Acoustic propagation in fluids: an unexpected example of Lorentzian geometry

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 partialₘu (sqrt-g g^mu nu partialₙu psi) /sqrt-g = 0. The acoustic metric --- g₌ₔ ₍ₔ (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.