Let us consider a three-dimensional irrotational and inviscid fluid system with a stationary sink-type of flow – a “draining bathtub” flow – (see Figures 1 and 2). The details of the configuration are not important for the following discussion, only the fact that there is a spherically-symmetric fluid flow accelerating towards a central sink, that sink being surrounded by a sphere acting as a sonic horizon. Then, as we have discussed in Section 2, linearizing the Euler and continuity equations leads to a massless scalar field theory over a black-hole–like spacetime. (We are assuming that the hydrodynamic regime remains valid up to arbitrarily-short length scales; for instance, we are neglecting the existence of MDR.) To be specific, let us choose the geometry of the canonical acoustic black-hole spacetime described in [624]:

In this expression we have used the Schwarzschild time coordinate instead of the lab time ; is constant. If we expand the field in spherical harmonics, we obtain the following equation for the radial part of the field: where Here is a “tortoise” coordinate.In a normal mode analysis one requires boundary conditions such that the field is regular everywhere, even at infinity. However, if one is analysing the solutions of the linear field theory as a way of probing the stability of the background configuration, one can consider less restrictive boundary conditions. For instance, one can consider the typical boundary conditions that lead to quasinormal modes: These modes are defined to be purely out-going at infinity and purely in-going at the horizon; but one does not require, for example, the modes to be normalizable. The quasinormal modes associated with this sink configuration have been analysed in [69]. The results found are qualitatively similar to those in the classical linear stability analysis of the Schwarzschild black hole in general relativity [619, 620, 521, 698, 447]. Of course, the gravitational field in general relativity has two dynamical degrees of freedom – those associated with gravitational waves – that have to be analysed separately; these are the “axial” and “polar” perturbations. In contrast, in the present situation we only have scalar perturbations. Nevertheless, the potentials associated with “axial” and “polar” perturbations of Schwarzschild spacetime, and that associated with scalar perturbations of the canonical acoustic black hole, produce qualitatively the same behaviour: There is a series of damped quasinormal modes – proving the linear stability of the system – with higher and higher damping rates.

An important point we have to highlight here is that, although in the linear regime the dynamical behaviour of the acoustic system is similar to general relativity, this is no longer true once one enters the nonlinear regime. The underlying nonlinear equations in the two cases are very different. The differences are so profound, that in the general case of acoustic geometries constructed from compressible fluids, there exist sets of perturbations that, independent of how small they are initially, can lead to the development of shocks, a situation completely absent in vacuum general relativity.

Now, given an approximately stationary, and at the very least metastable, classical black-hole-like configuration, a standard quantum mode analysis leads to the existence of Hawking radiation in the form of phonon emission. This shows, among other things, that quantum corrections to the classical behaviour of the system must make the configuration with a sonic horizon dynamically unstable against Hawking emission. As a consequence, in any system (analogue or general relativistic) with quantum fluctuations that maintain strict adherence to the equivalence principle (no MDR), it must then be impossible to create an isolated truly stationary horizon by merely setting up external initial conditions and letting the system evolve by itself. However, in an analogue system a truly stationary horizon can be set up by providing an external power source to stabilise it against Hawking emission. Once one compensates, by manipulating external forces, for the backreaction effects that in a physical general relativity scenario cause the horizon to shrink or evaporate, one would be able to produce, in principle, an analogue system exhibiting precisely a stationary horizon and a stationary Hawking flux.

Let us describe what happens when one takes into account the existence of MDR. Once again, a wonderful physical system that has MDR explicitly incorporated in its description is the Bose–Einstein condensate. The macroscopic wave function of the BEC behaves as a classical irrotational fluid but with some deviations when short length scales become involved. (For length scales on the order of, or shorter than, the healing length.) What are the effects of the MDR on the dynamical stability of a black-hole-like configuration in a BEC? The stability of a sink configuration in a BEC has been analysed in [231, 232] but taking the flow to be effectively one-dimensional. What these authors found is that these configurations are dynamically unstable: There are modes satisfying the appropriate boundary conditions such that the imaginary parts of their associated frequencies are positive. These instabilities are associated basically with the bound states inside the black hole. The dynamical tendency of the system to evolve is suggestively similar to that in the standard evaporation process of a black hole in semiclassical general relativity.

Before continuing with the discussion of the instability of configurations with horizons, and in order not to cause confusion between the different wording used when talking about the physics of BECs and the emergent gravitational notions on them, let us write down a quite loose but useful translation dictionary:

- The “classical” or macroscopic wave function of the BEC represents the classical spacetime of GR, but only when probed at long-enough wavelengths such that it behaves as pure hydrodynamics.
- The “classical” long-wavelength perturbations to a background solution of the Gross–Pitaevskii equation correspond to classical gravitational waves in GR. Of course, this analogy does not imply that these are spin 2 waves; it only points out that the perturbations are made from the same “substance” as the background configuration itself.
- The macroscopic wave function of the BEC, without the restriction of being probed only at long wavelengths, corresponds to some sort of semiclassical vacuum gravity. Its “classical” behaviour (in the sense that does not involve any probability notion) is already taking into account, in the form of MDR, its underlying quantum origin.
- The Bogoliubov quantum quasiparticles over the “classical” wave function correspond to a further step away from semiclassical gravity in that they are analogous to the existence of quantum gravitons over a (semiclassical) background spacetime.

At this point we would like to remark, once again, that the analysis based on the evolution of a BEC has to be used with care. For example, they cannot directly serve to shed light on what happens in the final stages of the evaporation of a black hole, as the BEC does not fulfil, at any regime, the Einstein equations. Summarizing:

- If the perturbations to the BEC background configuration have “classical seeds” (that is, are describable by the linearised Gross–Pitaevskii equation alone), then, one will have “classical” instabilities.
- If the perturbations have “quantum seeds” (that is, are described by the Bogoliubov equations), then, one will have “quantum” instabilities.

In the light of the acoustic analogies it is natural to ask whether there are other geometric configurations with horizons of interest, besides the sink type of configurations (these are the most similar to the standard description of black holes in general relativity, but probably not the simplest in terms of realizability in a real laboratory; for an entire catalogue of them see [37]). Here, let us mention four effectively one-dimensional configurations: a black hole with two asymptotic regions, a white hole with two asymptotic regions, a black-hole–white-hole in a straight line and the same in a ring (see Figures 17, 18, 19 and 20, respectively).

There are several classical instability analyses of these types of configurations in the literature [231, 232, 386, 29, 155, 199]. In these analyses one looks for the presence or absence of modes with a positive-imaginary-part eigenfrequency, under certain appropriate boundary conditions. The boundary condition in each asymptotic region can be described as outgoing, as in quasi-normal modes, or as convergent, meaning that at a particular instant of time the mode is exponentially damped towards the asymptotic region. Let us mention that in Lorentz invariant theory these two types of conditions are not independent: any unstable mode is at the same time both convergent and outgoing. However, in general, in dispersive theories, once the frequency is extended to the complex plane, these two types of conditions become, at least in principle, independent.

Under outgoing and convergent boundary conditions in both asymptotic regions, in [29] it was concluded that there are no instabilities in any of the straight line (non-ring) configurations. If one relaxed the convergence condition in the downstream asymptotic region, (the region that substitutes the unknown internal region, and so the region that might require a different treatment for more realistic black hole configurations), then the black hole is still stable, while the white hole acquires a continuous region of instability, and the black-hole–white-hole configuration shows up as a discrete set of unstable modes. The white-hole instability was previously identified in [386]. Let us mention here that the stable black-hole configuration has been also analyzed in terms of stable or quasinormal modes in [30]. It was found that, although the particular configurations analyzed (containing idealised step-like discontinuities in the flow) did not posses quasinormal modes in the acoustic approximation, the introduction of dispersion produced a continuous set of quasinormal modes at trans-Planckian frequencies.

Continuing with the analysis of instabilities, in contrast to [29], the more recent analysis in [155, 199] consider
only convergent boundary conditions in both asymptotic regions. They argue that the ingoing contributions
that these modes sometimes have always correspond to waves that do not carry energy, so that they have to
be kept in the analysis, as their ingoing character should not be interpreted as an externally-provoked
instability^{25}.
If this is confirmed, then the appropriate boundary condition for instability analysis under dispersion would
be just the convergent condition, as in non-dispersive theories.

Under these convergent conditions, the authors of [155, 199] show that the previously-considered black-hole and white-hole configurations in BECs are stable. (Let us remark that this does not mean that configurations with a more complicated internal region need be stable.) However, black-hole–white-hole configurations do show a discrete spectrum of instabilities. In these papers, one can find a detailed analysis of the strength of these instabilities, depending on the form and size of the intermediate supersonic region. For instance, it is necessary that the supersonic region acquire a minimum size so that the first unstable mode appears. (This feature was also observed in [29].) When the previous mode analysis is used in the context of a quantum field theory, as we mention in Section 5.1, one is led to the conclusion that black-hole–white-hole configurations emit particles in a self-amplified (or runaway) manner [150, 155, 199]. Although related to Hawking’s process, this phenomenon has a quite different nature. For example, there is no temperature associated with it.

When the black-hole–white-hole configuration is compactified in a ring, it is found that there are regions of stability and instability, depending on the parameters characterizing the configuration [231, 232]. We suspect that the stability regions appear because of specific periodic arrangements of the modes around the ring. Among other reasons, these arrangements are interesting because they could be easier to create in the laboratory with current technology, and their instabilities easier to detect than Hawking radiation itself.

To conclude this subsection, we would like to highlight that there is still much to be learned by studying the different levels of description of an analogue system, and how they influence the stability or instability of configurations with horizons.

Living Rev. Relativity 14, (2011), 3
http://www.livingreviews.org/lrr-2011-3 |
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