gauge-invariant method to identify background structures in tetrad gravity, linking teleparallel theories and general relativity to specific gauge groups.

Findings

Background structure can be identified via gauge connections.

Allowed structure groups include T_4 and U(2).

Perturbations are coordinate independent and observable.

Abstract

We deal with the problem of identifying a background structure and its perturbation in tetrad theories of gravity. Starting from a peculiar trivial principal bundle we define a metric which depends only on the gauge connection. We find the allowed four-dimensional structure groups; two of them turn out to be the translation group T_4 and the unitary group U(2). When the curvature vanishes the metric reduces to its background form which coincides with Minkowski flat metric for the T_4 case and with the Einstein static universe metric for the U(2) case. The perturbation has a coordinate independent definition and allows for the introduction of observables distinguished from those obtained from the metric alone. Finally, we show that any teleparallel theory of gravity, and hence general relativity, can be considered as a gauge theory over the groups introduced.