Lorentz-Violating Gravity Models and the Linearized Limit

Michael Seifert

doi:10.3390/sym10100490

Abstract

Many models in which Lorentz symmetry is spontaneously broken in a curved spacetime do so via a “Lorentz-violating” (LV) vector or tensor field, which dynamically takes on a vacuum expectation value and provides additional local geometric structure beyond the metric. The kinetic terms of such a field will not necessarily be decoupled from the kinetic terms of the metric, and will generically lead to a set of coupled equations for the perturbations of the metric and the LV field. In some models, however, the imposition of certain additional conditions can decouple these equations, yielding an “effective equation” for the metric perturbations alone. The resulting effective equation may depend on the metric in a gauge-invariant way, or it may be gauge-dependent. The only two known models yielding gauge-invariant effective equations involve differential forms; I show in this work that the obvious generalizations of these models do not yield gauge-invariant effective equations. Meanwhile, I show that a gauge-dependent effective equation may be obtained from any “tensor Klein–Gordon” model under similar assumptions. Finally, I discuss the implications of this work in the search for Lorentz-violating gravitational effects.

Highlights

The prospect of Lorentz symmetry violation has received a fair amount of attention in recent years [1]
A framework for the study of Lorentz symmetry violation in the context of gravity was developed by Kostelecký & Bailey [4], and much experimental work in the years since has searched for the effects described within this framework This framework attempts to follow the method taken by the particle physics sector of the Standard Model Extension (SME) by obtaining a Lorentz-violating equation of motion for the metric perturbations of a flat background, and examining the phenomenology of the modified metrics
The Euler–Lagrange equations for a model containing both a tensor field and a dynamical metric contain coupled kinetic terms, and if the tensor field takes on a vacuum expectation value in flat spacetime, this coupling will persist in the linearized equations about this background

Summary

Introduction

The prospect of Lorentz symmetry violation has received a fair amount of attention in recent years [1]. In the same sense that the original Standard Model action contains “all possible low-energy physics” that is consistent with the underlying gauge groups, locality, and Lorentz symmetry, the SME action contains. A broad class of models containing a Lorentz-violating field can be cast into a more general form which lies outside the minimal gravity sector of the SME. While it remains an open (and ill-defined) question whether these latter models are viable, these results suggest that a broader framework for Lorentz violation in the gravitational regime may be necessary

Linearized Equations for Lorentz-Violating Gravity

Tensor Klein–Gordon Fields

Discussion