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:
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.
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 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:
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:
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).
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:
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