Massive particles in acoustic space-times: Emergent inertia and passive gravity

I show that massive-particle dynamics can be simulated by a weak, external perturbation on a potential flow in an ideal fluid. The perturbation defining a particle is dictated in a small (spherical) region that is otherwise free to roam in the fluid. Here I take it as an external potential that couples to the fluid density or as a rigid distribution of sources with vanishing total outflux. The effective Lagrangian for such particles is shown to be of the form $m{c}^2\ell (U^2/c^2)$, where $\overrightarrow{U}$ is the velocity of the particle relative to the fluid and $c$ the speed of sound. This can serve as a model for emergent relativistic inertia à la Mach’s principle with $m$ playing the role of inertial mass, and also of analog gravity where $m$ is also the passive gravitational mass. The mass $m$ depends on the particle type and intrinsic structure (and on position if the background density is not constant), while $\ell$ is universal: For $D$-dimensional particles $\ell \propto F(1,1/2;D/2;U^2/c^2)$ ($F$ is the hypergeometric function). These particles have the following interesting dynamics: Particles fall in the same way in the analog gravitational field mimicked by the flow, independent of their internal structure, thus satisfying the weak equivalence principle. For $D\le 5$ they all have a relativistic limit with the acquired energy and momentum diverging as $U\to c$. For $D\le 7$ the null geodesics of the standard acoustic metric solve our equation of motion. Interestingly, for $D=4$ the dynamics is very nearly Lorentzian: $\ell \propto -m{c}^2{\gamma }^{-1}\lambda (\gamma )$ (up to a constant), with $\lambda =(1+{\gamma }^{-1}{)}^{-1}$ varying between $1/2$ and 1 ($\gamma$ is the “Lorentz factor” for the particle velocity relative to the fluid). The particles can be said to follow the geodesics of a generalized acoustic metric of a Finslerian type that shares the null geodesics with the standard acoustic metric. In vortex geometries, the ergosphere is automatically the static limit. As in the real world, in “black hole” geometries circular orbits do not exist below a certain radius that occurs outside the horizon. There is a natural definition of antiparticles, and I describe a mock particle vacuum in whose context one can discuss, e.g., particle Hawking radiation near event horizons.