Is there any symmetry left in gravity theories with explicit Lorentz violation?

TL;DR

This paper investigates whether gravity theories with explicit Lorentz violation retain any symmetry under local translations, finding that such symmetries are generally broken in the minimal Standard Model Extension's gravity sector.

Contribution

It demonstrates that local translations are not symmetries of explicit Lorentz-violating gravity theories, using a robust algorithm applied to the Standard Model Extension.

Findings

Local translations are not symmetries of the minimal gravity sector.

Explicit Lorentz violation breaks local translation invariance.

Spontaneous Lorentz violation might preserve local translations.

Abstract

It is well known that a theory with explicit Lorentz violation is not invariant under diffeomorphisms. On the other hand, for geometrical theories of gravity, there are alternative transformations, which can be best defined within the first-order formalism, and that can be regarded as a set of improved diffeomorphisms. These symmetries are known as local translations and, among other features, they are Lorentz covariant off shell. It is thus interesting to study if theories with explicit Lorentz violation are invariant under local translations. In this work, an example of such a theory, known as the minimal gravity sector of the Standard Model Extension, is analyzed. Using a robust algorithm, it is shown that local translations are not a symmetry of the theory. It remains to be seen if local translations are spontaneously broken under spontaneous Lorentz violation, which are regarded as a more natural alternative when spacetime is dynamic.