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

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 T4 and the unitary group U(2). When the curvature vanishes the metric reduces to its background form which coincides with the Minkowski flat metric for the T4 case and with the Einstein static universe metric for the U(2) case. The perturbation has a coordinate independent definition and allows 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.