Causality and Stability from Acoustic Geometry

TL;DR

The paper develops a covariant framework for sound-wave perturbations in derivative-interacting scalar-tensor theories that spontaneously break Lorentz invariance. By deriving an effective acoustic metric and its nonmetricity, it identifies two cones—the ray (N-cone) and momentum (P-cone)—that govern propagation, energy signs, and stability in a frame-dependent but physically meaningful way. It provides precise criteria for hyperbolicity, well-posedness of the initial-value problem, and Hamiltonian boundedness, including the role of acoustic Killing vectors and conserved currents. The authors classify all possible acoustic geometries, illustrate with Gordon’s metric, k-essence, and kinetic gravity braiding, and show how phenomena like Mach cones and Cherenkov radiation arise within this geometric picture. The results offer a robust, frame-aware understanding of stability and causality in a broad class of scalar-tensor theories, with implications for analogue gravity and EFT consistency bounds.

Abstract

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 non-metricity with respect to the metric defined by gravity. These acoustic geodesics correspond to motion with four-acceleration in the usual spacetime. We indicate the differences and duality between the phonons' canonical four-momenta and four-velocities, and the analogies with photons in media. 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 signaling 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. 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. 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.

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