Einstein-Aether Waves

TL;DR

The paper develops the linearized spectrum of Einstein-aether theory, a generally covariant model in which a unit timelike vector $u^a$ (the aether) couples to gravity and introduces four derivative constants $c_i$. Using a diffeomorphism-invariant action with a Lagrange multiplier enforcing $u^a u_a=1$, the authors derive the linearized field equations around Minkowski space, fix a practical gauge, and perform a plane-wave analysis to identify five propagating modes: two GR-like transverse-traceless metric modes and three coupled aether-metric modes (two transverse and one trace). The mode speeds $s$ are functions of the $c_i$ and reduce to the GR values in appropriate limits, with special parameter choices leading to unique behaviors (e.g., infinite speeds when $c_{14}=0$). These results establish the conditions under which the linearized equations are hyperbolic and provide a foundation for predicting observable deviations in gravitational-wave physics and precision tests of Lorentz violation in gravity.

Abstract

Local Lorentz invariance violation can be realized by introducing extra tensor fields in the action that couple to matter. If the Lorentz violation is rotationally invariant in some frame, then it is characterized by an ``aether'', i.e. a unit timelike vector field. General covariance requires that the aether field be dynamical. In this paper we study the linearized theory of such an aether coupled to gravity and find the speeds and polarizations of all the wave modes in terms of the four constants appearing in the most general action at second order in derivatives. We find that in addition to the usual two transverse traceless metric modes, there are three coupled aether-metric modes.