Instabilities in the Aether

TL;DR

This paper analyzes the stability of Lorentz-violating theories with a fixed-norm vector field, identifying only three stable kinetic terms and discussing their stability conditions and evolution obstacles.

Contribution

It classifies stable kinetic terms for aether models and clarifies conditions under which these theories are physically viable.

Findings

Only three kinetic terms are free of manifest instabilities.

Timelike sigma-model is stable with a fixed norm constraint.

Maxwell and scalar models are perturbatively stable but have evolution issues.

Abstract

We investigate the stability of theories in which Lorentz invariance is spontaneously broken by fixed-norm vector "aether" fields. Models with generic kinetic terms are plagued either by ghosts or by tachyons, and are therefore physically unacceptable. There are precisely three kinetic terms that are not manifestly unstable: a sigma model $(\partial_\mu A_\nu)2$, the Maxwell Lagrangian $F_{\mu\nu}F^{\mu\nu}$, and a scalar Lagrangian $(\partial_\mu A^\mu)2$. The timelike sigma-model case is well-defined and stable when the vector norm is fixed by a constraint; however, when it is determined by minimizing a potential there is necessarily a tachyonic ghost, and therefore an instability. In the Maxwell and scalar cases, the Hamiltonian is unbounded below, but at the level of perturbation theory there are fewer degrees of freedom and the models are stable. However, in these two theories there are obstacles to smooth evolution for certain choices of initial data.