Go to previous page Go up Go to next page

5.1 Hawking radiation

5.1.1 Basics

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 [159160]. 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:

  1. 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]).
  2. 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.

  1. 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.
  2. 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 [415Jump To The Next Citation Point]. 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 [199200]. 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 [19Jump To The Next Citation Point]. 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 10View Image), but without actually crossing into the supersonic regime [13Jump To The Next Citation Point], softening in this way the appearance of dynamical instabilities.
    View Image

    Figure 10: Velocity profile for a left going flow; the profile is dynamically modified with time so that it reaches the profile with a sonic point at the asymptotic future.
  3. 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 [380Jump To The Next Citation Point].
  4. 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, ~~ 3 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 T ~ 100 nK [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.
  5. 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 ([167Jump To The Next Citation Point168Jump To The Next Citation Point169Jump To The Next Citation Point], 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.

5.1.2 Trans-Planckian problem

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 [185Jump To The Next Citation Point]. 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 [185Jump To The Next Citation Point186Jump To The Next Citation Point377Jump To The Next Citation Point]. 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 [376Jump To The Next Citation Point]. 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 [185Jump To The Next Citation Point] and references therein, and the comments in [167Jump To The Next Citation Point168Jump To The Next Citation Point169Jump To The Next Citation Point]). 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. Modified dispersion relations:
The general feature that most of the work on this subject has focussed on is that in analogue models the quasi-particles propagating on the effective geometry are actually collective excitations of atoms. This generically implies that their dispersion relation will be a relativistic one only at low energies (large scales),23 and in each case there will be some short length scale (e.g., intermolecular distance for a fluid, coherence length for a superfluid, healing length for a BEC) beyond which deviations will be non-negligible. In general such microphysics induced corrections to the dispersion relation take the form

E2 = c2 (m2c2 + k2 + D(k, K)) (247)
where K is the scale that describes the transition to the full microscopic system (what we might call the “analogue Planck scale”). In general the best one can do is to expand D(k, K) around k = 0, obtaining an infinite power series (of which it will be safe to retain only the lowest order terms), although in some special models (like BEC) the series is automatically finite due to intrinsic properties of the system. (In any case one can see that most of the analogue models so far considered lead to modifications of the form ± k3/K2 or ± k4/K2.) Depending on the sign in front of the modification the group velocity at high energy can be larger (+) or smaller (-) than the low energy speed of light c. These cases are usually referred in the literature as “superluminal” and “subluminal” dispersion relations.

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 [185Jump To The Next Citation Point186Jump To The Next Citation Point377Jump To The Next Citation Point378Jump To The Next Citation Point88Jump To The Next Citation Point], and the question of whether such phenomenology could be applied to the case of real black holes (see e.g., [50Jump To The Next Citation Point18888Jump To The Next Citation Point299Jump To The Next Citation Point]).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 K. 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., [380Jump To The Next Citation Point] for further details). Subluminal dispersion relations:
This was the case originally considered by Unruh [377Jump To The Next Citation Point378Jump To The Next Citation Point],

1/n w = K (tanh(k/K)n) , (248)
and by Corley and Jacobson [88Jump To The Next Citation Point]
2 2 4 2 w = k - k /K , (249)
where both dispersion relations are given in the co-moving frame.

The key feature is that in the presence of a subluminal modification the group velocity of the modes increases with k only up to some turning point (which is equivalent to saying that the group velocity does not asymptote to c, which could be the speed of sound, but instead is upper bounded). For values of k beyond the turning point the group velocity decreases to zero (for (248View Equation)) or becomes imaginary (for (249View Equation)). In the latter case this can be interpreted as signifying the breakdown of the regime where the dispersion relation (249View Equation) 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 [378Jump To The Next Citation Point50Jump To The Next Citation Point8887Jump To The Next Citation Point170330380Jump To The Next Citation Point] and that the Hawking result can be recovered for k « K where k is the black hole surface gravity. Superluminal dispersion relations:
The case of a superluminal dispersion relation is quite different and, as we have seen, has some experimental interest given that this is the kind of dispersion relation that arises in some promising analogue models (e.g., BECs). In this situation, the outgoing modes are actually seen as originating from behind the horizon. This implies that these modes somehow originate from the singularity (which can be a region of high turbulence in acoustic black hole analogues), and hence it would seem that not much can be said in this case. However it is possible to show that if one imposes vacuum boundary conditions on these modes near the singularity, then it is still possible to recover the Hawking result, i.e., thermal radiation outside the hole [87Jump To The Next Citation Point]. It is particularly interesting to note that this recovering of the standard result is not always guaranteed in the presence of superluminal dispersion relations. Corley and Jacobson [90Jump To The Next Citation Point] in fact discovered a very peculiar type of instability due to such superluminal dispersion in the presence of black holes with inner horizons. The net result of the investigation carried out in [90Jump To The Next Citation Point] is that the compact ergo-region characterizing such configurations is unstable to self-amplifying Hawking radiation. The presence of such an instability seems to be confirmed by the analysis carried on in [136Jump To The Next Citation Point137Jump To The Next Citation Point] where a Bose-Einstein condensate analogue black hole was considered. General conditions for Hawking radiation:
Is it possible to reduce the rather complex phenomenology just described to a few basic assumptions that must be satisfied in order to recover Hawking radiation in the presence of Lorentz violating dispersion relations? A tentative answer is given in [380Jump To The Next Citation Point], where the robustness of the Hawking result is considered for general modified (subluminal as well as superluminal) dispersion relations. The authors of [380Jump To The Next Citation Point] assume that the geometrical optics approximation breaks down only in the proximity of the event horizon (which is equivalent to saying that the particle production happens only in such a region). Here, the would-be trans-Planckian modes are converted in sub-Planckian ones. Then, they try to identify the minimal set of assumptions that guarantees that such “converted modes” are generated in their ground states (with respect to a freely falling observer), as this is a well known condition in order to recover Hawking’s result. They end up identifying three basic assumptions that guarantee such emergence of modes in the ground state at the horizon. First, the preferred frame selected by the breakdown of Lorentz invariance must be the freely falling one instead of the rest frame of the static observer at infinity (which coincides in this limit with the laboratory observer). Second, the Planckian excitations must start off in the ground state with respect to freely falling observers. Finally, they must evolve in an adiabatic way (i.e., the Planck dynamics must be much faster than the external sub-Planckian dynamics). Of course, although several systems can be found in which such conditions hold, it is also possible to show [380] that realistic situations in which at least one of these assumptions is violated can be imagined. It is hence still an open question whether real black hole physics does indeed satisfy such conditions, and whether it is hence robust against modifications induced by the violation of Lorentz invariance. Open issues:
In spite of the remarkable insight given by the models discussed above (based on modified dispersion relations) it is not possible to consider them fully satisfactory in addressing the trans-Planckian problem. In particular it was soon recognised [89Jump To The Next Citation Point189Jump To The Next Citation Point] that in this framework it is not possible to explain the origin of the short wavelength incoming modes which are “progenitors” of the outgoing modes after bouncing off in the proximity of the horizon. For example, in the Unruh model (248View Equation), one can see that if one keeps tracking a “progenitor” incoming mode back in time, then its group velocity (in the co-moving frame) drops to zero as its frequency becomes more and more blue shifted (up to arbitrarily large values), just the situation one was trying to avoid. This is tantamount to saying that the trans-Planckian problem has been moved from the region near the horizon out to the region near infinity. In the Corley-Jacobson model (249View Equation) this unphysical behaviour is removed thanks to the presence of the physical cutoff K. However it is still true that in tracking the incoming modes back in time one finally sees a wave packet so blue shifted that |k|= K. At this point one can no longer trust the dispersion relation (249View Equation) (which anyway in realistic analogue models is emergent and not fundamental), and hence the model has no predictive power regarding the ultimate origin of the relevant incoming modes.

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 (248View Equation) or (249View Equation) 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. Solid state and lattice models:
It was to overcome this type of issue that alternative ways of introducing an ultra-violet cutoff due to the microphysics were considered [318319Jump To The Next Citation Point89Jump To The Next Citation Point]. In particular in [319Jump To The Next Citation Point] the transparency of the refractive medium at high frequencies has been used to introduce an effective cutoff for the modes involved in Hawking radiation in a classical refractive index analogue model (see Section 4.1.3 of this review). In this model an event horizon for the electromagnetic field modes can be simulated by a surface of singular electric and magnetic permeabilities. This would be enough to recover Hawking radiation but it would imply the unphysical assumption of a refractive index which is valid at any frequency. However it was shown in [319] that the Hawking result can be recovered even in the case of a dispersive medium which becomes transparent above some fixed frequency K (which we can imagine as the plasma frequency of the medium), the only (crucial) assumption being again that the “trans-Planckian” modes with k > K are in their ground state near the horizon.

An alternative avenue was considered in [89Jump To The Next Citation Point]. 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 2p/l where l 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 [89Jump To The Next Citation Point] 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 1/k, 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 [89189].

5.1.3 Horizon stability

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 1View Image, 2View Image). 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 [389Jump To The Next Citation Point]:

( ) ( ) 2 2 r40 2 r40 - 1 2 2( 2 2 2) ds = - c 1 - r4 dt + 1 - r4 dr + r dh + sin h df . (250)
In this expression we have used the Schwarzschild time coordinate t instead of the lab time t; c is constant. If we expand the field in spherical harmonics,
-iwt xlm(r) flm(t,r,h,f) =_ e ---r--- Ylm(h, f), (251)
we obtain the following equation for the radial part of the field:
2 ( 2 ) w-- x = - -d---+ V (r) x; (252) c2 dr*2 l
( 4) [ 4] V (r) = 1 - r0 l(l-+-1)-+ 4r0- . (253) l r4 r2 r6
r* =_ r - (r /4){ln(r + r)/(r - r ) + 2arctan r/r } (254) 0 0 0 0
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 [31Jump To The Next Citation Point]. The results found are qualitatively similar to those in the classical linear stability analysis of the Schwarzschild black hole in general relativity [384385317431267]. 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 [136Jump To The Next Citation Point137Jump To The Next Citation Point] 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 [134Jump To The Next Citation Point]. The formulation of effective gravitational theories that incorporate some sort of MDR at high energies is currently under investigation (see for example [2472]); 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:

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 [508791]. (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.


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 11View Image and 12View Image, respectively).

View Image

Figure 11: One-dimensional velocity profile with a black hole horizon and a white hole horizon.
View Image

Figure 12: One-dimensional velocity profile in a ring; the fluid flow exhibits two sonic horizons, one of black hole type and the other of white hole type.
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 [136137]. 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.

5.1.4 Analogue spacetimes as background gestalt

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

  Go to previous page Go up Go to next page