5.1 Hawking radiation

5.1.1 Basics

As is well known, in 1974 Stephen Hawking announced that quantum mechanically black holes should emit radiation with a spectrum approximately that of a black body [270, 271]. We shall not re-derive the existence of Hawking radiation from scratch, but will instead assume a certain familiarity with at least the basics of the phenomenon. (See, for instance, [95Jump To The Next Citation Point] or [309].) The collapse of a distribution of matter will end up forming an evaporating black hole emitting particles from its horizon toward future null 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’s 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 [629]. So, in principle, to be able to reproduce Hawking radiation in a laboratory one would have to fulfill 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 [612Jump To The Next Citation Point]).
  2. To configure the analogue geometry such that it includes some sort of 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 such as to result in the emission of a thermal distribution of field particles. 20

This is a straightforward 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 [606]. 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 caution. This is the 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 on the order of the critical Landau speed, which in these cases coincides with the sound speed [359]) 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 liquid helium three [657Jump 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 [326, 327]. In the case of BECs, a way to suppress the formation ofquantised 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 [48Jump 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 14View Image), but without actually crossing into the supersonic regime [37Jump To The Next Citation Point], softening in this way the appearance of dynamical instabilities.
    View Image

    Figure 14: 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 [613Jump To The Next Citation Point, 35Jump 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 of 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 on 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 ∼ 100nK [48Jump To The Next Citation Point]. 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 backreaction in the Hawking effect. See, for instance, the arguments presented by Helfer ([278Jump To The Next Citation Point, 279Jump To The Next Citation Point, 280Jump 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 UV robustness

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 near 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 [307Jump To The Next Citation Point]. However there are (at least) two problems in doing so:

  1. One has 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 arbitrarily large boosts that is as 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 the 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 ideas regarding the nature of spacetime at ultra-short distances.
  2. Worse, even given these assumptions, “one cannot boost away an s-wave”. That is, given the expected isotropy of Hawking radiation, a boost in any given direction could, at most, tame the trans-Planckian problem only in that specific direction. Indeed, the problem is then not ameliorated in directions orthogonal to the boost, and would become even worse on the opposite side of the black hole.

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 [307Jump To The Next Citation Point, 308Jump To The Next Citation Point, 608Jump 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 [607Jump 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 [307Jump To The Next Citation Point] and references therein, and the comments in [278Jump To The Next Citation Point, 279Jump To The Next Citation Point, 280Jump To The Next Citation Point]). Indeed there are alternative ways through which the effect of the short-scale 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 focused 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),21 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 + Δ(k, K ) , (303 )
where K is the scale that describes the transition to the full microscopic system (what we might call – within this section – the “analogue Planck scale”).

In general, the best one can do is to expand Δ (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 3 ±k ∕K 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 to in the literature as “superluminal” and “subluminal” dispersion relations.

Most of the work on the trans-Planckian problem in the 1990s focused on studying the effect on Hawking radiation due to such modifications of the dispersion relations at high energies in the case of acoustic analogues [307Jump To The Next Citation Point, 308Jump To The Next Citation Point, 608Jump To The Next Citation Point, 148Jump 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., [94Jump To The Next Citation Point, 310, 148Jump To The Next Citation Point, 490Jump To The Next Citation Point]).22 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. It is also important to stress that the mechanism by which the Hawking radiation is recovered is conceptually rather different depending on the type of dispersion relation considered. We concisely summarise here the main results (but see, e.g., [613Jump To The Next Citation Point] for further details).

Subluminal dispersion relations:
This was the case originally considered by Unruh [608Jump To The Next Citation Point],

n 1∕n ω = K (tanh (k∕K ) ) , (304 )
and by Corley and Jacobson [148Jump To The Next Citation Point]
ω2 = k2 − k4∕K2, (305 )
where both dispersion relations are given in the co-moving frame.
View Image

Figure 15: The picture shows a subsonic dispersion relation as a relation ω − vk = ±c |k|Γ k. In particular we plot a dispersion of the type ∘ -------2---2- Γ k = K tanh (k ∕K ) originally employed by Unruh. For supersonic velocities the dispersion relation has two real roots. For subsonic velocities and ω greater than a critical frequency ωc, the dispersion relation has four real roots.

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 Equation (304View Equation)) or becomes imaginary (for Equation (305View Equation)). See Figure 15View Image for a schematic representation of this effect (for Equation (304View Equation)). In the latter case, this can be interpreted as signifying the breakdown of the regime where the dispersion relation (305View 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 a combination of ingoing modes dragged into the black hole by the flow. Stepping further back in time it is seen that such modes are located at larger and larger distances from the horizon, and tend to have very high wave numbers far away at early times. The important point is that one of the modes has “negative norm”. (That is, the norm is negative in the appropriate Klein–Gordon-like inner product [35Jump To The Next Citation Point].) In this way 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 [608Jump To The Next Citation Point, 94Jump To The Next Citation Point, 148, 147Jump To The Next Citation Point, 283, 538, 613Jump To The Next Citation Point, 413Jump To The Next Citation Point, 412Jump To The Next Citation Point] and that the Hawking result, specifically the presence of a Planckian spectrum of particles at a temperature T ≃ κ∕(2π) moving outwards toward the asymptotic region, can be recovered for K ≫ κ where κ is the black-hole surface gravity. In fact, the condition to recover a Planckian spectrum is a bit more subtle. In the mode conversion process every frequency experiences the horizon as located at a different position, and so, having a different surface gravity (a rainbow geometry). To have an approximately Planckian spectrum one would need this frequency-dependent surface gravity to stay almost constant for frequencies below and around κ (see, for example, the analysis in [413Jump To The Next Citation Point, 412Jump To The Next Citation Point, 199Jump To The Next Citation Point]). This condition seems to be easily realizable. In addition, we would like to point out another subtlety of these analogue black-hole configurations: It is necessary that the constant flow velocity reached at the asymptotic region is different from zero. For the particular dispersion relation in Equation (305View Equation), |v | ≥ κ ∕K so that the Planckian form is not cut down at frequencies around κ.

Let us mention here (more details in Section 6.1) that the classical counterpart of the above-described mode-conversion mechanism has recently been observed for the first time in a wave tank experiment [682Jump To The Next Citation Point]. It is remarkable that the exponential factor associated with a black body spectrum is clearly observed even with the inherent noise of the experiment. We will have to wait for the repetition of the same sort of experiment in a more explicitly quantum system to observe the spontaneous production of particles with a Planckian spectrum.

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 (see Figure 16View Image). 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 black hole [147].

View Image

Figure 16: The picture shows a supersonic dispersion relation as a relation ω − vk = ±c |k|Γ k. In particular, we plot a dispersion of the type ∘ ----------- Γ k = 1 + k2 ∕K2. For supersonic velocities the dispersion relation has two real roots. For supersonic velocities and ω less than a critical frequency ωc, the dispersion relation has four real roots.

The understanding of the physics behind the presence of Hawking radiation in superluminal dispersive theories has greatly improve recently through the detailed analysis of 1+1 stationary configurations possessing one black or white horizon connecting two asymptotic regions [413Jump To The Next Citation Point] (for previous work dealing with mode conversion through a horizon, see [386Jump To The Next Citation Point]). One of the asymptotic regions corresponds to the asymptotic region of a black-hole spacetime and is subsonic; the other asymptotic region is supersonic and replaces the internal singularity. Once the appropriate acoustic geometry is defined, this analysis considers a Klein–Gordon field equation in this geometry, modified by a ∂4 x term that gives rise to the quartic dispersion relation 2 2 4 2 ω = k + k ∕K. For this setup it has been shown that the relevant Bogoliubov coefficients have the form ′ βω′,i;ω,j = δ(ω − ω )βω;ij, with β ω;ij a 3 × 3 matrix. To recover from these coefficients a Planckian spectrum of particles at the external asymptotic region, the relevant condition happens to be not κ ≪ K but κ ≪ ωc where ωc = Kf (vint−asym ) and with f(⋅) a specific function of v int−asym, the value of the flow velocity at the internal (or supersonic) asymptotic region [413Jump To The Next Citation Point]. In all cases, for ω > ωc the Bogoliubov coefficients are exactly zero. When vint−asym is just above c = 1, that is, when the flow is only “slightly supersonic”, the function f(⋅) can become very small f(vint− asym) ∝ (vint−asym − 1)3∕2, and thus also the critical frequency ωc at which particle production is cut off. If this happens at frequencies comparable with κ the whole Planckian spectrum will be truncated and distorted. Therefore, to recover a Planckian spectrum of particles at the external asymptotic region one needs to have a noticeable supersonic region.

As with subsonic dispersion, the existence of a notion of rainbow geometry makes modes of different frequency experience different surface gravities. One can think of the distortion of the Planckian spectrum as having a running κ (ω ) that, for these configurations, interpolates between its low energy, or geometric value, κ(ω = 0) = κ0 and a value of zero for ω → ωc. This transition is remarkably sudden for the smooth profiles analyzed. If κ0 ≪ ωc, then κ (ω) will stay constant and equal to κ 0 throughout the relevant part of the spectrum reproducing Hawking’s result. However, in general terms one can say that the spectrum takes into account the characteristic of the profile deep inside the supersonic region (the analogue of the black hole interior). The existence of superluminal modes makes it possible to obtain information from inside the (low energy) horizon.

In these analyses based on stationary configurations, the quantum field was always assumed to be in the in-vacuum state. Is this vacuum state, which has thermal properties, in terms of out-observers? However, it is interesting to realise that the dispersive character of the theory allows one to select for these systems a different, perfectly regular state, which is empty of both incoming and outgoing particles in the external asymptotic region [35]. This state can be interpreted as the regular generalization to a dispersive theory of the Boulware state for a field in a stationary black hole (let us recall that this state is not regular at the horizons). This implies that, in principle, one could set up a semi-classically stable acoustic black hole geometry. Another important result of the analysis in [413] is that stationary white holes do also Hawking radiate and in a very similar way to black holes.

These analyses have been repeated for the specific case of the fluctuations of a BEC [412Jump To The Next Citation Point] with identical qualitative results. The reasons for this is that, although the Bogoliubov–de Gennes system of equations is different from the modified scalar field equation analyzed in previous papers, they share the same quartic dispersion relation. Apart from the more formal treatments of BECs, Carusotto et al. reported [119Jump To The Next Citation Point, 118Jump To The Next Citation Point] the numerical observation of the Hawking effect in simulations in which a black-hole horizon has been created dynamically from an initially homogeneous flow. The observation of the effect has been through the calculation of two-point correlation functions (see Section 5.1.6 below). These simulations strongly suggest that any non-quasi-static formation of a horizon would give place to Hawking radiation.

It is particularly interesting to note that this recovery of the standard result is not always guaranteed in the presence of superluminal dispersion relations. Corley and Jacobson [150Jump 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 [150Jump 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 was also identified in the dynamical analysis carried on in [231Jump To The Next Citation Point, 232Jump To The Next Citation Point, 29Jump To The Next Citation Point] where Bose–Einstein condensate analogue black holes were considered. As we have already mentioned, the spectrum associated with the formation of a black hole horizon in a superluminal dispersive theory depends, in a more or less obvious fashion, on the entire form of the internal velocity profile. In the case in which the internal region contains an additional white horizon, the resulting spectrum is completely changed. These configurations, in addition to a steady Hawking flux, produce a self-amplified particle emission; from this feature arises their name “black-hole lasers”. In the recent analyses in [155Jump To The Next Citation Point, 199Jump To The Next Citation Point] it has been shown that the complete set of modes to be taken into account in these configurations is composed of a continuous sector with real frequencies, plus a discrete sector with complex frequencies of positive imaginary part. These discrete frequencies encode the unstable behaviour of these configurations, and are generated as resonant modes inside the supersonic cavity encompassed between the two horizons.

5.1.3 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 [613Jump 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 [613Jump 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 into 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.

Of course, although several systems can be found in which such conditions hold, it is also possible to show [613] that realistic situations in which at least one of these assumptions is violated can be imagined. Hence, it is still an open question whether real black hole physics does indeed satisfy such conditions, and whether it is therefore robust against modifications induced by the violation of Lorentz invariance.

5.1.4 Source of the Hawking quanta

There is a point of view (not universally shared within the community) that asserts that the trans-Planckian problem also makes it clear that the ray-optics limit cannot be the whole story behind Hawking radiation. Indeed, it is precisely the ray optics approximation that leads to the trans-Planckian problem. Presumably, once one goes beyond ray optics, to the wave optics limit, it will be the region within a wavelength or so of the horizon (possibly the region between the horizon and the unstable circular photon orbit) that proves to be quantum-mechanically unstable and will ultimately be the “origin” of the Hawking photons. If this picture is correct, then the black-hole particle production is a low-frequency and low-wavenumber process. See, for instance, [563Jump To The Next Citation Point, 610Jump To The Next Citation Point, 611Jump To The Next Citation Point]. Work along these lines is continuing.

5.1.5 Which surface gravity?

One issue that has become increasingly important, particularly in view of recent experimental advances, is the question of exactly which particular definition of surface gravity is the appropriate one for controlling the temperature of the Hawking radiation. In standard general relativity with Killing horizons there is no ambiguity, but there is already considerable maneuvering room once one goes to evolving horizons in general relativity, and even more ambiguities once one adopts modified dispersion relations (as is very common in analogue spacetimes).

Already at the level of time-dependent systems in standard general relativity there are two reasonably natural definitions of surface gravity, one in terms of the inaffinity of null geodesics skimming along the event horizon, and another in terms of the peeling properties of those null geodesics that escape the black hole to reach future null infinity. It is this latter definition that is relevant for Hawking-like fluxes from non-stationary systems (e.g., evaporating black holes) and in such systems it never coincides with the inaffinity-based definition of the surface gravity except possibly at asymptotic future-timelike infinity i+. Early comments along these lines can be found in [94, 95Jump To The Next Citation Point]; more recently this point was highlighted in [42, 41Jump To The Next Citation Point].

Regarding analogue models of gravity, the conclusions do not change when working in the hydrodynamic regime (where there is a strict analogy with GR). This point was implicitly made in [37Jump To The Next Citation Point, 41] and clearly stressed in [412Jump To The Next Citation Point]. If we now add modified dispersion relations, there are additional levels of complication coming from the distinction between “group velocity horizons”, and “phase velocity horizons”, and the fact that null geodesics have to be replaced by modified characteristic curves. The presence of dispersion also makes explicit that the crucial notion underlying Hawking emission is the “peeling” properties of null ray characteristics. For instance, the relevant “peeling” surface gravity for determining Hawking fluxes has to be determined locally, in the vicinity of the Killing horizon, and over a finite frequency range. (See for instance [612, 505Jump To The Next Citation Point, 504Jump To The Next Citation Point, 532Jump To The Next Citation Point, 66Jump To The Next Citation Point, 682Jump To The Next Citation Point, 412Jump To The Next Citation Point] for some discussion of this and related issues.) This “surface gravity” is actually an emergent quantity coming from averaging the naive surface gravity (the slope of the cv profile) on a finite region around the would-be Killing horizon associated with the acoustic geometry [200]. Work on these important issues is ongoing.

5.1.6 How to detect Hawking radiation: Correlations

While the robustness of Hawking radiation against UV violations of the acoustic Lorentz invariance seems a well-established feature by now (at least in static or stationary geometries), its strength is indubitably a main concern for a future detection of this effect in a laboratory. As we have seen, the Hawking temperature in acoustic systems is simply related to the gradient of the flow velocity at the horizon (see Equation (61View Equation)). This gradient cannot be made arbitrarily large and, for the hydrodynamic approximation to hold, one actually needs it to be at least a few times the typical coherence length (e.g., the healing length for a BEC) of the superfluid used for the experiment. This implies that in a cold system, with low speed of sound, like a BEC, the expected power loss due to the Hawking emission could be estimated to be on the order of P ≈ 10 −48 W (see, e.g., [48Jump To The Next Citation Point]): arguably too faint to be detectable above the thermal phonon background due to the finite temperature of the condensate (alternatively, one can see that the Hawking temperature is generally below the typical temperature of the BEC, albeit they are comparable and both in the nano-Kelvin range). Despite this, it is still possible that a detection of the spontaneous quantum particle creation can be obtained via some other feature rather than the spectrum of the Hawking flux. A remarkable possibility in this sense is offered by the fact that vacuum particle creation leads generically to a spectrum, which is (almost) Planckian but not thermal (in the sense that all the higher order field correlators are trivial combinations of the two-point one).23

Indeed, particles created by the mode mixing (Bogoliubov) mechanism are generically in a squeezed state (in the sense that the in vacuum appears as a squeezed state when expressed in terms of the out vacuum) [313] and such a state can be distinguished from a real thermal one exactly by the nontrivial structure of its correlators. This discrimination mechanism was suggested a decade ago in the context of dynamical Casimir effect explanations of Sonoluminescence [65], and later envisaged for analogue black holes in [48Jump To The Next Citation Point], but was finally investigated and fully exploited only recently in a stream of papers focussed on the BEC set up [22, 119, 412, 520, 19, 118Jump To The Next Citation Point, 190, 496, 512Jump To The Next Citation Point, 564, 604]. The outcome of such investigations (carried out taking into account the full Bogoliubov spectrum) is quite remarkable as it implies that indeed, while the Hawking flux is generically out-powered by the condensate intrinsic thermal bath, it is, in principle, possible to have a clear cut signature of the Hawking effect by looking at the density-density correlator for phonons on both sides of the acoustic horizon. In fact, the latter will show a definite structure totally absent when the flow is always subsonic or always supersonic. Even more remarkably, it was shown, both via numerical simulations as well as via a detailed analytical investigation, that a realistic finite temperature background does not spoil the long distance correlations which are intrinsic to the Hawking effect (and, indeed, for non-excessively large condensate temperatures the correlations can be amplified).

This seems to suggest that for the foreseeable future the correlation pattern will offer the most amenable route for obtaining a clean signature of the (spontaneous) Hawking effect in acoustic analogues. (For stimulated Hawking emission, see [682Jump To The Next Citation Point].) Finally, it is interesting to add that the correlator analysis can be applied to a wider class of analogue systems, in particular it has been applied to analogue black holes based on cold atoms in ion rings [290], or extended to the study of the particle creation in time-varying external fields (dynamical Casimir effect) in Bose–Einstein condensates (where the time-varying quantity is the scattering length via a Feshbach resonance) [118]. As we shall see, such studies are very interesting for their possible application as cosmological particle production simulations (possibly including Lorentz-violations effects). (See Section 5.4, and [512].)

5.1.7 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 [149Jump To The Next Citation Point, 311Jump 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 (304View 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 (305View 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 (305View Equation) (which in realistic analogue models is emergent and not fundamental anyway), 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 (304View Equation) or (305View Equation) one is using. In fact, the Killing frequency is conserved on a static background; thus, the incoming modes must have the same frequency as the outgoing ones. Hence, in the case of strictly Lorentz invariant dispersion relations there can be no mode-mixing and particle creation. This is why one actually has 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 yet fully capable of solving the trans-Planckian problem. Ultimately, these issues underpin the analysis by Schützhold and Unruh regarding the spatial “origin” of the Hawking quanta [563, 610, 611].

5.1.8 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 [522, 523Jump To The Next Citation Point, 149Jump To The Next Citation Point]. In particular, in [523Jump 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 4.1.5). 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 [523] 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 [149Jump 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 2π∕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 [149Jump 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 with 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.24 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 ∕κ, and hence unnoticeable to long-wavelength modes and relevant only for those with wavelengths on 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 [149, 311].

5.1.9 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:

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