Causality and stability from acoustic geometry

A bstract In scalar-tensor theories with derivative interactions, backgrounds spontaneously break local Lorentz invariance. We study the motion of perturbations of the scalar, “phonons”, on these anisotropic time-dependent backgrounds in curved spacetimes. The phonons propagate on null geodesics of an effective acoustic spacetime, which has its own metric and a connection featuring nonmetricity with respect to the metric defined by gravity. These acoustic geodesics correspond to motion with four-acceleration in the usual spacetime. We stress the differences and duality between the phonons’ canonical four-momenta and four-velocities, and point out analogies with photons in a medium. For an arbitrary moving observer, we covariantly define the phonon’s energy, relative phase velocity, effective refraction index and mass tensor. We point out that true instabilities (ghosts, gradient) are observer independent, being identified by the acoustic metric’s signature and determinant. However, apparent instabilities, such as complex phonon energies, can stem from an ill-posed Cauchy problem in certain observer frames. Negative phonon energies appear for supersonic observers, not indicating true instabilities, but leading to Cherenkov radiation. We extend this local picture to a global foliation, deriving the condition for a spatial slice to be a Cauchy surface for a well-posed initial value problem. The action for perturbations yields an acoustically conserved asymmetric energy-momentum tensor (EMT), not conserved in the usual spacetime. Yet, with a timelike acoustic Killing vector, this EMT forms a current conserved in both the acoustic and usual spacetimes, with the acoustic Hamiltonian functional as its conserved charge. This Hamiltonian is bounded if the foliation’s comoving observer is subsonic. Otherwise, for a Killing vector timelike in both metrics, an alternative conserved charge that bounds motion exists.