Acoustic geometry for general relativistic barotropic irrotational fluid flow

Abstract

Acoustic spacetimes", in which techniques of differential geometry are used to investigate sound propagation in moving fluids, have attracted considerable attention over the last few decades. Most of the models currently considered in the literature are based on non-relativistic barotropic irrotational fluids, defined in a flat Newtonian background. The extension, first to special relativistic barotropic fluid flow, and then to general relativistic barotropic fluid flow in an arbitrary background, is less straightforward than it might at first appear. In this article we provide a pedagogical and simple derivation of the general relativistic "acoustic spacetime" in an arbitrary (d+1) dimensional curved-space background.

Analogue Gravity

“…Therefore, we finally have the (contravariant) acoustic metric and (covariant) acoustic metric In the non-relativistic limit p 0 ≪ ϱ 0 and , where is the average mass of the particles making up the fluid (which by the barotropic assumption is a time-independent and position-independent constant). So in the non-relativistic limit we recover the standard result for the conformal factor [639] Under what conditions is the fully general relativistic discussion of this section necessary? (The non-relativistic analysis is, after all, the basis of the bulk of the work in “analogue spacetimes”, and is perfectly adequate for many purposes.)…”

“…One proceeds by combining the relativistic Euler equation, the relativistic energy equation, an assumed barotropic equation of state, and assuming a relativistic irrotational flow of the form [639] In this situation the relativistic Bernoulli equation can be shown to be where we emphasize that ϱ is now the energy density (not the mass density), and the total particle number density can be shown to be After linearization around some suitable background [448, 72, 639], the perturbations in the scalar velocity potential Θ can be shown to satisfy a dimension-independent d’Alembertian equation which leads to the identification of the relativistic acoustic metric as The dimension-dependence now comes from solving this equation for . Therefore, we finally have the (contravariant) acoustic metric and (covariant) acoustic metric In the non-relativistic limit p 0 ≪ ϱ 0 and , where is the average mass of the particles making up the fluid (which by the barotropic assumption is a time-independent and position-independent constant).…”

Aharonov–Bohm effect for a fermion field in a planar black hole “spacetime”

“…These advances are important for a better insight into the understanding of quantum gravity. In addition, the study of a relativistic version of acoustic black holes was presented in [31][32][33][34]. Furthermore, the acoustic black hole metrics obtained from a relativistic fluid in a noncommutative spacetime [35] and Lorentz-violating Abelian Higgs model [36,37] have been considered.…”

One-Dimensional Mass-Spring Chains Supporting Elastic Waves with Non-Conventional Topology

“…Energy is added to extrinsic topological systems to break time reversal symmetry [15][16][17][18][19][20]. A common example of an extrinsic approach is that of time-reversal symmetry breaking of acoustic waves by moving fluids [21][22][23][24][25][26][27][28][29]. Recently, extrinsic topological phononic crystals have demonstrated the astonishing property of non-reciprocity and backscattering-immune edge states and bulk states establishing classical equivalents of topological electronic insulators.…”