Causal Structure of Acoustic Spacetimes

Causal structure of acoustic spacetimes Carlos Barceló ∗, Stefano Liberati †, Sebastiano Sonego ‡, and Matt Visser § ∗Instituto de Astrof´ısica de Andaluc´ıa (CSIC), Camino Bajo de Huétor 24, 18008 Granada, Spain †International School for Advanced Studies (SISSA), Via Beirut 2-4, 34013 Trieste, Italy and INFN, Trieste, Italy ‡Universitàdi Udine, Via delle Scienze 208, 33100 Udine, Italy § School of Mathematical and Computing Sciences, Victoria University of Wellington, New Zealand 8 August 2004; V2: 1 Sept 2004; LATEX-ed February 7, 2008; gr-qc/0408022 Abstract The so-called “analogue models of general relativity” provide a number of specific physical systems, well outside the traditional realm of general relativity, that nevertheless are well-described by the differential geometry of curved spacetime. Specifically, the propagation of acoustic disturbances in moving fluids are described by “effective metrics” that carry with them notions of “causal structure” as determined by an exchange of sound signals. These acoustic causal structures serve as specific examples of what can be done in the presence of a Lorentzian metric without having recourse to the Einstein equations of general relativity. (After all, the un- derlying fluid mechanics is governed by the equations of traditional hydrodynamics, not by the Einstein equations.) In this article we take a careful look at what can be said about the causal structure of acoustic spacetimes, focusing on those containing sonic points or horizons, both with a view to seeing what is different from standard general relativity, and to seeing what the similarities might be. PACS: 02.40.Ma. 04.20.Cv, 04.20.Gz, 04.70.-s Keywords: analogue models, acoustic spacetime, causal structure, conformal structure, Penrose–Carter diagrams. ∗[email protected] †[email protected]; http://www.sissa.it/~liberati ‡[email protected] §[email protected]; http://www.mcs.vuw.ac.nz/~visser 1 Causal structure of acoustic spacetimes 2 1 Introduction Acoustic spacetimes are curved Lorentzian manifolds that are used to describe the propagation of sound in moving fluids [1, 2, 3, 4, 5, 6]. As such they are equipped with a Lorentzian spacetime metric (strictly speaking, a pseudo-metric) that is associated with the “sound cones” emanating from each event in the spacetime. Though the acoustic spacetimes are in general curved manifolds, with nonzero Riemann tensor, their curvature and evolution (their geometrodynamics) are not determined by the Einstein equations, but are instead implicit in the equations of traditional fluid mechanics [3, 5, 6]. The acoustic effective geometry must not be confused with the “real” spacetime geometry. Indeed, the “real” physical spacetime structure that we experience in a condensed matter laboratory is approximately Minkowskian. Moreover, under normal circumstances the ve- locities involved are so small in comparison with the velocity of light that we can perfectly well assume that the system is non-relativistic (Galilean), be it classical or quantum. Because of these features, the acoustic spacetimes play a rather special role with respect to traditional general relativity. They are examples of Lorentzian manifolds without “gravity” and their existence forces us to think deeply and carefully about the distinc- tion between kinematics and dynamics in general relativity — specifically how much of standard general relativity depends on the Einstein equations and how much of it depends on more general considerations that continue to hold independently of the Einstein equations? In particular, this forces us to think about the deep connections and fundamental differences between Lorentzian geometry, the Einstein equivalence principle, and full general relativity. Some features that one normally thinks of as intrinsically aspects of gravity, both at the classical and semiclassical levels (such as horizons and Hawking radiation), can in the context of acoustic manifolds be instead seen to be rather generic features of curved spacetimes and quantum field theory in curved spacetimes, that have nothing to do with gravity per se [1, 5, 7, 8]. In this article we will develop an entire bestiary of (1+1)-dimensional acoustic geometries, specifically chosen because of their naturalness from the point of view of flowing fluids. We will focus on the particularly interesting cases in which the acoustic geometries possess one or two sonic points. These geometries will be the starting point of a follow-up paper in which we will investigate their different effects in terms of curved-space quantum field theoretic vacuum polarization. After describing each of these geometries, we will investigate their global causal structure by the use of Carter–Penrose diagrams. These diagrams make it clear that because the acoustic geometries are not governed by the Einstein equations, their causal structure can be quite different from what is usually encountered in the context of general relativity. In this context, we will also discuss the notion of “maximal analytic extension” in these acoustic geometries on both mathematical and physical grounds. While mathematically the notion of analytic extension makes perfectly good sense, there are now good physical reasons for being cautious. This may have implications for physical gravity and in particular for the ability to characterize spacetime structure by a single well-behaved metric (as opposed to a multi-metric theory). Since this is the first step towards implementing any version of the equivalence principle, it strikes at the very foundations of general relativity. The geometrical analyses of acoustic spacetimes we are going to present are also interesting for two additional reasons: 1. Because we have a very specific and concrete physical picture for these acoustic Causal structure of acoustic spacetimes 3 spacetimes it is sometimes easier for a classically trained physicist to see what is going on, and to then use this as a starting point for investigations of the perhaps more formal causal structure in standard general relativity. 2. Conversely, relativists can adopt their training to ask questions in acoustics that might not normally occur to classically trained acoustic physicists. After dealing with some fundamental issues in section 2 we shall introduce the concept of null coordinates in section 3. In section 4 we develop a “zoo” of stationary acoustic spacetimes, focussing on situations with either one or two sonic points. Section 5 looks at the dynamical evolution of acoustic horizons as the fluid flow is switched on from zero flow to the fully developed flows considered in section 4. In sections 6 and 7 we investigate the global causal structure of the stationary and dynamical acoustic spacetimes by the use of Penrose–Carter diagrams, while the mathematical possibility of performing an analytical extension for acoustic spacetimes will be considered and discussed in section 8. Finally, our summary and conclusions are presented in section 9. 2 Fundamental features We start by pointing out that in acoustic spacetimes, as in general relativity, causal structure can be characterized in two complementary ways — in terms of the rays of geometrical acoustics/optics or in terms of the characteristics of the partial differential equations (wave equations) of physical acoustics/optics [6, 9]. At the level of geometrical acoustics we need only assume that: the speed of sound c, relative to the fluid, is well defined; • the velocity of the fluid ~v, relative to the laboratory, is well defined. • Then, relative to the laboratory, the velocity of a sound ray propagating, with respect to the fluid, along the direction defined by the unit vector ~n is d~x = c ~n + ~v, (2.1) dt which defines a sound cone in spacetime given by the condition ~n2 = 1, i.e., c2dt2 + (d~x ~v dt)2 =0 . (2.2) − − This is associated with a conformal class of Lorentzian metrics [5, 6] (c2 v2) ~v T g = Ω2 − − − , (2.3) " ~v I # − where Ω is an unspecified non-vanishing function. The virtues of the geometric approach are its extreme simplicity and the fact that the basic structure is dimension-independent. At the level of physical acoustics, setting up an acoustic spacetime is a little trickier. For technical reasons it is easiest to confine attention to an irrotational flow for a fluid Causal structure of acoustic spacetimes 4 with a barotropic equation of state, in which case it is relatively straightforward to derive, in any number of dimensions, a wave equation of the form [5, 6] ab ∂a f ∂bθ =0 . (2.4) Turning this into a statement about a metric requires the identification ( det g)1/2 gab = f ab , (2.5) − where, as usual, g denotes the matrix [g ], obtained by inverting g−1 [gab]. Defining ab ≡ now det f =1/ det[f ab] we have det g =( det f)−2/(d−2) , (2.6) − − and gab =( det f)1/(d−2) f ab . (2.7) − With this procedure one gets, for a fluid with mass density ρ, a metric of the form (2.3), with Ω2 equal to an unspecified positive constant multiplied by some power of ρ/c. ab However, the exponent in equation (2.7) indicates that g and gab are not defined in d = 2 (that is, in 1+1 dimensions). Fortunately this problem is formal rather than fundamental. One can always augment any interesting (1+1)-dimensional acoustic geometry by two extra flat space dimensions — which is after all exactly how such a geometry would actually be experimentally realised, letting the fluid flow along a thin pipe — and simply phrase physical questions in terms of the plane symmetric 3+1 geometry. Alternatively, one could forget the extra dimensions and simply ask questions based on the geometric acoustics approximation, which, after all, is quite sufficient for dealing with issues of causal structure. We mention in passing that attempts to include vorticity into the physical acoustics formalism lead to a more complicated mathematical structure of which the “effective metric” is only one part.

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