Is There Any Symmetry Left in Gravity Theories with Explicit Lorentz Violation?

Yuri Bonder,Cristóbal Corral

doi:10.3390/sym10100433

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.

Highlights

Conventional theories of gravity, when geometrical, are invariant under diffeomorphisms (Diff) and local Lorentz transformations (LLT)
The motivation to do this stems from the fact that within the most prominent approaches to quantum gravity, it has been argued that LLT may not be a fundamental symmetry [8,9,10,11,12], and it may be possible to look for empirical traces of quantum gravity by searching for violations of LLT
The so-called local translations are studied in a gravity theory with explicit Lorentz violation, which is introduced by a non-dynamical zero-form k ab cd

Summary

Introduction

Conventional theories of gravity, when geometrical, are invariant under diffeomorphisms (Diff) and local Lorentz transformations (LLT). The motivation to do this stems from the fact that within the most prominent approaches to quantum gravity, it has been argued that LLT may not be a fundamental symmetry [8,9,10,11,12], and it may be possible to look for empirical traces of quantum gravity by searching for violations of LLT. It should be stressed that the dominant position in the SME community is that, in the presence of gravity, Lorentz violation must occur spontaneously This is assumed to deal with the severe restrictions arising from the contracted Bianchi identity [15], i.e., the fact that the Einstein tensor is divergence free. Other alternatives to deal with these restrictions include a Stückelberg-like mechanism [32] and the use of Finsler geometries [33]

Gauge Theories of Gravity and Local Translations

Explicit Lorentz Violation in the Gravity Sector

Conclusions