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 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:
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:
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
instability25.
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.
http://www.livingreviews.org/lrr-2011-3 |
Living Rev. Relativity 14, (2011), 3
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