Acoustic propagation in fluids: an unexpected example of Lorentzian geometry

TL;DR

This paper demonstrates that sound wave propagation in non-homogeneous fluids can be described using a Lorentzian spacetime geometry, revealing a surprising connection between fluid dynamics and relativistic geometry.

Contribution

It shows that acoustic disturbances in fluids can be modeled with a Lorentzian metric, bridging classical fluid mechanics and relativistic geometric frameworks.

Findings

Acoustic wave equations can be expressed as a d'Alembertian in a Lorentzian metric.

The acoustic metric depends on fluid density, velocity, and sound speed.

Fluid dynamics in Newtonian fluids can exhibit relativistic geometric properties.

Abstract

It is a deceptively simple question to ask how acoustic disturbances propagate in a non--homogeneous flowing fluid. If the fluid is barotropic and inviscid, and the flow is irrotational (though it may have an arbitrary time dependence), then the equation of motion for the velocity potential describing a sound wave can be put in the (3+1)--dimensional form: d'Alembertian psi = 0. That is partial_mu(sqrt{-g} g^{mu nu} partial_nu psi)/sqrt{-g} = 0. The acoustic metric --- g_{mu nu}(t,x) --- governing the propagation of sound depends algebraically on the density, flow velocity, and local speed of sound. Even though the underlying fluid dynamics is Newtonian, non--relativistic, and takes place in flat space + time, the fluctuations (sound waves) are governed by a Lorentzian spacetime geometry.