Ja n 20 06 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 out-flux. The effective Lagrangian for such particles is shown to be of the form mc 2 ℓ(U 2 /c 2), where 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 a 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 ℓ is universal: For D dimensional particles ℓ ∝ 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 ≤ 5 they all have a relativistic limit with the acquired energy and momentum diverging as U → c. For D ≤ 7 the null geodesics of the standard acoustic metric solve our equation of motion. Interestingly, for D = 4 the dynamics is very nearly Lorentzian: ℓ ∝ −mc 2 γ −1 λ(γ) (up to a constant), with λ = (1 + γ −1) −1 varying between 1/2 to 1 (γ 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.