As is well known, in 1974 Stephen Hawking announced that quantum mechanically even a spherical distribution of matter collapsing to form a black hole should emit radiation; with a spectrum approximately that of a black body [159, 160]. A black hole will tend to evaporate by emitting particles from its horizon toward infinity. Hawking radiation is a quantum-field-in-curved-space effect. The existence of radiation emission is a kinematic effect that does not rely on Einstein equations. Therefore, one can aim to reproduce it in a condensed matter system. Within standard field theory, a minimal requirement for having Hawking radiation is the existence in the background configuration of an apparent horizon [394]. So, in principle, to be able to reproduce Hawking radiation in a laboratory one would have to fulfil at least two requirements:

- To choose an adequate analogue system; it has to be a quantum analogue model (see Section 4) such that its description could be separated into a classical effective background spacetime plus some standard relativistic quantum fields living on it (it can happen that the quantum fields do not satisfy the appropriate commutation or anti-commutation relations [379]).
- To configure the analogue geometry such that it includes an apparent horizon. That is, within an
appropriate quantum analogue model, the formation of an apparent horizon for the propagation
of the quantum fields should excite the fields as to result in the emission of a thermal distribution
of field particles.
^{22}.

This is a straight and quite naive translation of the standard Hawking effect derivation to the condensed matter realm. However, in reality, this translation process has to take into account additional issues that, as we are trying to convey, instead of problems, are where the interesting physics lies.

- The effective description of the quantum analogue systems as fields over a background geometry breaks down when probed at sufficiently short length scales. This could badly influence the main features of Hawking radiation. In fact, immediately after the inception of the idea that black holes radiate, it was realised that there was a potential problem with the calculation [375]. It strongly relies on the validity of quantum field theory on curved backgrounds up to arbitrary high energies. Following a wave packet with a certain frequency at future infinity backwards in time, we can see that it had to contain arbitrarily large frequency components with respect to a local free fall observer (well beyond the Planck scale) when it was close to the horizon. In principle any unknown physics at the Planck scale could strongly influence the Hawking process so that one should view it with suspicion. This is the so-called trans-Planckian problem of Hawking radiation. To create an analogue model exhibiting Hawking radiation will be, therefore, equivalent to giving a solution to the trans-Planckian problem.
- In order to clearly observe Hawking radiation, one should first be sure that there is no other source of instabilities in the system that could mask the effect. In analogue models such as liquid Helium or BECs the interaction of a radial flow (with speed of the order of the critical Landau speed, which in these cases coincides with the sound speed [213]) with the surface of the container (an electromagnetic potential in the BECs case) might cause the production of rotons and quantised vortices, respectively. Thus, in order to produce an analogue model of Hawking radiation, one has to be somewhat ingenious. For example in the liquid Helium case, instead of taking acoustic waves in a supersonic flow stream as the analogue model, it is preferable to use as analogue model ripplons in the interface between two different phases, A and B phases, of Helium-3 [415]. Another option is to start from a moving domain wall configuration. Then, the topological stability of the configuration prevents its destruction when creating a horizon [199, 200]. In the case of BECs a way to suppress the formation of quantised vortices is to take effectively one-dimensional configurations. If the transverse dimension of the flow is smaller than the healing length then there is no space for the existence of a vortex [19]. In either liquid Helium or BECs, there is also the possibility of creating an apparent horizon by rapidly approaching a critical velocity profile (see Figure 10), but without actually crossing into the supersonic regime [13], softening in this way the appearance of dynamical instabilities.
- Real analogue models cannot, strictly speaking, reproduce eternal black-hole configurations. An analogue model of a black hole has always to be created at some finite laboratory time. Therefore, one is forced to carefully analyse the creation process, as it can greatly influence the Hawking effect. Depending on the procedure of creation, one could end up in quite different quantum states for the field and only some of them might exhibit Hawking radiation. This becomes more important when considering that the analogue models incorporate modified dispersion relations. An inappropriate preparation, together with modified dispersion relation effects, could completely eliminate Hawking radiation [380].
- Another important issue is the need to characterise “how quantum” a specific analogue model is. Even though, strictly speaking, one could say that any system undergoes quantum fluctuations, the point is how important they are in its description. In trying to build an analogue model of Hawking’s quantum effect, the relative value of Hawking temperature with respect to the environment is going to tell us whether the system can be really thought as a quantum analogue model or as effectively classical. For example, in our standard cosmological scenario, for a black hole to radiate at temperatures higher than that of the Cosmic Microwave Background, K, the black hole should have a diameter of the order of micrometers or less. We would have to say that such black holes are no longer classical, but semiclassical. The black holes for which we have some observational evidence are of much higher mass and size, so their behaviour can be thought of as completely classical. Estimates of the Hawking temperature reachable in BECs yield [19]. This has the same order of magnitude of the temperature as the BECs themselves. This is telling us that regarding the Hawking process, BECs can be considered to be highly-quantum analogue models.
- There is also the very real question of whether one should trust semiclassical calculations at all when it comes to dealing with back-reaction in the Hawking effect. See for instance the arguments presented by Helfer ([167, 168, 169], and references therein).

Because of its importance, let us now review what we know about the effects of high-energy dispersion relations on the Hawking process.

We saw in the introduction to this section that the trans-Planckian problem of Hawking radiation was one of the strongest motivations for the modern research into analogue models of gravity. In fact it was soon realised that such models could provide a physical framework within which a viable solution of the problem could be found. Let us explain why and how.

As we have said, the requirement of a reservoir of ultra-high frequency modes nearby the horizon seems to indicate a possible (and worrisome) sensitivity of the black hole radiation to the microscopic structure of spacetime. Actually by assuming exact Lorentz invariance one could in principle always locally transform the problematic ultra high frequency modes to low energy ones via some appropriate Lorentz transformation [185]. However in doing so it would have to rely on the physics of reference frames moving ultra fast with respect to us, as the reference frame needed would move arbitrarily close to the speed of light. Hence we would have to apply Lorentz invariance in a regime of arbitrary large boosts yet untested and in principle never completely testable given the non-compactness of the boost subgroup. The assumption of an exact boost symmetry is linked to a scale-free nature of spacetime given that unbounded boosts expose ultra-short distances. Hence the assumption of exact Lorentz invariance needs, in the end, to rely on some idea on the nature of spacetime at ultra-short distances.

It was this type of reasoning that led in the nineties to a careful reconsideration of the crucial ingredients required for the derivation of Hawking radiation [185, 186, 377]. In particular investigators explored the possibility that spacetime microphysics could provide a short distance, Lorentz-breaking cutoff, but at the same time leave Hawking’s results unaffected at energy scales well below that set by the cutoff.

Of course ideas about a possible cutoff imposed by the discreteness of spacetime at the Planck scale had already been discussed in the literature well before Unruh’s seminal paper [376]. However such ideas were running into serious difficulties given that a naive short distance cutoff posed on the available modes of a free field theory results in a complete removal of the evaporation process (see e.g., Jacobson’s article [185] and references therein, and the comments in [167, 168, 169]). Indeed there are alternative ways through which the effect of the short scales physics could be taken into account, and analogue models provide a physical framework where these ideas could be put to the test. In fact analogue models provide explicit examples of emergent spacetime symmetries, they can be used to simulate black hole backgrounds, they may be endowed with quantizable perturbations and, in most of the cases, they have a well known microscopic structure. Given that Hawking radiation can be, at least in principle, simulated in such systems one might ask how and if the trans-Planckian problem is resolved in these cases.

Most of the work on the trans-Planckian problem in the nineties focussed on studying
the effect on Hawking radiation due to such modifications of the dispersion relations at
high energies in the case of acoustic analogues [185, 186, 377, 378, 88], and the question
of whether such phenomenology could be applied to the case of real black holes (see
e.g., [50, 188, 88, 299]).^{24}
In all the aforementioned works Hawking radiation can be recovered under some suitable assumptions, as
long as neither the black hole temperature nor the frequency at which the spectrum is considered are too
close to the scale of microphysics . However, the applicability of these assumptions to the real case of
black hole evaporation is an open question. Also, in the case of the analogue models the mechanism by
which the Hawking radiation is recovered is not always the same. We concisely summarise here the main
results (but see e.g., [380] for further details).

The key feature is that in the presence of a subluminal modification the group velocity of the modes increases with only up to some turning point (which is equivalent to saying that the group velocity does not asymptote to , which could be the speed of sound, but instead is upper bounded). For values of beyond the turning point the group velocity decreases to zero (for (248)) or becomes imaginary (for (249)). In the latter case this can be interpreted as signifying the breakdown of the regime where the dispersion relation (249) can be trusted. The picture that emerged from these analyses concerning the origin of the outgoing Hawking modes at infinity is quite surprising. In fact, if one traces back in time an outgoing mode, as it approaches the horizon it decreases its group velocity below the speed of sound. At some point before reaching the horizon, the outgoing mode will appear as an ingoing one dragged into the black hole by the flow. Stepping further back in time it is seen that such a mode was located at larger and larger distances from the horizon, and tends to a very high energy mode far away at early times. In this case one finds what might be called a “mode conversion”, where the origin of the outgoing Hawking quanta seems to originate from ingoing modes which have “bounced off” the horizon without reaching trans-Planckian frequencies in its vicinities. Several detailed analytical and numerical calculations have shown that such a conversion indeed happens [378, 50, 88, 87, 170, 330, 380] and that the Hawking result can be recovered for where is the black hole surface gravity.

These conclusions regarding the impossibility of clearly predicting the origin at early times of the modes ultimately to be converted into Hawking radiation are not specific to the particular dispersion relations (248) or (249) one is using. The Killing frequency is in fact conserved on a static background, thus the incoming modes must have the same frequency as the outgoing ones, hence there can be no mode-mixing and particle creation. This is why one has actually to assume that the WKB approximation fails in the proximity of the horizon and that the modes are there in the vacuum state for the co-moving observer. In this sense the need for these assumptions can be interpreted as evidence that these models are not fully capable of solving the trans-Planckian problem.

An alternative avenue was considered in [89]. There a lattice description of the background was used for
imposing a cutoff in a more physical way with respect to the continuum dispersive models previously
considered. In such a discretised spacetime the field takes values only at the lattice points, and wavevectors
are identified modulo where is the lattice characteristic spacing, correspondingly one obtains a
sinusoidal dispersion relation for the propagating modes. Hence the problem of recovering a smooth
evolution of incoming modes to outgoing ones is resolved by the intrinsically regularised behaviour of the
wave vectors field. In [89] the authors explicitly considered the Hawking process for a discretised version
of a scalar field, where the lattice is associated to the free-fall coordinate system (taken as
the preferred system). With such a choice it is possible to preserve a discrete lattice spacing.
Furthermore the requirement of a fixed short distance cutoff leads to the choice of a lattice spacing
constant at infinity, and that the lattice points are at rest at infinity and fall freely into the black
hole.^{25}
In this case the lattice spacing grows in time and the lattice points spread in space as they fall toward the
horizon. However this time dependence of the lattice points is found to be of order , and hence
unnoticeable to long wavelength modes and relevant only for those with wavelengths of the order of the
lattice spacing. The net result is that on such a lattice long wavelength outgoing modes are seen to originate
from short wavelength incoming modes via a process analogous to the Bloch oscillations of accelerated
electrons in crystals [89, 189].

Although closely related, as we will soon see, we have to distinguish carefully between the mode analysis of a linear field theory (with or without modified dispersion relations - MDR) over a fixed background and the stability analysis of the background itself.

Let us consider a three-dimensional irrotational and inviscid fluid system with a stationary sink-type of flow (see Figures 1, 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 [389]:

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 have 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 to this sink configuration have been analysed in [31]. The results found are qualitatively similar to those in the classical linear stability analysis of the Schwarzschild black hole in general relativity [384, 385, 317, 431, 267]. 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 in 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 non-linear 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, independently 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. Moreover, in an analogue system 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 external means - any truly stationary horizon must be provided with an external power source to stabilise it against Hawking emission. That is, in an analogue system one could in principle, by manipulating external forces, compensate for the backreaction effects that in a physical general relativity scenario cause the horizon to shrink (or evaporate) and thus become non-stationary.

Let us describe what happens when one takes into account the existence of MDR. 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 of 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 [136, 137] but taking the flow to be effectively one-dimensional. What they 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. This observation alone could make us question whether the first signatures of a quantum theory underlying general relativity might show up in precisely this manner. Interest in this question is reinforced by a specific analysis in the “loop quantum gravity” approach to quantizing gravity that points towards the existence of fundamental MDR at high energies [134]. The formulation of effective gravitational theories that incorporate some sort of MDR at high energies is currently under investigation (see for example [247, 2]); these exciting developments are however beyond the scope of this review.

Before continuing with the discussion on the stability 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.
- 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 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.

Now continuing the discussion, what happens when treating the perturbations to the background BEC flow as quantum excitations (Bogoliubov quasiparticles)? What we certainly know is that the analysis of modes in a collapsing-to-form-a-black-hole background spacetime leads to the existence of radiation emission very much like Hawking emission [50, 87, 91]. (This is why it is said that Hawking’s process is robust against modifications of the physics at high energies.) The comparison of these calculations with that of Hawking (without MDR), tells us that the main modification to Hawking’s result is that now the Hawking flux of particles would not last forever but would vanish after a long enough time (this is why, in principle, we can dynamically create a configuration with a sonic horizon in a BEC). The emission of quantum particles reinforces the idea that the supersonic sink configurations are unstable.

Summarizing:

- If the perturbation 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 [13]). Here, we are going to specifically mention two effectively one-dimensional black hole-white hole configurations, one in a straight line and one in a ring (see Figures 11 and 12, respectively).

A quantum mode analysis of the black hole-white hole configuration in a straight line, taking into account the existence of superluminal dispersion relations (similar to those in a BEC), led to the conclusion that the emission of particles in this configuration proceeds in a self-amplified (or runaway) manner [90]. We can understand this effect as follows: At the black horizon a virtual pair of phonons are converted into real phonons, the positive energy phonon goes towards infinity while the negative energy pair falls beyond the black horizon. However, the white horizon makes this negative energy phonon bounce back towards the black horizon (thanks to superluminal motion) stimulating the emission of additional phonons. Although related to Hawking’s process this phenomenon has a quite different nature: For example, there is no temperature associated with it. A stability analysis of a configuration like this in a BEC would lead to strong instabilities. This same configuration, but compactified into a ring configuration, has been dynamically analysed in [136, 137]. What they found is that there are regions of stability and instability depending on the parameters characterizing the configuration. 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 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.

In addition, among the many papers using analogue spacetimes as part of their background mindset when addressing these issues we mention:

- “Top down” calculations of Hawking radiation starting from some idealised model of quantum gravity [1, 201, 276, 277].
- “Bottom up” calculations of Hawking radiation starting from curved space quantum field theory [22, 23, 61, 62, 71, 111, 112, 128, 253, 254, 255, 256, 257, 298, 321].
- Trans-Hawking versions of Hawking radiation, either as reformulations of the physics, or as alternative scenarios [30, 68, 130, 131, 150, 156, 167, 168, 169, 233, 289, 291, 325, 329, 333, 334, 340, 341, 350, 364].
- Black hole entropy viewed in the light of analogue spacetimes [94].
- Hawking radiation interpreted as a statement about particles travelling along complex spacetime trajectories [292, 350, 360].

http://www.livingreviews.org/lrr-2005-12 |
© Max Planck Society and the author(s)
Problems/comments to |