Analogue Gravity
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Abstract
Analogue gravity is a research programme which investigates analogues of general relativistic gravitational fields within other physical systems, typically but not exclusively condensed matter systems, with the aim of gaining new insights into their corresponding problems. Analogue models of (and for) gravity have a long and distinguished history dating back to the earliest years of general relativity. In this review article we will discuss the history, aims, results, and future prospects for the various analogue models. We start the discussion by presenting a particularly simple example of an analogue model, before exploring the rich history and complex tapestry of models discussed in the literature. The last decade in particular has seen a remarkable and sustained development of analogue gravity ideas, leading to some hundreds of published articles, a workshop, two books, and this review article. Future prospects for the analogue gravity programme also look promising, both on the experimental front (where technology is rapidly advancing) and on the theoretical front (where variants of analogue models can be used as a springboard for radical attacks on the problem of quantum gravity).
1 Introduction
Analogies have played a very important role in physics and mathematics — they provide new ways of looking at problems that permit crossfertilization of ideas among different branches of science. A carefully chosen analogy can be extremely useful in focusing attention on a specific problem, and in suggesting unexpected routes to a possible solution. In this review article we will focus on “analogue gravity”, the development of analogies (typically but not always based on condensed matter physics) to probe aspects of the physics of curved spacetime — and in particular to probe aspects of curved space quantum field theory, and to obtain lessons of potential relevance on the road towards a theory of quantum gravity.And I cherish more than anything else the Analogies, my most trustworthy masters.
They know all the secrets of Nature, and they ought to be least neglected in Geometry.
— Johannes Kepler
The most wellknown of these analogies is the use of sound waves in a moving fluid as an analogue for light waves in a curved spacetime. Supersonic fluid flow can then generate a “dumb hole”, the acoustic analogue of a “black hole”, and the analogy can be extended all the way to mathematically demonstrating the presence of phononic Hawking radiation from the acoustic horizon. This particular example provides (at least in principle) a concrete laboratory model for curvedspace quantum field theory in a realm that is technologically accessible to experiment.
There are many other “analogue models” that may be useful for this or other reasons — some of the analogue models are interesting for experimental reasons, others are useful for the way they provide new light on perplexing theoretical questions. The information flow is, in principle, bidirectional and sometimes insights developed within the context of general relativity can be used to understand aspects of the analogue model.
Of course, analogy is not identity, and we are in no way claiming that the analogue models we consider are completely equivalent to general relativity — merely that the analogue model (in order to be interesting) should capture and accurately reflect a sufficient number of important features of general relativity (or sometimes special relativity). The list of analogue models is extensive, and in this review we will seek to do justice both to the key models, and to the key features of those models.
1.1 Overview

Discuss the flowing fluid analogy in some detail.

Summarise the history and motivation for various analogue models.

Discuss the many physics issues various researchers have addressed.

Provide a representative catalogue of extant models.

Discuss the main physics results obtained to date, both on the theoretical and experimental sides.

Outline some of the many possible directions for future research.

Summarise the current state of affairs.
1.2 Motivations

Partly to use condensed matter to gain insight into classical general relativity.

Partly to use condensed matter to gain insight into curvedspace quantum field theory.

Partly to develop an observational window on curvedspace quantum field theory.

Partly to use classical general relativity to gain insight into condensed matter physics.

Partly to gain insight into new and radicallydifferent ways of dealing with “quantum/emergent gravity”.
1.3 Going further

The book “Artificial Black Holes”, edited by Novello, Visser, and Volovik [470].

The archival website for the “Analogue models” workshop: — http://www.msor.victoria.ac.nz/∼visser/Analog/

The book “The Universe in a Helium Droplet”, by Volovik [660].

The Physics Reports article, “Superfluid analogies of cosmological phenomena”, by Volovik [655].

The review article by Balbinot, Fabbri, Fagnocchi, and Parentani [21] (focussing largely on backreaction and shortdistance effects).

The Lecture Notes in Physics volume on “Quantum Analogies” edited by Unruh and Schützhold [614].
2 The Simplest Example of an Analogue Spacetime
Acoustics in a moving fluid is the simplest and cleanest example of an analogue model [607, 622, 626, 624]. The basic physics is simple, the conceptual framework is simple, and specific computations are often simple (whenever, that is, they are not impossibly hard).^{1}
2.1 Background

Geometrical acoustics.

Physical acoustics.
2.2 Geometrical acoustics

The speed of sound c, relative to the fluid, is well defined.

The velocity of the fluid v, relative to the laboratory, is well defined.
The virtues of the geometric approach are its extreme simplicity and the fact that the basic structure is dimensionindependent. Moreover, this logic rapidly (and relatively easily) generalises to more complicated physical situations.^{3}
2.3 Physical acoustics
Comment. It is quite remarkable that even though the underlying fluid dynamics is Newtonian, nonrelativistic, and takes place in flat spaceplustime, the fluctuations (sound waves) are governed by a curved (3+1)dimensional Lorentzian (pseudoRiemannian) spacetime geometry. For practitioners of general relativity this observation describes a very simple and concrete physical model for certain classes of Lorentzian spacetimes, including (as we shall later see) black holes. On the other hand, this discussion is also potentially of interest to practitioners of continuum mechanics and fluid dynamics in that it provides a simple concrete introduction to Lorentzian differential geometric techniques.
Since this is a subtle issue that we have seen cause considerable confusion in the past, let us be even more explicit by asking the rhetorical question: “How can we tell the difference between a wind gust and a sound wave?” The answer is that the difference is to some extent a matter of convention — sufficiently lowfrequency longwavelength disturbances (wind gusts) are conventionally lumped in with the average bulk motion. Higherfrequency, shorterwavelength disturbances are conventionally described as acoustic disturbances. If you wish to be hypertechnical, we can introduce a highpass filter function to define the bulk motion by suitably averaging the exact fluid motion. There are no deep physical principles at stake here — merely an issue of convention. The place where we are making a specific physical assumption that restricts the validity of our analysis is in the requirement that the amplitude of the highfrequency shortwavelength disturbances be small. This is the assumption underlying the linearization programme, and this is why sufficiently highamplitude sound waves must be treated by direct solution of the full equations of fluid dynamics.
We have presented the theorem and proof, which closely follows the discussion in [624], in considerable detail because it is a standard template that can be readily generalised in many ways. This discussion can then be used as a starting point to initiate the analysis of numerous and diverse physical models.
2.4 General features of the acoustic metric

Observe that the signature of this effective metric is indeed (−, +, +, +), as it should be to be regarded as Lorentzian.
 Observe that in physical acoustics it is the inverse metric density,that is of more fundamental significance for deriving the wave equation than is the metric g_{ μν } itself. (This observation continues to hold in more general situations where it is often significantly easier to calculate the tensor density f^{ μν } than it is to calculate the effective metric g_{ μν }.)$${f^{\mu \nu}} = \sqrt { g} {g^{\mu \nu}}$$(36)
 It should be emphasised that there are two distinct metrics relevant to the current discussion:
 The physical spacetime metric is, in this case, just the usual flat metric of Minkowski space:$${\eta _{\mu \nu}} \equiv {({\rm{diag}}[  c_{{\rm{light}}}^2,1,1,1])_{\mu \nu}}.$$(37)
(Here c_{light} is the speed of light in vacuum.) The fluid particles couple only to the physical metric η_{ μν }. In fact the fluid motion is completely nonrelativistic, so that ‖v_{0}‖ ≪ c_{light}, and it is quite sufficient to consider Galilean relativity for the underlying fluid mechanics.

Sound waves on the other hand, do not “see” the physical metric at all. Acoustic perturbations couple only to the effective acoustic metric g_{ μν }.


It is quite remarkable that (to the best of our knowledge) a version of this acoustic metric was first derived and used in Moncrief’s studies of the relativistic hydrodynamics of accretion flows surrounding black holes [448]. Indeed, Moncrief was working in the more general case of a curved background “physical” metric, in addition to a curved “effective” metric. We shall come back to this work later on, in our historical section. (See also Section 4.1.2.)

However, the geometry determined by the acoustic metric does inherit some key properties from the existence of the underlying flat physical metric. For instance, the topology of the manifold does not depend on the particular metric considered. The acoustic geometry inherits the underlying topology of the physical metric — ordinary \(\mathfrak{R}^4\) — with possibly a few regions excised (due to whatever hardwall boundary conditions one might wish to impose on the fluid). In systems constrained to have effectively less than 3 spacelike dimensions one can reproduce more complicated topologies (consider, for example, an effectively onedimensional flow in a tubular ring).
 Furthermore, the acoustic geometry automatically inherits from the underlying Newtonian time parameter, the important property of “stable causality” [275, 670]. Note that$${g^{\mu \nu}}({\nabla _\mu}t)({\nabla _\nu}t) =  {1 \over {{\rho _0}c}} < 0.$$(38)
This precludes some of the more entertaining causalityrelated pathologies that sometimes arise in general relativity. (For a general discussion of causal pathologies in general relativity, see, for example, [275, 272, 273, 125, 274, 630]).

Other concepts that translate immediately are those of “ergoregion”, “trapped surface”, “apparent horizon”, and “event horizon”. These notions will be developed more fully in the following subsection.
 The properly normalised fourvelocity of the fluid isso that$${V^\mu} = {{(1;v_0^i)} \over {\sqrt {{\rho _0}c}}},$$(39)This fourvelocity is related to the gradient of the natural time parameter by$${g_{\mu \nu}}{V^\mu}{V^\nu} = g(V,V) =  1{.}$$(40)Thus the integral curves of the fluid velocity field are orthogonal (in the Lorentzian metric) to the constant time surfaces. The acoustic proper time along the fluid flow lines (streamlines) is$${\nabla _\mu}t = (1,0,0,0);\quad {\nabla ^\mu}t =  {{(1;v_0^i)} \over {{\rho _0}c}} =  {{{V^\mu}} \over {\sqrt {{\rho _0}c}}}.$$(41)and the integral curves are geodesics of the acoustic metric if and only if ρ_{0} c is position independent.$$\tau = \int {\sqrt {{\rho _0}c} {\rm{d}}t,}$$(42)

Observe that in a completely general (3+1)dimensional Lorentzian geometry the metric has 6 degrees of freedom per point in spacetime. (4 × 4 symmetric matrix ⇒ 10 independent components; then subtract 4 coordinate conditions).
In contrast, the acoustic metric is more constrained. Being specified completely by the three scalars Φ_{0}(t, x), ρ_{0}(t, x), and c(t, x), the acoustic metric has, at most, 3 degrees of freedom per point in spacetime. The equation of continuity actually reduces this to 2 degrees of freedom, which can be taken to be Φ_{0}(t, x) and c(t, x).
Thus, the simple acoustic metric of this section can, at best, reproduce some subset of the generic metrics of interest in general relativity.

A point of notation: Where the general relativist uses the word “stationary” the fluid dynamicist uses the phrase “steady flow”. The generalrelativistic word “static” translates to a rather messy constraint on the fluid flow (to be discussed more fully below).

Finally, we should emphasise that in Einstein gravity the spacetime metric is related to the distribution of matter by the nonlinear EinsteinHilbert differential equations. In contrast, in the present context, the acoustic metric is related to the distribution of matter in a simple algebraic fashion.
2.4.1 Horizons and ergoregions
In the next two subsections we shall undertake to more fully explain some of the technical details underlying the acoustic analogy. Concepts and quantities such as horizons, ergoregions and “surface gravity” are important features of standard general relativity, and analogies are useful only insofar as they adequately preserve these notions.
A trapped surface in acoustics is defined as follows: Take any closed twosurface. If the fluid velocity is everywhere inwardpointing and the normal component of the fluid velocity is everywhere greater than the local speed of sound, then no matter what direction a sound wave propagates, it will be swept inward by the fluid flow and be trapped inside the surface. The surface is then said to be outertrapped. (For comparison with the usual situation in general relativity see [275, pp. 319–323] or [670, pp. 310–311].) Innertrapped surfaces (antitrapped surfaces) can be defined by demanding that the fluid flow is everywhere outwardpointing with supersonic normal component. It is only because of the fact that the background Minkowski metric provides a natural definition of “at rest” that we can adopt such a simple and straightforward definition. In ordinary general relativity we need to develop considerable additional technical machinery, such as the notion of the “expansion” of bundles of ingoing and outgoing null geodesics, before defining trapped surfaces. That the above definition for acoustic geometries is a specialization of the usual one can be seen from the discussion in [275, pp. 262–263]. The acoustic trapped region is now defined as the region containing outer trapped surfaces, and the acoustic (future) apparent horizon as the boundary of the trapped region. That is, the acoustic apparent horizon is the twosurface for which the normal component of the fluid velocity is everywhere equal to the local speed of sound. (We can also define antitrapped regions and past apparent horizons but these notions are of limited utility in general relativity.)^{6}
The fact that the apparent horizon seems to be fixed in a foliationindependent manner is only an illusion due to the way in which the analogies work. A particular fluid flow reproduces a specific “metric”, (a matrix of coefficients in a specific coordinate system, not a “geometry”), and, in particular, a specific foliation of spacetime. (Only “internal” observers see a “geometry”; see the discussion in Section 7.4). The same “geometry” written in different coordinates would give rise to a different fluid flow (if at all possible, as not all coordinate representations of a fixed geometry give rise to acoustic metrics) and, therefore, to a different apparent horizon.
In all stationary geometries the apparent and event horizons coincide, and the distinction is immaterial. In timedependent geometries the distinction is often important. When computing the surface gravity, we shall restrict attention to stationary geometries (steady flow). In fluid flows of high symmetry (spherical symmetry, plane symmetry), the ergosphere may coincide with the acoustic apparent horizon, or even the acoustic event horizon. This is the analogue of the result in general relativity that for static (as opposed to stationary) black holes the ergosphere and event horizon coincide. For many more details, including appropriate null coordinates and CarterPenrose diagrams, both in stationary and timedependent situations, see [37].
2.4.2 Surface gravity
Because of the definition of event horizon in terms of phonons (null geodesics) that cannot escape the acoustic black hole, the event horizon is automatically a null surface, and the generators of the event horizon are automatically null geodesics. In the case of acoustics, there is one particular parameterization of these null geodesics that is “most natural”, which is the parameterization in terms of the Newtonian time coordinate of the underlying physical metric. This allows us to unambiguously define a “surface gravity” even for nonstationary (timedependent) acoustic event horizons, by calculating the extent to which this natural time parameter fails to be an affine parameter for the null generators of the horizon. (This part of the construction fails in general relativity where there is no universal natural timecoordinate unless there is a timelike Killing vector — this is why extending the notion of surface gravity to nonstationary geometries in general relativity is so difficult.)
When it comes to explicitly calculating the surface gravity in terms of suitable gradients of the fluid flow, it is nevertheless very useful to limit attention to situations of steady flow (so that the acoustic metric is stationary). This has the added bonus that for stationary geometries the notion of “acoustic surface gravity” in acoustics is unambiguously equivalent to the general relativity definition. It is also useful to take cognizance of the fact that the situation simplifies considerably for static (as opposed to merely stationary) acoustic metrics.
2.4.2.1 Static acoustic spacetimes
Since this is a static geometry, the relationship between the Hawking temperature and surface gravity may be verified in the usual fasttrack manner — using the Wick rotation trick to analytically continue to Euclidean space [245]. If you don’t like Euclidean signature techniques (which are in any case only applicable to equilibrium situations) you should go back to the original Hawking derivations [270, 271].^{8}
One final comment to wrap up this section: The coordinate transform we used to put the acoustic metric into the explicitly static form is perfectly good mathematics, and from the general relativity point of view is even a simplification. However, from the point of view of the underlying Newtonian physics of the fluid, this is a rather bizarre way of deliberately desynchronizing your clocks to take a perfectly reasonable region — the boundary of the region of supersonic flow — and push it out to “time” plus infinity. From the fluid dynamics point of view this coordinate transformation is correct but perverse, and it is easier to keep a good grasp on the physics by staying with the original Newtonian time coordinate.
2.4.2.2 Stationary (nonstatic) acoustic spacetimes
If the fluid flow does not satisfy the integrability condition, which allows us to introduce an explicitly static coordinate system, then defining the surface gravity is a little trickier.
This is in agreement with the previous calculation for static acoustic black holes, and insofar as there is overlap, is also consistent with results of Unruh [607, 608], Reznik [523], and the results for “dirty black holes” [621]. From the construction it is clear that the surface gravity is a measure of the extent to which the Newtonian time parameter inherited from the underlying fluid dynamics fails to be an affine parameter for the null geodesics on the horizon.^{10}
Again, the justification for going into so much detail on this specific model is that this style of argument can be viewed as a template — it will (with suitable modifications) easily generalise to more complicated analogue models.
2.4.3 Example: vortex geometry
(If these flow velocities are nonzero, then following the discussion of [641] there must be some external force present to set up and maintain the background flow. Fortunately it is easy to see that this external force affects only the background flow and does not influence the linearised fluctuations we are interested in.)
In conformity with previous comments, the vortex fluid flow is seen to possess an acoustic metric that is stably causal and which does not involve closed timelike curves. At large distances it is possible to approximate the vortex geometry by a spinning cosmic string [646], but this approximation becomes progressively worse as the core is approached. Trying to force the existence of closed timelike curves leads to the existence of evanescent waves in what would be the achronal region, and therefore to the breakdown of the analogue model description [8].
2.4.4 Example: slab geometry
If we set c = 1 and ignore the x, y coordinates and conformal factor, we have the toy model acoustic geometry discussed in many papers. (See for instance the early papers by Unruh [608, p. 2828, Equation (8)], Jacobson [310, p. 7085, Equation (4)], Corley and Jacobson [148], and Corley [145].) Depending on the velocity profile one can simulate black holes, white holes or black holewhite hole pairs, interesting for analyzing the black hole laser effect [150] or aspects of the physics of warpdrives [197].
In this situation one must again invoke an external force to set up and maintain the fluid flow. Since the conformal factor is regular at the event horizon, we know that the surface gravity and Hawking temperature are independent of this conformal factor [315]. In the general case it is important to realise that the flow can go supersonic for either of two reasons: The fluid could speed up, or the speed of sound could decrease. When it comes to calculating the “surface gravity” both of these effects will have to be taken into account.
2.4.5 Example: PainlevéGullstrand geometry
As emphasised by Kraus and Wilczek, the PainlevéGullstrand line element exhibits a number of features of pedagogical interest. In particular the constanttime spatial slices are completely flat. That is, the curvature of space is zero, and all the spacetime curvature of the Schwarzschild geometry has been pushed into the timetime and timespace components of the metric.
Given the PainlevéGullstrand line element, it might seem trivial to force the acoustic metric into this form: Simply take ρ and c to be constants, and set \(\upsilon = \sqrt {2GM/r}\). While this certainly forces the acoustic metric into the PainlevéGullstrand form, the problem with this is that this assignment is incompatible with the continuity equation ∇ · (ρv) ≠ 0 that was used in deriving the acoustic equations.
2.4.6 Causal structure
We can now turn to another aspect of acoustic black holes, i.e., their global causal structure, which we shall illustrate making use of the CarterPenrose conformal diagrams [445, 275]. A systematic study in this sense was performed in [37] for 1+1 geometries (viewed either as a dimensional reduction of a physical 3+1 system, or directly as geometrical acoustic metrics). The basic idea underlying the conformal diagram of any noncompact 1+1 manifold is that its metric can always be conformally mapped to the metric of a compact geometry, with a boundary added to represent events at infinity. Since compact spacetimes are in some sense “finite”, they can then properly be drawn on a sheet of paper, something that is sometimes very useful in capturing the essential features of the geometry at hand.
The basic steps in the acoustic case are the same as in standard general relativity: Starting from the coordinates (t, x) as in Equation (35), one has to introduce appropriate null coordinates (analogous to the EddingtonFinkelstein coordinates) (u, v), then, by exponentiation, null Kruskallike coordinates (U, W), and finally compactify by means of a new coordinate pair \(\mathcal{U},\, \mathcal{W}\) involving a suitable function mapping an infinite range to a finite one (typically the arctan function). We shall explicitly present only the conformal diagrams for an acoustic black hole, and a black holewhite hole pair, as these particular spacetimes will be of some relevance in what follows. We redirect the reader to [37] for other geometries and technical details.
2.4.6.1 Acoustic black hole
For the case of a single isolated blackhole horizon we find the CarterPenrose diagram of Figure 8. (In the figure we have introduced an aspect ratio different from unity for the coordinates \(\mathcal{U}\) and \(\mathcal{W}\), in order to make the various regions of interest graphically more clear.) As we have already commented, in the acoustic spacetimes, with no periodic identifications, there are two clearlydifferentiated notions of asymptotia, “right” and “left”. In all our figures we have used subscripts “right” and “left” to label the different null and spacelike infinities. In addition, we have denoted the different sonicpoint boundaries with \(\Im _{{\rm{right}}}^ \pm\) or \(\Im _{{\rm{left}}}^ \pm\) depending on whether they are the starting point (− sign) or the ending point (+ sign) of the null geodesics in the right or left parts of the diagram.
In contradistinction to the CarterPenrose diagram for the Schwarzschild black hole (which in the current context would have to be an eternal black hole, not one formed via astrophysical stellar collapse) there is no singularity. On reflection, this feature of the conformal diagram should be obvious, since the fluid flow underlying the acoustic geometry is nowhere singular.
Note that the event horizon \(\mathcal{H}\) is the boundary of the causal past of future right null infinity; that is, \({\mathcal H} = {\dot J^ }\left({\Im _{{\rm{right}}}^ \pm} \right)\), with standard notations [445].
2.4.6.2 Acoustic blackholewhitehole pair
2.5 Cosmological metrics
In a cosmological framework the key items of interest are the FriedmannRobertsonWalker (FRW) geometries, more properly called the FriedmannLemaîtreRobertsonWalker (FLRW) geometries. The simulation of such geometries has been considered in various works such as [46, 47, 105, 106, 194, 195, 196, 328, 403, 674, 675, 676, 683, 677] with a specific view to enhancing our understanding of “cosmological particle production” driven by the expansion of the universe.
2.5.1 Explosion
We can either let the explosion take place more or less spherically symmetrically, or through a pancakelike configuration, or through a cigarlike configuration.
2.5.1.1 Threedimensional explosion
2.5.1.2 Twodimensional explosion
2.5.1.3 Onedimensional explosion
2.5.2 Varying the effective speed of light
2.6 Regaining geometric acoustics
Up to now, we have been developing general machinery to force acoustics into Lorentzian form. This can be justified either with a view to using fluid mechanics to teach us more about general relativity, or to using the techniques of Lorentzian geometry to teach us more about fluid mechanics.
2.7 Generalizing the physical model

Adding external forces.

Working in truly (1+1) or (2+1) dimensional systems.

Adding vorticity, to go beyond the irrotational constraint.
2.7.1 External forces
Adding external forces is (relatively) easy, an early discussion can be found in [624] and more details are available in [641]. The key point is that with an external force one can to some extent shape the background flow (see for example the discussion in [249]). However, upon linearization, the fluctuations are insensitive to any external force.
2.7.2 The role of dimension
This situation would be appropriate, for instance, when dealing with surface waves or excitations confined to a particular substrate.
d = 1: The naive form of the acoustic metric in (1+1) dimensions is illdefined, because the conformal factor is raised to a formally infinite power. This is a side effect of the wellknown conformal invariance of the Laplacian in 2 dimensions. The wave equation in terms of the densitised inverse metric f^{ μν } continues to make good sense; it is only the step from f^{ μν } to the effective metric that breaks down. Acoustics in intrinsically (1+1) dimensional systems does not reproduce the conformallyinvariant wave equation in (1+1) dimensions.
Note that this issue only presents a difficulty for physical systems that are intrinsically onedimensional. A threedimensional system with plane symmetry, or a twodimensional system with line symmetry, provides a perfectly wellbehaved model for (1+1) dimensions, as in the cases d = 3 and d = 2 above.
2.7.3 Adding vorticity
For the preceding analysis to hold, it is necessary and sufficient that the flow locally be vorticity free, ∇ × v = 0, so that velocity potentials exist on an atlas of open patches. Note that the irrotational condition is automatically satisfied for the superfluid component of physical superfluids. (This point has been emphasised by Comer [143], who has also pointed out that in superfluids there will be multiple acoustic metrics — and multiple acoustic horizons — corresponding to first and second sound.) Even for normal fluids, vorticityfree flows are common, especially in situations of high symmetry. Furthermore, the previous condition enables us to handle vortex filaments, where the vorticity is concentrated into a thin vortex core, provided we do not attempt to probe the vortex core itself. It is not necessary for the velocity potential Φ to be globally defined.
Though physically important, dealing with situations of distributed vorticity is much more difficult, and the relevant wave equation is more complicated in that the velocity scalar is now insufficient to completely characterise the fluid flow.^{14} An approach similar to the spirit of the present discussion, but in terms of Clebsch potentials, can be found in [502]. The eikonal approximation (geometrical acoustics) leads to the same conformal class of metrics previously discussed, but in the realm of physical acoustics the wave equation is considerably more complicated than a simple d’Alembertian. (Roughly speaking, the vorticity becomes a source for the d’Alembertian, while the vorticity evolves in response to gradients in a generalised scalar potential. This seems to take us outside the realm of models of direct interest to the general relativity community.)^{15}
2.8 Simple Lagrangian metamodel
As a first (and rather broad) example of the very abstract ways in which the notion of an acoustic metric can be generalised, we start from the simple observation that irrotational barotropic fluid mechanics can be described by a Lagrangian, and ask if we can extend the notion of an acoustic metric to all (or at least some wide class of) Lagrangian systems?
It is important to realise just how general the result is (and where the limitations are): It works for any Lagrangian depending only on a single scalar field and its first derivatives. The linearised PDE will be hyperbolic (and so the linearised equations will have wavelike solutions) if and only if the effective metric g_{ μν } has Lorentzian signature ±[−, +^{ d }]. Observe that if the Lagrangian contains nontrivial second derivatives you should not be too surprised to see terms beyond the d’Alembertian showing up in the linearised equations of motion.
As a specific example of the appearance of effective metrics due to Lagrangian dynamics we reiterate the fact that inviscid irrotational barotropic hydrodynamics naturally falls into this scheme (which is why, with hindsight, the derivation of the acoustic metric presented earlier in this review was so relatively straightforward). In inviscid irrotational barotropic hydrodynamics the lack of viscosity (dissipation) guarantees the existence of a Lagrangian; which a priori could depend on several fields. Since the flow is irrotational v = − ∇Φ is a function only of the velocity potential, and the Lagrangian is a function only of this potential and the density. Finally, the equation of state can be used to eliminate the density leading to a Lagrangian that is a function only of the single field Φ and its derivatives [44].
2.9 Going further
 Working with specific fluids.

Superfluids.

BoseEinstein condensates.

 Abstract generalizations.

Normal modes in generic systems.

Multiple signal speeds.

We next turn to a brief historical discussion, seeking to place the work of the last two decades into its proper historical perspective.
3 History
From the point of view of the general relativity community the history of analogue models can reasonably neatly (but superficially) be divided into an “historical” period (essentially pre1981) and a “modern” period (essentially post1981). We shall in the next Section 3.1 focus more precisely on the early history of analogue models, and specifically those that seem to us to have had a direct historical connection with the sustained burst of work carried out in the last 20 years.
3.1 Historical period
Of course, the division into pre1981 and post1981 articles is at a deeper level somewhat deceptive. There have been several analogue models investigated over the years, with different aims, different levels of sophistication, and ultimately different levels of development. Armed with a good library and some hindsight it is possible to find interesting analogues in a number of places.^{16}
3.1.1 Optics — the Gordon metric
After that, there was sporadic interest in effective metric techniques. An historicallyimportant contribution was one of the problems in the wellknown book “The Classical Theory of Fields” by Landau and Lifshitz [373]. See the end of Chapter 10, Paragraph 90, and the problem immediately thereafter: “Equations of electrodynamics in the presence of a gravitational field”. Note that in contrast to Gordon, here the interest is in using dielectric media to mimic a gravitational field.
3.1.2 Acoustics
After the pioneering hydrodynamical paper by White in 1973 [687], which studied acoustic ray tracing in nonrelativistic moving fluids, there were several papers in the 1980s using an acoustic analogy to investigate the propagation of shockwaves in astrophysical situations, most notably those of Moncrief [448] and Matarrese [433, 434, 432]. In particular, in Moncrief’s work [448] the linear perturbations of a relativistic perfect fluid on a Schwarzschild background were studied, and it was shown that the wave equation for such perturbations can be expressed as a relativistic wave equation on some effective (acoustic) metric (which can admit acoustic horizons). In this sense [448] can be seen as a precursor to the later works on acoustic geometries and acoustic horizons. Indeed, because they additionally permit a general relativistic Schwarzschild background, the results of Moncrief [448] are, in some sense, more general than those considered in the mainstream acoustic gravity papers that followed.
In spite of these impressive results, we consider these papers to be part of the “historical period” for the main reason that such works are philosophically orthogonal to modern developments in analogue gravity. Indeed the main motivation for such works was the study of perfect fluid dynamics in accretion flows around black holes, or in cosmological expansion, and in this context the description via an acoustic effective background was just a tool in order to derive results concerning conservation laws and stability. This is probably why, even if temporally, [448] predates Unruh’s 1981 paper by one year, and while [433, 434, 432] postdate Unruh’s 1981 paper by a few years, there seems to have not been any crossconnection.
3.1.3 Surface waves
Somewhat ironically, 1983 marked the appearance of some purely experimental results on surface waves in water obtained by Badulin et al. [17]. At the time these results passed unremarked by the relativity community, but they are now of increasing interest, and are seen to be precursors of the theoretical work reported in [560, 531] and the modern experimental work reported in [532, 682].
3.2 Modern period
3.2.1 The years 1981–1999
The key event in the “modern” period (though largely unrecognised at the time) was the 1981 publication of Unruh’s paper “Experimental black hole evaporation” [607], which implemented an analogue model based on fluid flow, and then used the power of that analogy to probe fundamental issues regarding Hawking radiation from “real” generalrelativistic black holes.
We believe that Unruh’s 1981 article represents the first observation of the now widely established fact that Hawking radiation has nothing to do with general relativity per se, but that Hawking radiation is instead a fundamental curvedspace quantum field theory phenomenon that occurs whenever a horizon is present in an effective geometry.^{17} Though Unruh’s 1981 paper was seminal in this regard, it lay largely unnoticed for many years.
Some 10 years later Jacobson’s article “Blackhole evaporation and ultrashort distances” [307] used Unruh’s analogy to build a physical model for the “transPlanckian modes” believed to be relevant to the Hawking radiation process. Progress then sped up with the relatively rapid appearance of [308] and [608]. (This period also saw the independent rediscovery of the fluid analogue model by one of the present authors [622], and the first explicit consideration of superfluids in this regard [143].)
The later 1990s then saw continued work by Jacobson and his group [309, 310, 148, 146, 150, 322], with new and rather different contributions coming in the form of the solidstate models considered by Reznik [523, 522]. [285] is an attempt at connecting Hawking evaporation with the physics of collapsing bubbles. This was part of a more general programme aimed at connecting blackhole thermodynamics with perfectfluid thermodynamics [286]. This period also saw the introduction of the more general class of superfluid models considered by Volovik and his collaborators [644, 645, 359, 183, 649, 647, 648, 326, 651, 652], more precise formulations of the notions of horizon, ergosphere, and surface gravity in analogue models [624, 626], and discussions of the implications of analogue models regarding BekensteinHawking entropy [625, 626]. Optical models were considered in [389]. Finally, analogue spacetimes based on special relativistic acoustics were considered in [72].
By the year 2000, articles on one or another aspect of analogue gravity were appearing at the rate of over 20 per year, and it becomes impractical to summarise more than a selection of them.
3.2.2 The year 2000
Key developments in 2000 were the introduction, by Garay and collaborators, of the use of BoseEinstein condensates as a working fluid [231, 232], and the extension of those ideas by the present authors [43]. Further afield, the transPlanckian problem also reared its head in the context of cosmological inflation, and analogue model ideas previously applied to Hawking radiation were reused in that context [341, 457].
That year also marked the appearance of a review article on superfluid analogues [655], more work on “nearhorizon” physics [210], and the transference of the idea of analogueinspired “multiple metric” theories into cosmology, where they can be used as the basis for a precise definition of what is meant by a VSL (“variable speed of light”) cosmology [58]. Models based on nonlinear electrodynamics were investigated in [26], ^{3}HeA based models were reconsidered in [316, 653], and “slow light” models in quantum dielectrics were considered in [390, 391, 382]. The most radical proposal to appear in 2000 was that of Laughlin et al. [131]. Based on taking a superfluid analogy rather literally, they mooted an actual physical breakdown of general relativity at the horizon of a black hole [131].
Additionally, the workshop on “Analogue models of general relativity”, held at CBPF (Rio de Janeiro) gathered some 20 international participants and greatly stimulated the field, leading ultimately to the publication of a book [470] in 2002.
3.2.3 The year 2001
This year saw more applications of analogueinspired ideas to cosmological inflation [179, 441, 440, 343, 459], to neutron star cores [116], and to the cosmological constant [656, 658].
Closer to the heart of the analogue programme were the development of a “normal mode” analysis in [44, 45, 637], the development of dielectric analogues in [557], speculations regarding the possibly emergent nature of Einstein gravity [50, 637], and further developments regarding the use of ^{3}HeA [178] as an analogue for electromagnetism. Experimental proposals were considered in [48, 637, 539]. Vorticity was discussed in [502], and the use of BECs as a model for the breakdown of Lorentz invariance in [636]. Analogue models based on nonlinear electrodynamics were discussed in [169]. Acoustics in an irrotational vortex were investigated in [207].
The excitation spectrum in superfluids, specifically the fermion zero modes, were investigated in [654, 303], while the relationship between rotational friction in superfluids and superradiance in rotating spacetimes was discussed in [104]. More work on “slow light” appeared in [91]. The possible role of Lorentz violations at ultrahigh energy was emphasised in [312].
3.2.4 The year 2002
“What did we learn from studying acoustic black holes?” was the title and theme of Parentani’s article in 2002 [492], while Schützhold and Unruh developed a rather different fluidbased analogy based on gravity waves in shallow water [560, 560]. Superradiance was investigated in [57], while the propagation of phonons and quasiparticles was discussed in [209, 208]. More work on “slow light” appeared in [211, 509]. Applications to inflationary cosmology were developed in [460], while analogue spacetimes relevant to braneworld cosmologies were considered in [28].
The stability of an acoustic white hole was investigated in [386], while further developments regarding analogue models based on nonlinear electrodynamics were presented by Novello and collaborators in [170, 171, 468, 464, 214]. Though analogue models lead naturally to the idea of highenergy violations of Lorentz invariance, it must be stressed that definite observational evidence for violations of Lorentz invariance is lacking — in fact, there are rather strong constraints on how strong any possible Lorentz violating effect might be [318, 317].
3.2.5 The year 2003
This year saw further discussion of analogueinspired models for blackhole entropy and the cosmological constant [661, 668], and the development of analogue models for FRW geometries [195, 194, 46, 177, 403]. There were several further developments regarding the foundations of BECbased models in [47, 196], while analogue spacetimes in superfluid neutron stars were further investigated in [117].
Effective geometry was the theme in [466], while applications of nonlinear electrodynamics (and its effective metric) to cosmology were presented in [467]. Superradiance was further investigated in [56, 54], while the limitations of the “slow light” analogue were explained in [612]. Vachaspati argued for an analogy between phase boundaries and acoustic horizons in [615]. Emergent relativity was again addressed in [378]. The review article by Burgess [98] emphasised the role of general relativity as an effective field theory — the sine qua non for any attempt at interpreting general relativity as an emergent theory. The lecture notes by Jacobson [313] give a nice introduction to Hawking radiation and its connection to analogue spacetimes.
3.2.6 The year 2004
The year 2004 saw the appearance of some 30 articles on (or closely related to) analogue models. Effective geometries in astrophysics were discussed by Perez Bergliaffa [501], while the physical realizability of acoustic Hawking radiation was addressed in [159, 616]. More cosmological issues were raised in [616, 675], while a specifically astrophysical use of the acoustic analogy was invoked in [160, 161, 162]. BECbased horizons were again considered in [249, 247], while backreaction effects were the focus of attention in [25, 23, 344]. More issues relating to the simulation of FRW cosmologies were raised in [204, 206].
Unruh and Schützhold discussed the universality of the Hawking effect [613], and a new proposal for possibly detecting Hawking radiation in an electromagnetic wave guide [562]. The causal structure of analogue spacetimes was considered in [37], while quasinormal modes attracted attention in [69, 392, 112, 452]. Two dimensional analogue models were considered in [101].
There were attempts at modeling the Kerr geometry [641], and generic “rotating” spacetimes [132], a proposal for using analogue models to generate massive phonon modes in BECs [640, 642], and an extension of the usual formalism for representing weakfield gravitational lensing in terms of an analogue refractive index [81]. Finally, we mention the development of yet more strong observational bounds on possible ultrahighenergy Lorentz violation [319, 320].
3.2.7 The year 2005
The year 2005 saw continued and vigorous activity on the analogue model front. More studies of the superresonance phenomenon appeared [55, 193, 347, 578], and a minisurvey was presented in [111]. Quasinormal modes again received attention in [135], while the Magnus force was reanalysed in terms of the acoustic geometry in [699]. Singularities in the acoustic geometry are considered in [102], while backreaction has received more attention in [559].
The original version of this Living Review appeared in May of 2005 [49], and since then activity has, if anything, increased. Work on BECrelated models included [401, 402, 678, 642, 679, 605, 205], while additional work on superradiance [136], the background fluid flow [103], and quasinormal modes [537] also appeared. Dynamical phase transitions were considered in [551], and astrophysical applications to accretion flow in [2, 165]. The connection between white hole horizon and the classical notion of a “hydraulic jump” was explored in [662] and in [572]. A “spacetime condensate” point of view was advocated in [299], and analogue applications to “quantum teleportation” were considered in [243]. A nice survey of analogue ideas and backreaction effects was presented in [21] (and related articles [23, 25]). Finally, we mention the appearance in 2005 of another Living Review, one that summarises and systematises the very stringent bounds that have been developed on possible ultrahighenergy Lorentz violation [435].
3.2.8 The year 2006
A key article, which appeared in 2006, involved the “inverse” use of the acoustic metric to help understand hydrodynamic fluid flow in quarkgluon plasma [123]. The relationship between modified dispersion relations and Finsler spacetimes was discussed in [251]. Backreaction effects were again considered in [20].
Using analogue ideas as backdrop, Markopoulou developed a pregeometric model for quantum gravity in [421]. Analogue implications visavis entanglement entropy were discussed in [226, 225]. A microscopic analysis of the microtheory underlying acoustic Hawking radiation in a “piston” geometry appeared in [248]. Volovik extended and explained his views on quantum hydrodynamics as a model for quantum gravity in [664]. Applications to the cosmological constant were considered in [543]. More BECrelated developments appeared in [680, 38, 39, 29, 24, 188, 189]. An analogue model based on a suspended “shoestring” was explored in [282]. Superresonance was again discussed in [352]. More analogueinspired work on blackhole accretion appeared in [164], while “ripplons” (quantised surface capillary waves) were discussed in [663]. Modified dispersion relations again attracted attention [250], and analogue inspired ideas concerning constrained systems were explored in [357]. Quasinormal frequencies were considered in [134]. Finally, we emphasise particularly the realization that the occurrence of Hawkinglike radiation does not require the presence of an event horizon or even a trapped region [38, 39].
3.2.9 The year 2007
Emergent geometry [16, 14, 15, 618] was an important theme in 2007, as were efforts at moving beyond the semiclassical description [491]. BECbased analogue models were adapted to investigating “signature change events” in [684]. Acoustic crosssections were considered in [156]. “Rimfall” was discussed in [609].
Analogueinspired ideas were adapted to the study of gravitational collapse in [40], while the importance of nonlocal correlations in the Hawking flux was emphasized in [22]. Quantum field theoretic anomalies were considered in [345], while entanglement entropy was investigated in [324]. The specific shape of the de Laval nozzle needed to acoustically reproduce linearised perturbations of the Schwarzschild geometry was discussed in [1], and quasinormal frequencies were investigated in [692]. Superradiance and disclinations were considered in [167]. Theoretical aspects of the circular hydraulic jump were investigated in [519].
The use of analogue spacetimes as “toy models” for quantum gravity was emphasized in [643, 632]. Within the optics community Philbin, Leonhardt, and coworkers initiated the study of “fibreoptic black holes” [505, 504]. Within the fluid dynamics community, wavetank experiments were initiated [532] by Rousseaux and coworkers, who demonstrated the presence of negative phasevelocity waves. Dissipationinduced breakdown of Lorentz invariance was considered by Parentani in [494, 495], and BECbased models continued to attract attention [676, 30, 31, 333]. While analyzing quark matter, acoustic metrics were found to be useful in [419] (see also [418]).
Analogues based on ion traps were considered by Schützhold in [558], while a toy model for backreaction was explored in [414]. Further afield, analogue models were used to motivate a “AbrahamLorentz” interpretation of relativity in terms of a physicallyreal aether and physicallyreal LorentzFitzGerald contraction [36]. In a similar vein analogue models were used to motivate a counterfactual counterhistorical approach to the Bohm versus Copenhagen interpretations of quantum physics [463]. Analogueinspired ideas regarding the possible “localization” of the origin of the Hawking flux were investigated by Unruh in [610]. Additionally, an analogue inspired analysis of accretion appeared in [163], while astrophysical constraints on modified dispersion relations were improved and extended in [409, 394].
3.2.10 The year 2008
This year saw the introduction of “quantum graphity” [358, 356], an analogueinspired model for quantum gravity. The BEC theme continued to generate attention [368], in particular regarding cosmological particle production [677, 683], and Hawking radiation [119]. A minireview appeared in [553]. The theme of “emergence” was also represented in articles such as [255, 252, 329]. Localization of the Hawking radiation was again addressed in [563], while sensitivity of the Hawking flux to the presence of superluminal dispersion was considered in [33]. Astrophysical constraints on modified dispersion relations were again considered in [408, 410], while applications to quark matter were investigated in [415, 417]. Possible applications to hightemperature superconductivity were reported in [444]. Quantum field theoretic anomalies in an acoustic geometry were considered in [60, 691], while (2+1) acoustic blackhole thermodynamics were investigated in [346]. Gravitational collapse was again discussed in [40].
Gibbons and coworkers used analoguebased ideas in their analysis of general stationary spacetimes, demonstrating that the spatial slices of stationary spacetimes are best thought of as a special class of Finsler spaces, in particular, Randers spaces [246]. Attempts at developing a generally useful notion of Finsler spacetime were discussed in [573, 574], with Finslerian applications to the Higgs mechanism being investigated in [569]. Signature change, now not in a BEC context, was again addressed in [686]. Blackhole lasers were considered in [388], and the fluidgravity correspondence in [5]. Backreaction of the Hawking flux was again considered in [556], while the analogue physics of a “photon fluid” was considered in [420]. In [73] analogue ideas were applied to polytrope models of Newtonian stars, while superradiance was considered in [598].
3.2.11 The year 2009
This year saw intriguing and unexpected relations develop between analogue spacetimes and Hořava gravity [584, 585, 634, 666, 570]. These connections seem primarily related to the way Hořava’s projectability condition interleaves with the ADM decomposition of the metric, and to the manner in which Hořava’s distinguished spacetime foliation interleaves with the preferred use of PainlevéGullstrandlike coordinates.
The theme of emergent gravity continues to play a role [300, 395, 571], as does the theme of nontrivial dispersion relations [413, 412, 411, 396, 437]. A variant of quantum graphity was further developed in [268], and a matrix model implementation of analogue spacetime was developed in [220]. Quasinormal modes were considered in [174, 700], with a survey appearing in [70]. Acoustic scattering was considered in [173]. Applications to quark matter were again investigated in [416]. In [340], the universe was interpreted as a “soap film”. Nonlinear electrodynamics was again considered in [260]. A model based on liquid crystals appeared in [500]. Attempts at including backreaction in a cosmological fluid context were investigated in [451, 450]. Possible experimental implementations of acoustic black holes using circulating ion rings are discussed in [290], while ultrashort laser pulses are considered in [187]. Signature change events were again considered in [685], while analogueinspired lessons regarding the fundamental nature of time were investigated in [330] and [254]. Noncanonical quantum fields were considered in [304].
In [217, 218] analogue ideas are implemented in an unusual direction: fluid dynamics is used to model aspects of quantum field theory. That the transPlanckian and information loss problems are linked is argued in [397]. The BEC paradigm for acoustic geometry is again discussed in [331] and [520], while universal aspects of superradiant scattering are considered in [524]. Most remarkably, a BECbased black hole analogue was experimentally realised in [369]. (See Sections 6.2 and 7.13.)
3.2.12 The year 2010
This current year has already (September) seen some 50 articles appear that can legitimately claim to have either direct or tangential relationships to the analogue spacetime programme. Being necessarily very selective, we first mention work related to “entropic” attempts at understanding the “emergence” of general relativity and the spacetime “degrees of freedom” from the quantum regime [483, 482, 130, 590, 355]. The use of correlations as a potential experimental probe has been theoretically investigated in [19, 118, 190, 496, 512, 564, 604], while an analysis of optimality conditions for the detection of HawkingUnruh radiation appeared in [13]. Entanglement issues were explored in [427]. More work on “emergent horizons” has appeared in [552, 554, 555]. Relativistic fluids have been revisited in [639], with specific applications to relativistic BECs being reported in [191]. Possible measurement protocols for Hawking radiation in ionic systems were discussed in [291]. In [34] analogue spacetimes were used to carefully separate the notion of “emergent manifold” from that of “emergent curvature”. Quantum graphity was again considered in [110].
Stepfunction discontinuities in BECs were considered in [186, 438]. (Signature change can be viewed as an extreme case of stepfunction discontinuity [684, 685, 686].) Black holes induced by dielectric effects, and their associated Hawking radiation, were considered in [63]. The acoustic geometry of polytrope rotating Newtonian stars was considered in [74]. Random fluids were investigated in [365]. Finsler spacetime geometries were again considered in [575], while the relationship between analogue spacetimes and foundational mathematical relativity was discussed in [139]. Further afield, analogue spacetimes were used as an aid to understanding “warp drive” spacetimes [32].
Quantum sound in BECs was again investigated in [35], BECbased particle creation in [367], and BECbased black hole lasers in [199]. Optical effective geometries in Kerr media were discussed in [100]. 2+1 dimensional drainingbathtub geometries were probed in [477]. Theoretical and historical analyses of surface waves in a wave tank were presented in [531]. Finally, we mention the stunning experimental verification by Weinfurtner et al. of the existence of classical stimulated Hawking radiation in a wave flume [682], and the experimental detection of photons associated with a phase velocity horizon by Belgiorno et al. [66].
3.2.13 The future?
Interest in analogue models, analogue spacetimes, and analogue gravity is intense and shows no signs of abating. Interest in these ideas now extends far beyond the general relativity community, and there is significant promise for direct laboratorybased experimental input. We particularly wish to encourage the reader to keep an eye out for future developments regarding the possible experimental verification of the existence of Hawking radiation or the closely related Unruh radiation.
3.3 Going further

Analoguebased “geometrical” interpretations of pseudomomentum, Iordanskii forces, Magnus forces, and the acoustic AharanovBohm effect [227, 592, 593, 594, 595, 650].

An analogueinspired interpretation of the Kerr spacetime [267].

The use of analogies to clarify the Newtonian limit of general relativity [602], to provide heuristics for motivating interest in specific spacetimes [525, 631], and to discuss a simple interpretation of the notion of a horizon [473].

Discrete [583, 281, 436] and noncommutative [140] spacetimes partially influenced and flavoured by analogue ideas.

Analoguebased hints on how to implement “doubly special relativity” (DSR) [361, 362, 363, 599], and a cautionary analysis of why this might be difficult [561].

Possible blackhole phase transitions placed in an analogue context [591].

Attempts at deriving inertia and passive gravity, (though not active gravity), from analogue ideas [442].

Applications of analogue ideas in braneworld [241] and KaluzaKlein [242] settings.

Analogue inspired views on the “mathematical universe” [332].

Cosmological structure formation viewed as noise amplification [568].

Discussions of unusual topology, “acoustic wormholes”, and unusual temporal structure [453, 455, 514, 581, 582, 701, 702].

Abstract quantum field theoretic considerations of the Unruh effect [695].

The interpretation of blackhole entropy in terms of universal “near horizon” behaviour of quantum fields living on spacetime [114, 115].

Numerous suggestions regarding possible transPlanckian physics [12, 59, 121, 128, 129, 284, 440, 528, 600].

Numerous suggestions regarding a minimum length in quantum gravity [71, 78, 93, 144, 230, 293, 294, 295, 361, 362, 363, 405, 407, 406, 579].

Standard quantum field theory physics reformulated in the light of analogue models [9, 10, 203, 215, 393, 398, 399, 400, 422, 471, 472, 486, 485, 487, 493, 515, 695].

Standard general relativity supplemented with analogue viewpoints and insights [354, 376, 422, 456, 325, 475].

The discussion of, and argument for, a possible reassessment of fundamental features of quantum physics and general relativity [11, 259, 342, 377, 400, 488, 535, 544].

Nonstandard viewpoints on quantum physics and general relativity [157, 292, 479, 529, 527, 545, 546, 547, 548].

Soliton physics [497], defect physics [172], and the Fizeau effect [454], presented with an analogue flavour.

Analogueinspired models of blackhole accretion [516, 517, 518].

Cosmological horizons from an analogue spacetime perspective [244].

Analogueinspired insights into renormalization group flow [107].

An analysis of “wave catastrophes” inspired by analogue models [348].

Improved numerical techniques for handling wave equations [688], and analytic techniques for handling wave tails [80], partially based on analogue ideas.
There is not much more that we can usefully say here. We have doubtless missed some articles of historical importance, but with a good library or a fast Internet connection the reader will be in as good a position as we are to find any additional historical articles.
4 A Catalogue of Models
 Classical models:

Classical sound.

Sound in relativistic hydrodynamics.

Water waves (gravity waves).

Classical refractive index.

Normal modes.

 Quantum models:

BoseEinstein condensates (BECs).

The heliocentric universe.
(Helium as an exemplar for just about anything.)

Slow light.

4.1 Classical models
4.1.1 Classical sound
Sound in a nonrelativistic moving fluid has already been extensively discussed in Section 2, and we will not repeat such discussion here. In contrast, sound in a solid exhibits its own distinct and interesting features, notably in the existence of a generalization of the normal notion of birefringence — longitudinal modes travel at a different speed (typically faster) than do transverse modes. This may be viewed as an example of an analogue model which breaks the “light cone” into two at the classical level; as such this model is not particularly useful if one is trying to simulate special relativistic kinematics with its universal speed of light, though it may be used to gain insight into yet another way of “breaking” Lorentz invariance (and the equivalence principle).
4.1.2 Sound in relativistic hydrodynamics

when working in a nontrivial curved general relativistic background;

whenever the fluid is flowing at relativistic speeds;

less obviously, when the internal degrees of freedom of the fluid are relativistic, even if the overall fluid flow is nonrelativistic. (That is, in situations where it is necessary to distinguish the energy density from the mass density ρ; this typically happens in situations where the fluid is strongly selfcoupled — for example in neutron star cores or in relativistic BECs [191]. See Section 4.2.)
4.1.3 Shallow water waves (gravity waves)
The main advantage of this model is that the velocity of the surface waves can very easily be modified by changing the depth of the basin. This velocity can be made very slow, and consequently, the creation of ergoregions should be relatively easier than in other models. As described here, this model is completely classical given that the analogy requires long wavelengths and slow propagation speeds for the gravity waves. Although the latter feature is convenient for the practical realization of analogue horizons, it is a disadvantage in trying to detect analogue Hawking radiation as the relative temperature will necessarily be very low. (This is why, in order to have a possibility of experimentally observing (spontaneous) Hawking evaporation and other quantum phenomena, one would need to use ultracold quantum fluids.) However, the gravity wave analogue can certainly serve to investigate the classical phenomena of mode mixing that underlies the quantum processes.
It is this particular analogue model (and its extensions to finite depth and surface tension) that underlies the experimental [532] and theoretical [531] work of Rousseaux et al., the historicallyimportant experimental work of Badulin et al. [17], and the very recent experimental verification by Weinfurtner et al. of the existence of classical stimulated Hawking radiation [682].
4.1.4 More general water waves
4.1.5 Classical refractive index
4.1.5.1 Eikonal approximation
4.1.5.2 3 degenerate eigenvalues
4.1.5.3 2 distinct eigenvalues
If \(\tilde{\epsilon}\) has two distinct eigenvalues then the determinant det(C^{ ij }) factorises into a trivial factor of ω^{2} and two quadratics. Each quadratic corresponds to a distinct effective metric. This is the physical situation encountered in uniaxial crystals, where the ordinary and extraordinary rays each obey distinct quadratic dispersion relations [82]. From the point of view of analogue models this corresponds to a twometric theory.
4.1.5.4 3 distinct eigenvalues
If \(\tilde{\epsilon}\) has three distinct eigenvalues then the determinant det(C^{ ij }) is the product of a trivial factor of ω^{2} and a nonfactorizable quartic. This is the physical situation encountered in biaxial crystals [82, 638], and it seems that no meaningful notion of the effective Riemannian metric can be assigned to this case. (The use of Finsler geometries in this situation is an avenue that may be worth pursuing [306]. But note some of the negative results obtained in [573, 574, 575].)
4.1.5.5 Abstract linear electrodynamics
Hehl and coworkers have championed the idea of using the linear constitutive relations of electrodynamics as the primary quantities, and then treating the spacetime metric (even for flat space) as a derived concept. See [474, 276, 371, 277].
4.1.5.6 Nonlinear electrodynamics
4.1.5.7 Summary
The propagation of photons in a dielectric medium characterised by 3 × 3 permeability and permittivity tensors constrained by ϵ ∝ μ is equivalent (at the level of geometric optics) to the propagation of photons in a curved spacetime manifold characterised by the ultrastatic metric (198), provided one only considers wavelengths that are sufficiently long for the macroscopic description of the medium to be valid. If, in addition, one takes a fluid dielectric, by controlling its flow one can generalise the Gordon metric and again reproduce metrics of the PainlevéGullstrand type, and therefore geometries with ergoregions. If the proportionality constant relating ϵ ∝ μ is position independent, one can make the stronger statement (187) which holds true at the level of physical optics. Recently this topic has been revitalised by the increasing interest in (classical) metamaterials.
4.1.6 Normal mode metamodels
We have already seen how linearizing the EulerLagrange equations for a single scalar field naturally leads to the notion of an effective spacetime metric. If more than one field is involved the situation becomes more complicated, in a manner similar to that of geometrical optics in uniaxial and biaxial crystals. (This should, with hindsight, not be too surprising since electromagnetism, even in the presence of a medium, is definitely a Lagrangian system and definitely involves more than one single scalar field.) A normal mode analysis based on a general Lagrangian (many fields but still first order in derivatives of those fields) leads to a concept of refringence, or more specifically multirefringence, a generalization of the birefringence of geometrical optics. To see how this comes about, consider a straightforward generalization of the onefield case.
 Suppose that \({f^{\mu \nu}}_{AB}\) factorisesThen$${f^{\mu \nu}}_{AB} = {h_{AB}}\,{f^{\mu \nu}}.$$(221)The Monge cones and normal cones are then true geometrical cones (with the N sheets lying directly on top of one another). The normal modes all see the same spacetime metric, defined up to an unspecified conformal factor by g^{ μν } ∝ f^{ μν }. This situation is the most interesting from the point of view of general relativity. Physically, it corresponds to a singlemetric theory, and mathematically it corresponds to a strict algebraic condition on the \({f^{\mu \nu}}_{AB}\).$$Q(x,k) = \det ({h_{AB}})\,{[{f^{\mu \nu}}\,{k_\mu}\,{k_\nu}]^N}.$$(222)
 The next most useful situation corresponds to the commutativity condition:If this algebraic condition is satisfied, then for all spacetime indices μν and αβ the \(g_A^{\mu \nu} \propto \bar f_A^{\mu \nu}\) can be simultaneously diagonalised in field space leading to$${f^{\mu \nu}}_{AB}\,{f^{\alpha \beta}}_{BC} = {f^{\alpha \beta}}_{AB}\,{f^{\mu \nu}}_{BC};\quad {\rm{that}}\,{\rm{is}}\quad [\,{f^{\mu \nu}},{f^{\alpha \beta}}\,] = 0.$$(223)and then$${\bar f^{\mu \nu}}{\,_{AB}} = {\rm{diag}}\{\bar f_1^{\mu \nu},\bar f_2^{\mu \nu},\bar f_3^{\mu \nu}, \ldots, \bar f_N^{\mu \nu}\}$$(224)This case corresponds to an Nmetric theory, where up to an unspecified conformal factor \({f^{\mu \nu}}_{AB}\). This is the natural generalization of the twometric situation in biaxial crystals.$$Q(x,k) = \prod\limits_{A = 1}^N {[\bar f_A^{\mu \nu}\,{k_\mu}\,{k_\nu}]}.$$(225)

If \({f^{\mu \nu}}_{AB}\) is completely general, satisfying no special algebraic condition, then Q(x,k) does not factorise and is, in general, a polynomial of degree 2N in the wave vector k_{ μ }. This is the natural generalization of the situation in biaxial crystals. (And for any deeper analysis of this situation one will almost certainly need to adopt pseudoFinsler techniques [306]. But note some of the negative results obtained in [573, 574, 575].)
 1.
that there is a significant difference between the levels of physical normal modes (wave equations), and geometrical normal modes (dispersion relations), and
 2.
that the densitised inverse metric is in many ways more fundamental than the metric itself.
4.2 Quantum models
4.2.1 BoseEinstein condensates
We have seen that one of the main aims of research in analogue models of gravity is the possibility of simulating semiclassical gravity phenomena, such as the Hawking radiation effect or cosmological particle production. In this sense systems characterised by a high degree of quantum coherence, very cold temperatures, and low speeds of sound offer the best test field. One could reasonably hope to manipulate these systems to have Hawking temperatures on the order of the environment temperature (∼ 100 nK) [48]. Hence it is not surprising that in recent years BoseEinstein condensates (BECs) have become the subject of extensive study as possible analogue models of general relativity [231, 232, 45, 48, 47, 195, 194].
4.2.1.1 Lorentz breaking in BEC models — the eikonal approximation
 1.
It is interesting to recognize that the dispersion relation (271) is exactly in agreement with that found in 1947 by Bogoliubov [79] (reprinted in [508]; see also [374]) for the collective excitations of a homogeneous Bose gas in the limit T→0 (almost complete condensation). In his derivation Bogoliubov applied a diagonalization procedure for the Hamiltonian describing the system of bosons.
 2.
Coincidentally this is the same dispersion relation that one encounters for shallowwater surface waves in the presence of surface tension. See Section 4.1.4.
 3.
Because of the explicit momentum dependence of the comoving phase velocity and comoving group velocity, once one goes to high momentum the associated effective metric should be thought of as one of many possible “rainbow metrics” as in Section 4.1.4. See also [643]. (At low momentum one, of course, recovers the hydrodynamic limit with its uniquely specified standard metric.)
 4.
It is easy to see that Equation (271) actually interpolates between two different regimes depending on the value of the wavelength λ = 2π/‖k‖ with respect to the “acoustic Compton wavelength” λ_{ c } = h/(mc_{s}). (Remember that c_{s} is the speed of sound; this is not a standard particle physics Compton wavelength.) In particular, if we assume v_{0} = 0 (no background velocity), then, for large wavelengths λ≫λ_{ c }, one gets a standard phonon dispersion relation ω ≈ c‖k‖. For wavelengths λ ≪ λ_{ c } the quasiparticle energy tends to the kinetic energy of an individual gas particle and, in fact, ω ≈ ℏ^{2}k^{2}/(2m).
We would also like to highlight that in relative terms, the approximation by which one neglects the quartic terms in the dispersion relation gets worse as one moves closer to a horizon where v_{0} = −c_{s}. The nondimensional parameter that provides this information is defined byAs we will discuss in Section 5.2, this is the reason why sonic horizons in a BEC can exhibit different features from those in standard general relativity.$$\delta \equiv {{\sqrt {1 + {{\lambda _c^2} \over {4{\lambda ^2}}}}  1} \over {(1  {v_0}/{c_{\rm{s}}})}} \simeq {1 \over {(1  {v_0}/{c_{\rm{s}}})}}{{\lambda _c^2} \over {8{\lambda ^2}}}.$$(272)  5.The dispersion relation (271) exhibits a contribution due to the background flow \(\upsilon _0^i\,{k_i}\), plus a quartic dispersion at high momenta. The group velocity isIndeed, with hindsight, the fact that the group velocity goes to infinity for large k was preordained: After all, we started from the generalised nonlinear Schrödinger equation, and we know what its characteristic curves are. Like the diffusion equation the characteristic curves of the Schrödinger equation (linear or nonlinear) move at infinite speed. If we then approximate this generalised nonlinear Schrödinger equation in any manner, for instance by linearization, we cannot change the characteristic curves: For any wellbehaved approximation technique, at high frequency and momentum we should recover the characteristic curves of the system we started with. However, what we certainly do see in this analysis is a suitably large region of momentum space for which the concept of the effective metric both makes sense, and leads to finite propagation speed for mediumfrequency oscillations.$$v_g^i = {{\partial \omega} \over {\partial {k_i}}} = v_0^i \pm {{\left({{c^2} + {{{\hbar ^2}} \over {2{m^2}}}{k^2}} \right)} \over {\sqrt {{c^2}{k^2} + {{\left({{\hbar \over {2m}}{k^2}} \right)}^2}}}}{k^i}.$$(273)
4.2.1.2 Relativistic BEC extension
BoseEinstein condensation can occur not only for nonrelativistic bosons but for relativistic ones as well. The main differences between the thermodynamical properties of these condensates at finite temperature are due both to the different energy spectra and also to the presence, for relativistic bosons, of antibosons. These differences result in different conditions for the occurrence of BoseEinstein condensation, which is possible, e.g., in two spatial dimensions for a homogeneous relativistic Bose gas, but not for its nonrelativistic counterpart — and also, more importantly for our purposes, in the different structure of their excitation spectra.
Dispersion relation of gapless and gapped modes in different regimes. Note that we have \(c_{\mathcal S}^2 = {c^2}b/(1 + b)\), and \(c_{{\mathcal S},{\rm{gap}}}^2 = {c^2}(2 + b)/(1 + b)\), while m_{eff} = 2(μ/c^{2})(1 + b)^{3/2}/(2 + b).
Gapless  Gapped  

b ≪ 1  b ≫ 1  
\(k\, \ll {{m{u^0}(1 + b)} \over \hbar}\)  \(k\, \ll {{2m{c^0}} \over \hbar}\)  \({\omega ^2} = c_{\mathcal S}^2{k^2}\)  \({\omega ^2} = c_{\mathcal S}^2{k^2}\)  \({\omega ^2} = {{m_{{\rm{eff}}}^2c_{{\mathcal S},{\rm{gap}}}^4} \over {{\hbar ^2}}} + c_{{\mathcal S},{\rm{gap}}}^2{k^2}\) 
\({{2m{c^0}} \over \hbar} \ll k \ll {{m{u^0}} \over \hbar}\)  \(\hbar \omega = {{{{(\hbar ck)}^2}} \over {2\mu}}\)  
\(k\, \gg {{m{u^0}(1 + b)} \over \hbar}\)  ω^{2} = c^{2}k^{2} 
4.2.2 The heliocentric universe
Helium is one of the most fascinating elements provided by nature. Its structural richness confers on helium a paradigmatic character regarding the emergence of many and varied macroscopic properties from the microscopic world (see [660] and references therein). Here, we are interested in the emergence of effective geometries in helium, and their potential use in testing aspects of semiclassical gravity.
Helium four, a bosonic system, becomes superfluid at low temperatures (2.17 K at vapour pressure). This superfluid behaviour is associated with condensation in the vacuum state of a macroscopically large number of atoms. A superfluid is automatically an irrotational and inviscid fluid, so, in particular, one can apply to it the ideas worked out in Section 2. The propagation of classical acoustic waves (scalar waves) over a background fluid flow can be described in terms of an effective Lorentzian geometry: the acoustic geometry. However, in this system one can naturally go considerably further, into the quantum domain. For long wavelengths, the quasiparticles in this system are quantum phonons. One can separate the classical behaviour of a background flow (the effective geometry) from the behaviour of the quantum phonons over this background. In this way one can reproduce, in laboratory settings, different aspects of quantum field theory over curved backgrounds. The speed of sound in the superfluid phase is typically on the order of cm/sec. Therefore, at least in principle, it should not be too difficult to establish configurations with supersonic flows and their associated ergoregions.
Helium three, the fermionic isotope of helium, in contrast, becomes superfluid at much lower temperatures (below 2.5 milliK). The reason behind this rather different behaviour is the pairing of fermions to form effective bosons (Cooper pairing), which are then able to condense. In the ^{3}HeA phase, the structure of the fermionic vacuum is such that it possesses two Fermi points, instead of the more typical Fermi surface. In an equilibrium configuration one can choose the two Fermi points to be located at {p_{ x } = 0, p_{ y } = 0, p_{ z } = ±p_{ F }} (in this way, the zaxis signals the direction of the angular momentum of the pairs). Close to either Fermi point the spectrum of quasiparticles becomes equivalent to that of Weyl fermions. From the point of view of the laboratory, the system is not isotropic, it is axisymmetric. There is a speed for the propagation of quasiparticles along the zaxis, c_{‖} ≃ cm/sec, and a different speed, c_{⊥} ≃ 10^{−5} c_{‖}, for propagation perpendicular to the symmetry axis. However, from an internal observer’s point of view this anisotropy is not “real”, but can be made to disappear by an appropriate rescaling of the coordinates. Therefore, in the equilibrium case, we are reproducing the behaviour of Weyl fermions over Minkowski spacetime. Additionally, the vacuum can suffer collective excitations. These collective excitations will be experienced by the Weyl quasiparticles as the introduction of an effective electromagnetic field and a curved Lorentzian geometry. The control of the form of this geometry provides the sought for gravitational analogy.
The advantage of using surface waves instead of bulk waves in superfluids is that one could create horizons without reaching supersonic speeds in the bulk fluid. This could alleviate the appearance of dynamical instabilities in the system, that in this case are controlled by the strength of the interaction of the ripplons with bulk degrees of freedom [657, 659].
4.2.3 Slow light in fluids
The geometrical interpretation of the motion of light in dielectric media leads naturally to conjecture that the use of flowing dielectrics might be useful for simulating general relativity metrics with ergoregions and black holes. Unfortunately, these types of geometry require flow speeds comparable to the group velocity of the light. Since typical refractive indexes in nondispersive media are quite close to unity, it is then clear that it is practically impossible to use them to simulate such general relativistic phenomena. However recent technological advances have radically changed this state of affairs. In particular the achievement of controlled slowdown of light, down to velocities of a few meters per second (or even down to complete rest) [617, 338, 96, 353, 506, 603, 565], has opened a whole new set of possibilities regarding the simulation of curvedspace metrics via flowing dielectrics.
But how can light be slowed down to these “snaillike” velocities? The key effect used to achieve this takes the name of Electromagnetically Induced Transparency (EIT). A laser beam is coupled to the excited levels of some atom and used to strongly modify its optical properties. In particular one generally chooses an atom with two longlived metastable (or stable) states, plus a higher energy state that has some decay channels into these two lower states. The coupling of the excited states induced by the laser light can affect the transition from a lower energy state to the higher one, and hence the capability of the atom to absorb light with the required transition energy. The system can then be driven into a state where the transitions between each of the lower energy states and the higher energy state exactly cancel out, due to quantum interference, at some specific resonant frequency. In this way the higherenergy level has null averaged occupation number. This state is hence called a “dark state”. EIT is characterised by a transparency window, centered around the resonance frequency, where the medium is both almost transparent and extremely dispersive (strong dependence on frequency of the refractive index). This in turn implies that the group velocity of any light probe would be characterised by very low real group velocities (with almost vanishing imaginary part) in proximity to the resonant frequency.
In any case, the existence of this ADM form already tells us that an ergoregion will always appear once the norm of the effective velocity exceeds the effective speed of light (which for slow light is approximately c/α, where α can be extremely large due to the huge dispersion in the transparency window around the resonance frequency ω_{0}). However, a trapped surface (and hence an optical black hole) will form only if the inward normal component of the effective flow velocity exceeds the group velocity of light. In the slow light setup so far considered such a velocity turns out to be \(u = c/(2\sqrt {\alpha)}\).
The realization that ergoregions and event horizons can be simulated via slow light may lead one to the (erroneous) conclusion that this is an optimal system for simulating particle creation by gravitational fields. However, as pointed out by Unruh in [470, 612], such a conclusion would turn out to be overenthusiastic. In order to obtain particle creation through “mode mixing”, (mixing between the positive and negative norm modes of the incoming and outgoing states), an inescapable requirement is that there must be regions where the frequency of the quanta as seen by a local comoving observer becomes negative.
In a flowing medium this can, in principle, occur thanks to the tilting of the dispersion relation due to the Doppler effect caused by the velocity of the flow Equation (291); but this also tells us that the negative norm mode must satisfy the condition ω_{0} − u · k < 0, but this can be satisfied only if the velocity of the medium exceeds ω_{0}/k, which is the phase velocity of the probe light, not its group velocity. This observation suggests that the existence of a “phase velocity horizon” is an essential ingredient (but not the only essential ingredient) in obtaining Hawking radiation. A similar argument indicates the necessity for a specific form of “group velocity horizon”, one that lies on the negative norm branch. Since the phase velocity in the slow light setup we are considering is very close to c, the physical speed of light in vacuum, not very much hope is left for realizing analogue particle creation in this particular laboratory setting.
However, it was also noticed by Unruh and Schützhold [612] that a different setup for slow light might deal with this and other issues (see [612] for a detailed summary). In the setup suggested by these authors there are two strongbackground counterpropagating control beams illuminating the atoms. The field describing the beat fluctuations of this electromagnetic background can be shown to satisfy, once the dielectric medium is in motion, the same wave equation as that on a curved background. In this particular situation the phase velocity and the group velocity are approximately the same, and both can be made small, so that the previously discussed obstruction to mode mixing is removed. So in this new setup it is concretely possible to simulate classical particle creation such as, e.g., superradiance in the presence of ergoregions.
Nonetheless, the same authors showed that this does not open the possibility for a simulation of quantum particle production (e.g., Hawking radiation). This is because that effect also requires the commutation relations of the field to generate the appropriate zeropoint energy fluctuations (the vacuum structure) according to the Heisenberg uncertainty principle. This is not the case for the effective field describing the beat fluctuations of the system we have just described, which is equivalent to saying that it does not have a proper vacuum state (i.e., analogue to any physical field). Hence, one has to conclude that any simulation of quantum particle production is precluded.
4.2.4 Slow light in fibre optics
In addition to the studies of slow light in fluids, there has now been a lot of work done on slow light in a fibreoptics setting [505, 504, 64, 63], culminating in recent experimental detection of photons apparently associated with a phasevelocity horizon [66]. The key issue here is that the Kerr effect of nonlinear optics can be used to change the refractive index of an optical fibre, so that a “carrier” pulse of light traveling down the fibre carries with it a region of high refractive index, which acts as a barrier to “probe” photons (typically at a different frequency). If the relative velocities of the “carrier” pulse and “probe” are suitably arranged then the arrangement can be made to mimic a blackholewhitehole pair. This system is described more fully in Section 6.4.
4.2.5 Lattice models
The quantum analogue models described above all have an underlying discrete structure: namely the atoms they are made of. In abstract terms one can also build an analogue model by considering a quantum field on specific lattice structures representing different spacetimes. In [310, 149, 322] a fallinglattice blackhole analogue was put forward, with a view to analyzing the origin of Hawking particles in blackhole evaporation. The positions of the lattice points in this model change with time as they follow freely falling trajectories. This causes the lattice spacing at the horizon to grow approximately linearly with time. By definition, if there were no horizons, then for long wavelengths compared with the lattice spacing one would recover a relativistic quantum field theory over a classical background. However, the presence of horizons makes it impossible to analyze the field theory only in the continuum limit, it becomes necessary to recall the fundamental lattice nature of the model.
4.2.6 Graphene
4.3 Going further
We feel that the catalogue we have just presented is reasonably complete and covers the key items. For additional background on many of these topics, we would suggest sources such as the books “Artificial Black Holes” [470] and “The Universe in a Helium Droplet” [660]. For more specific detail, check this review’s bibliography, and use SPIRES (or the beta version of INSPIRE) to check for recent developments.
5 Phenomenology of Analogue Models
Of course, the entire motivation for looking at analogue models is to be able to learn more physics. One could start studying analogue models with the idea of seeing whether it is possible (either theoretically or in practice) to reproduce in the laboratory various gravitational phenomena whose real existence in nature cannot be currently checked. These are phenomena that surpass our present (and foreseeable) observational capabilities, but yet, we believe in their existence because it follows from seemingly strong theoretical arguments within the standard frameworks of general relativity and field theory in curved space. However, the interest of this approach is not merely to reproduce these gravitational phenomena in some formal analogue model, but to see which departures from the ideal case show up in real analogue models, and to analyse whether similar deviations are likely to appear in real gravitational systems.
When one thinks about emergent gravitational features in condensedmatter systems, one immediately realises that these features only appear in the lowenergy regime of the analogue systems. When these systems are probed at high energies (short length scales) the effective geometrical description of the analogue models break down, as one starts to be aware that the systems are actually composed of discrete pieces (atoms and molecules). This scenario is quite similar to what one expects to happen with our geometrical description of the Universe, when explored with microscopic detail at the Planck scale.
That is, the study of analogue models of general relativity is giving us insights as to how the standard theoretical picture of different gravitational phenomena could change when taking into account additional physical knowledge coming from the existence of an underlying microphysical structure. Quite robustly, these studies are telling us already that the first deviations from the standard general relativity picture can be encoded in the form of highenergy violations of Lorentz invariance in particle dispersion relations. Beyond these first deviations, the analogue models of general relativity provide wellunderstood examples (the underlying physics is well known) in which a description in terms of fields in curved spacetimes shows up as a lowenergyregime emergent phenomena.
The analogue models are being used to shed light on these general questions through a number of specific routes. Let us now turn to discussing several specific physics issues that are being analysed from this perspective.
5.1 Hawking radiation
5.1.1 Basics
 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 anticommutation relations [612]).
 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}
 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 transPlanckian problem of Hawking radiation. To create an analogue model exhibiting Hawking radiation will be, therefore, equivalent to giving a solution to the transPlanckian 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 [657]. 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 onedimensional 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 [48]. 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 14), but without actually crossing into the supersonic regime [37], softening in this way the appearance of dynamical instabilities.
 3.
Real analogue models cannot, strictly speaking, reproduce eternal blackhole 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 [613, 35].
 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 ∼ 100 nK [48]. 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 highlyquantum 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 ([278, 279, 280], and references therein).
Because of its importance, let us now review what we know about the effects of highenergy dispersion relations on the Hawking process.
5.1.2 UV robustness
We saw in the introduction to this section that the transPlanckian 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.
 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 noncompactness of the boost subgroup. The assumption of an exact boost symmetry is linked to the scalefree nature of spacetime given that unbounded boosts expose ultrashort distances. Hence, the assumption of exact Lorentz invariance needs, in the end, to rely on some ideas regarding the nature of spacetime at ultrashort distances.
 2.
Worse, even given these assumptions, “one cannot boost away an swave”. That is, given the expected isotropy of Hawking radiation, a boost in any given direction could, at most, tame the transPlanckian 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 [307, 308, 608]. In particular investigators explored the possibility that spacetime microphysics could provide a shortdistance, Lorentzbreaking 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 [607]. However, such ideas were running into serious difficulties given that a naive shortdistance 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 [307] and references therein, and the comments in [278, 279, 280]). Indeed there are alternative ways through which the effect of the shortscale 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 blackhole backgrounds; they may be endowed with quantizable perturbations and, in most of the cases, they have a wellknown microscopic structure. Given that Hawking radiation can be, at least in principle, simulated in such systems, one might ask how and if the transPlanckian problem is resolved in these cases.
5.1.2.1 Modified dispersion relations
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 lowestorder 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 ±k^{3}/K or ±k^{4}/K^{2}.) 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 transPlanckian 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 [307, 308, 608, 148], and the question of whether such phenomenology could be applied to the case of real black holes (see e.g., [94, 310, 148, 490]).^{22} In all the aforementioned works, Hawking radiation can be recovered under some suitable assumptions, as long as neither the blackhole 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., [613] for further details).
5.1.2.2 Subluminal dispersion relations
Let us mention here (more details in Section 6.1) that the classical counterpart of the abovedescribed modeconversion mechanism has recently been observed for the first time in a wave tank experiment [682]. 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.
5.1.2.3 Superluminal dispersion relations
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 [413] (for previous work dealing with mode conversion through a horizon, see [386]). One of the asymptotic regions corresponds to the asymptotic region of a blackhole 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 KleinGordon field equation in this geometry, modified by a \(\partial _x^4\) term that gives rise to the quartic dispersion relation ω^{2} = k^{2} + k^{4}/K^{2}. For this setup it has been shown that the relevant Bogoliubov coefficients have the form β_{ ω′,i;ω,j } = δ(ω − ω′)β_{ ω;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 ≪ ω_{ c } where ω_{ c } = Kf(v_{int−asym}) and with f(·) a specific function of v_{int−asym}, the value of the flow velocity at the internal (or supersonic) asymptotic region [413]. In all cases, for ω > ω_{ c } the Bogoliubov coefficients are exactly zero. When v_{int−asym} is just above c = 1, that is, when the flow is only “slightly supersonic”, the function f(·) can become very small f(v_{int−asym}) ∝ (v_{int−asym} − 1)^{3/2}, and thus also the critical frequency ω_{ c } 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) = ω_{0} and a value of zero for ω → ω_{ c }. This transition is remarkably sudden for the smooth profiles analyzed. If κ_{0} ≪ ω_{ c }, 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 invacuum state. Is this vacuum state, which has thermal properties, in terms of outobservers? 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 semiclassically 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 [412] with identical qualitative results. The reasons for this is that, although the Bogoliubovde 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 [119, 118] the numerical observation of the Hawking effect in simulations in which a blackhole horizon has been created dynamically from an initially homogeneous flow. The observation of the effect has been through the calculation of twopoint correlation functions (see Section 5.1.6 below). These simulations strongly suggest that any nonquasistatic 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 [150] 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 [150] is that the compact ergoregion characterizing such configurations is unstable to selfamplifying Hawking radiation. The presence of such an instability was also identified in the dynamical analysis carried on in [231, 232, 29] where BoseEinstein 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 selfamplified particle emission; from this feature arises their name “blackhole lasers”. In the recent analyses in [155, 199] 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

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 subPlanckian dynamics).
5.1.4 Source of the Hawking quanta
There is a point of view (not universally shared within the community) that asserts that the transPlanckian problem also makes it clear that the rayoptics limit cannot be the whole story behind Hawking radiation. Indeed, it is precisely the ray optics approximation that leads to the transPlanckian 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 quantummechanically unstable and will ultimately be the “origin” of the Hawking photons. If this picture is correct, then the blackhole particle production is a lowfrequency and lowwavenumber process. See, for instance, [563, 610, 611]. 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 timedependent 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 Hawkinglike fluxes from nonstationary systems (e.g., evaporating black holes) and in such systems it never coincides with the inaffinitybased definition of the surface gravity except possibly at asymptotic futuretimelike infinity i^{+}. Early comments along these lines can be found in [94, 95]; more recently this point was highlighted in [42, 41].
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 [37, 41] and clearly stressed in [412]. 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, 505, 504, 532, 66, 682, 412] 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 wouldbe 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 wellestablished 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 (61)). 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., [48]): 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 nanoKelvin 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 twopoint 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 [48], 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, 118, 190, 496, 512, 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 outpowered 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 densitydensity 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 nonexcessively 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 [682].) 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 timevarying external fields (dynamical Casimir effect) in BoseEinstein condensates (where the timevarying 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 Lorentzviolations 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 transPlanckian problem. In particular, it was soon recognised [149, 311] 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 (302), one can see that if one keeps tracking a “progenitor” incoming mode back in time, then its group velocity (in the comoving 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 transPlanckian problem has been moved from the region near the horizon out to the region near infinity. In the CorleyJacobson model (303) 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 (303) (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 (302) or (303) 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 modemixing 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 comoving 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 transPlanckian 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 ultraviolet cutoff due to the microphysics were considered [522, 523, 149]. In particular, in [523] 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 “transPlanckian” modes with k > K are in their ground state near the horizon.
An alternative avenue was considered in [149]. 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 intrinsicallyregularised behaviour of the wave vectors field. In [149] the authors explicitly considered the Hawking process for a discretised version of a scalar field, where the lattice is associated with the freefall 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 shortdistance 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 longwavelength 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

“Top down” calculations of Hawking radiation starting from some idealised model of quantum gravity [4, 335, 461, 462].

“Bottom up” calculations of Hawking radiation starting from curved space quantum field theory [52, 53, 108, 109, 124, 184, 185, 216, 428, 430, 429, 431, 489, 526].

TransHawking versions of Hawking radiation, either as reformulations of the physics, or as alternative scenarios [62, 120, 222, 223, 257, 266, 278, 279, 280, 384, 478, 480, 530, 536, 541, 542, 549, 550, 567, 591].

Black hole entropy viewed in the light of analogue spacetimes [158].

Hawking radiation interpreted as a statement about particles traveling along complex spacetime trajectories [481, 567, 586].
5.2 Dynamical stability of horizons
Although the two issues are very closely related, as we will soon see, we have to carefully distinguish between the stability analysis of the modes of a linear field theory (with or without modified dispersion relations — MDR) over a fixed background, and the stability analysis of the background itself.
5.2.1 Classical stability of the background (no MDR)
In a normal mode analysis one requires boundary conditions such that the field is regular everywhere, even at infinity. However, if one is analysing the solutions of the linear field theory as a way of probing the stability of the background configuration, one can consider less restrictive boundary conditions. For instance, one can consider the typical boundary conditions that lead to quasinormal modes: These modes are defined to be purely outgoing at infinity and purely ingoing at the horizon; but one does not require, for example, the modes to be normalizable. The quasinormal modes associated with this sink configuration have been analysed in [69]. The results found are qualitatively similar to those in the classical linear stability analysis of the Schwarzschild black hole in general relativity [619, 620, 521, 698, 447]. Of course, the gravitational field in general relativity has two dynamical degrees of freedom — those associated with gravitational waves — that have to be analysed separately; these are the “axial” and “polar” perturbations. In contrast, in the present situation we only have scalar perturbations. Nevertheless, the potentials associated with “axial” and “polar” perturbations of Schwarzschild spacetime, and that associated with scalar perturbations of the canonical acoustic black hole, produce qualitatively the same behaviour: There is a series of damped quasinormal modes — proving the linear stability of the system — with higher and higher damping rates.
An important point we have to highlight here is that, although in the linear regime the dynamical behaviour of the acoustic system is similar to general relativity, this is no longer true once one enters the nonlinear regime. The underlying nonlinear equations in the two cases are very different. The differences are so profound, that in the general case of acoustic geometries constructed from compressible fluids, there exist sets of perturbations that, independent of how small they are initially, can lead to the development of shocks, a situation completely absent in vacuum general relativity.
5.2.2 Semiclassical stability of the background (no MDR)
Now, given an approximately stationary, and at the very least metastable, classical blackholelike configuration, a standard quantum mode analysis leads to the existence of Hawking radiation in the form of phonon emission. This shows, among other things, that quantum corrections to the classical behaviour of the system must make the configuration with a sonic horizon dynamically unstable against Hawking emission. As a consequence, in any system (analogue or general relativistic) with quantum fluctuations that maintain strict adherence to the equivalence principle (no MDR), it must then be impossible to create an isolated truly stationary horizon by merely setting up external initial conditions and letting the system evolve by itself. However, in an analogue system a truly stationary horizon can be set up by providing an external power source to stabilise it against Hawking emission. Once one compensates, by manipulating external forces, for the backreaction effects that in a physical general relativity scenario cause the horizon to shrink or evaporate, one would be able to produce, in principle, an analogue system exhibiting precisely a stationary horizon and a stationary Hawking flux.
Let us describe what happens when one takes into account the existence of MDR. Once again, a wonderful physical system that has MDR explicitly incorporated in its description is the BoseEinstein condensate. The macroscopic wave function of the BEC behaves as a classical irrotational fluid but with some deviations when short length scales become involved. (For length scales on the order of, or shorter than, the healing length.) What are the effects of the MDR on the dynamical stability of a blackholelike configuration in a BEC? The stability of a sink configuration in a BEC has been analysed in [231, 232] but taking the flow to be effectively onedimensional. What these authors found is that these configurations are dynamically unstable: There are modes satisfying the appropriate boundary conditions such that the imaginary parts of their associated frequencies are positive. These instabilities are associated basically with the bound states inside the black hole. The dynamical tendency of the system to evolve is suggestively similar to that in the standard evaporation process of a black hole in semiclassical general relativity.
5.2.3 Classical stability of the background (MDR in BECs)

The “classical” or macroscopic wave function of the BEC represents the classical spacetime of GR, but only when probed at longenough wavelengths such that it behaves as pure hydrodynamics.

The “classical” longwavelength perturbations to a background solution of the GrossPitaevskii equation correspond to classical gravitational waves in GR. Of course, this analogy does not imply that these are spin 2 waves; it only points out that the perturbations are made from the same “substance” as the background configuration itself.

The macroscopic wave function of the BEC, without the restriction of being probed only at long wavelengths, corresponds to some sort of semiclassical vacuum gravity. Its “classical” behaviour (in the sense that does not involve any probability notion) is already taking into account, in the form of MDR, its underlying quantum origin.

The Bogoliubov quantum quasiparticles over the “classical” wave function correspond to a further step away from semiclassical gravity in that they are analogous to the existence of quantum gravitons over a (semiclassical) background spacetime.

If the perturbations to the BEC background configuration have “classical seeds” (that is, are describable by the linearised GrossPitaevskii equation alone), then, one will have “classical” instabilities.

If the perturbations have “quantum seeds” (that is, are described by the Bogoliubov equations), then, one will have “quantum” instabilities.
5.2.4 Black holes, white holes, and rings
Under outgoing and convergent boundary conditions in both asymptotic regions, in [29] it was concluded that there are no instabilities in any of the straight line (nonring) configurations. If one relaxed the convergence condition in the downstream asymptotic region, (the region that substitutes the unknown internal region, and so the region that might require a different treatment for more realistic black hole configurations), then the black hole is still stable, while the white hole acquires a continuous region of instability, and the blackholewhitehole configuration shows up as a discrete set of unstable modes. The whitehole instability was previously identified in [386]. Let us mention here that the stable blackhole configuration has been also analyzed in terms of stable or quasinormal modes in [30]. It was found that, although the particular configurations analyzed (containing idealised steplike discontinuities in the flow) did not posses quasinormal modes in the acoustic approximation, the introduction of dispersion produced a continuous set of quasinormal modes at transPlanckian frequencies.
Continuing with the analysis of instabilities, in contrast to [29], the more recent analysis in [155, 199] consider only convergent boundary conditions in both asymptotic regions. They argue that the ingoing contributions that these modes sometimes have always correspond to waves that do not carry energy, so that they have to be kept in the analysis, as their ingoing character should not be interpreted as an externallyprovoked instability^{25}. If this is confirmed, then the appropriate boundary condition for instability analysis under dispersion would be just the convergent condition, as in nondispersive theories.
Under these convergent conditions, the authors of [155, 199] show that the previouslyconsidered blackhole and whitehole configurations in BECs are stable. (Let us remark that this does not mean that configurations with a more complicated internal region need be stable.) However, blackholewhitehole configurations do show a discrete spectrum of instabilities. In these papers, one can find a detailed analysis of the strength of these instabilities, depending on the form and size of the intermediate supersonic region. For instance, it is necessary that the supersonic region acquire a minimum size so that the first unstable mode appears. (This feature was also observed in [29].) When the previous mode analysis is used in the context of a quantum field theory, as we mention in Section 5.1, one is led to the conclusion that blackholewhitehole configurations emit particles in a selfamplified (or runaway) manner [150, 155, 199]. Although related to Hawking’s process, this phenomenon has a quite different nature. For example, there is no temperature associated with it.
When the blackholewhitehole configuration is compactified in a ring, it is found that there are regions of stability and instability, depending on the parameters characterizing the configuration [231, 232]. We suspect that the stability regions appear because of specific periodic arrangements of the modes around the ring. Among other reasons, these arrangements are interesting because they could be easier to create in the laboratory with current technology, and their instabilities easier to detect than Hawking radiation itself.
To conclude this subsection, we would like to highlight that there is still much to be learned by studying the different levels of description of an analogue system, and how they influence the stability or instability of configurations with horizons.
5.3 Superradiance
Again, these processes have a purely kinematical origin, so they are perfectly suitable for being reproduced in an analogue model. Regarding these processes, the simplest geometry that one can reproduce, thinking of analogue models based on fluid flows, is that of the draining bathtub of Section 2. Of course, this metric does not exactly correspond to Kerr geometry, nor even to a section of it [641, 633]. However, it is qualitatively similar. It can be used to simulate both Penrose’s classical process and quantum superradiance, as these effects do not depend on the specific multipole decomposition of Kerr’s geometry, but only on its rotating character. A specific experimental setup has been put forward by Schützhold and Unruh using gravity waves in a shallow basin mimicking an ideal draining bathtub [560]. Equivalent to what happens with Kerr black holes, this configuration is classically stable in vacuum (in the linear regime) [69]. A word of caution is in order here: Interactions of the gravity surface waves with bulk waves (neglected in the analysis) could cause the system to become unstable [657]. This instability has no counterpart in standard general relativity (though it might have one in braneworld theories). Superresonant scattering of waves in this rotating sink configuration, or in a simple purely rotating vortex, could in principle be observed in this and other analogue models. There are already several articles dealing with this problem [55, 57, 56, 112, 193, 392]. Most recently, see [524], where necessary and sufficient conditions for superradiance were investigated.
A related phenomenon one can consider is the blackhole bomb mechanism [513]. One would only have to surround the rotating configuration by a mirror for it to become grossly unstable. What causes the instability is that those ingoing waves that are amplified when reflected in the ergosphere would then in turn be reflected back toward the ergoregion, due to the exterior mirror, thus being amplified again, and so on.
An interesting phenomenon that appears in many condensed matter systems is the existence of quantised vortices. The angular momentum of these vortices comes in multiples of some fundamental unit (typically ℏ or something proportional to ℏ). The extraction of rotational energy by a Penrose process in these cases could only proceed via finiteenergy transitions. This would supply an additional specific signature to the process. In such a highly quantum configuration, it is also important to look for the effect of having highenergy dispersion relations. For example, in BECs, the radius of the ergoregion of a single quantised vortex is on the order of the healing length, so one cannot directly associate an effective Lorentzian geometry with this portion of the configuration. Any analysis that neglects the highenergy terms is not going to give any sensible result in these cases.
5.4 Cosmological particle production

Fedichev and Fischer [195, 194] have investigated WKB estimates of the cosmological particle production rate and (1+1) dimensional cosmologies, both in expanding BECs.

Lidsey [403], and Fedichev and Fischer [196] have focussed on the behaviour of cigarlike condensates in grosslyasymmetric traps.

Barceló et al. [46, 47] have focussed on BECs and tried to mimic FLRW behaviour as closely as possible, both via free expansion, and via external control of the scattering length using a Feshbach resonance.

Fischer and Schützhold [206] propose the use of twocomponent BECs to simulate cosmic inflation.

Weinfurtner [674, 675] has concentrated on the approximate simulation of de Sitter spacetimes.

Weinfurtner, Jain, et al. have undertaken both numerical [328] and general theoretical [677, 683] analyses of cosmological particle production in a BECbased FLRW universe.
An interesting sideeffect of the original investigation, is that birefringence can now be used to model “variable speed of light” (VSL) geometries [58, 181]. Since analogue models quite often lead to two or more “excitation cones”, rather than one, it is quite easy to obtain a bimetric or multimetric model. If one of these metrics is interpreted as the “gravitational” metric and the other as the “photon” metric, then VSL cosmologies can be given a mathematically welldefined and precise meaning [58, 181].
5.5 Bose novae: an example of the reverse flow of information?
As we have seen in the previous Sections ?? and 5.4, analogue models have in the past been very useful in providing new, condensedmatterphysicsinspired ideas about how to solve longstanding problems of semiclassical gravity. In closing this section, it is interesting to briefly discuss what perhaps represents, so far, the only attempt to use analogue models in the reverse direction; that is to import wellknown concepts of semiclassical gravity into condensed matter frameworks.
The phenomenon we are referring to is the “Bose nova” [175]. This is an experiment dealing with a gas of a few million ^{85}Rb atoms at a temperature of about 3 nK. The condensate is rendered unstable by exploiting the possibility of tuning the interaction (more precisely the scattering length) between the atoms via a magnetic field. Reversing the sign of the interaction, making it attractive, destabilises the condensate. After a brief waiting time (generally called t_{collapse}), the condensate implodes and loses a sizable fraction of its atoms in the form of a “nova burst”. If left to evolve undisturbed, the number of atoms in the burst stabilises and a remnant condensate is left. However, if the condensate interaction is again made repulsive after some time evolve, before the condensate has sufficient time to stabilise, then the formation of “jets” of atoms is observed, these jets being characterised by lower kinetic energy and a distinct shape with respect to the burst emission.
Interestingly, an elegant explanation of such a phenomenology was proposed in [105, 106], based on the wellknown semiclassical gravity analysis of particle creation in an expanding universe. In fact, the dynamics of quantum excitations over the collapsing BEC were shown to closely mimic that for quantum excitations in a timereversed (collapsing instead of expanding) scenario for cosmological particle creation. This is not so surprising as the quantum excitations above the BEC ground state feel a timevarying background during the collapse, and, as a consequence, one then expects squeezing of the vacuum state and mode mixing, which are characteristic of quantum field theory in variable external fields.
However, the analogy is even deeper than this. In fact, in [105, 106] a key role in explaining the observed burst and jets is played by the concepts of “frozen” versus “oscillating” modes — borrowed from cosmology — (although with a reverse dynamics with respect to the standard (expanding) cosmological case). In the case of Bose novae, the modes, which are amplified, are those for which the physical frequency is smaller than the collapse rate, while modes with higher frequencies remain basically unaffected and their amplitudes obey a harmonic oscillator equation. As the collapse rate decreases, more and more modes stop growing and start oscillating, which is equivalent to a creation of particles from the quantum vacuum. In the case of a sudden stop of the collapse by a new reversal of the sign of the interaction, all of the previously growing modes are suddenly converted into particles, explaining in this way the generation of jets and their lower energy (they correspond to modes with lower frequencies with respect to those generating the bursts).
Although this simple model cannot explain all the details of the Bose novae phenomenology, we think it is remarkable how far it can go in explaining several observed features by exploiting the language and techniques so familiar to quantum cosmology. In this sense, the analysis presented in [105, 106] primarily shows a possible new application of analogue models, where they could be used to lend ideas and techniques developed in the context of gravitational physics to the explanation of condensed matter phenomena.
5.6 Romulan cloaking devices
A wonderful application of analogue gravity techniques is the design of cloaking devices [385, 498, 387]. How to achieve invisibility, or more properly, low observability has been a matter of extensive study for decades. With the appearance of a technology capable of producing and controlling metamaterials and plasmonic structures [3], cloaking is becoming a real possibility.
To achieve cloaking, one needs to ensure that light rays (beyond geometric optics it is impossible to produce perfect invisibility [449, 690]) effectively behave as if they were propagating in Minkowski spacetime, although in reality they are bending around the invisible compact region. One way of producing this is, precisely, to make rays propagate in Minkowski spacetime but using nonCartesian coordinates. Take the Minkowski metric in some Cartesian coordinates x′, η_{ μν }, and apply a diffeomorphism, which is different from the identity only inside the compact region. One then obtains a different representation g_{ μν } (x) of the flat geometry. Now take the x to be the Cartesian coordinates of the real laboratory spacetime, and build this metric with the metamaterial. By construction, the scattering process with the compact region will not change the directions of the rays, making everything within the compact region invisible. Recent implementations of these ideas have investigated the concept of a “spacetime cloak” or “history editor” that cloaks a particular event, not a particular region [439].
To end this brief account we would like to highlight the broad scope of application of these ideas: Essentially the cloaking techniques can be applied to any sort of wave, from acoustic cloaking [133] to earthquake damage prevention [192]. More radically, and with enough civil engineering, one might adapt the ideas of Berry [67, 68] to antitsunami cloaking.
5.7 Going further
For more details on the transPlanckian problem, some of the key papers are the relatively early papers of Unruh [608] and Jacobson [307, 308]. For superradiance and cosmological issues (especially particle production) there seems to be considerable ongoing interest, and one should carefully check SPIRES (or the beta version of INSPIRE) for the most recent articles.
6 Experimental efforts
In recent years several efforts towards the detection of analogue Hawking radiation have been carried out with different physical systems. Here we report a brief list of the ongoing efforts. There is already definite evidence for mode conversion and negativenorm modes (group velocity opposite to phase velocity) [532], and more recently for stimulated Hawking emission [682]. There are very recent claims of photon detection from a phase velocity horizon in an optical fibre [66], and for whiteholelike behaviour in hydraulic jumps [334]. It seems that reliable and reproducible experimental probes of spontaneous quantum Hawking radiation might be just a few years in the future.
6.1 Wave tank experiments
As we have seen, waves in shallow water can be considered to be a particularly simple analogue gravity system. (See Section 4.1.3.) Experimentally, water basins are relatively cheap and easy to construct and handle. In particular, shallow water basins (more precisely, wave tanks or wave flumes) have acquired a prominent role in recent years. In particular, such technology underlay the 1983 work of Badulin et al. [17], and such wave tanks are currently used by groups in Nice, France [532] and Vancouver, Canada [682]. The Nice experiments have been carried out using a large wavetank 30 m long, 1.8 m wide and 1.8 m deep. The simplest setup with such a device is to send water waves (e.g., produced by a piston) against a fluid flow produced by a pump. To generate a waterwave horizon, a ramp is placed in the water, with positive and negative slopes separated by a flat section. When a train of waves is sent against the reverse fluid flow there will be a place where the flow speed equals the group velocity of the waves — there a group velocity horizon will be created. (Remember that in the shallowwater regime the low momentum comoving dispersion relation for surface water waves is ω^{2} = gk tanh(kh) where g denotes the gravitational acceleration of the Earth at the water surface and h is the height of the channel.) For incident waves moving against the flow it would be impossible to cross such a horizon, and in the sense that the system is the analogue of a whitehole horizon, the time reversal of a blackhole horizon. (In the engineering and fluid mechanics literature this effect is typically referred to as “wave blocking”.)
Remarkably, in 2007 Rousseaux et al. [532] reported the first direct observation of negativefrequency waves, converted from positivefrequency waves in a moving medium, albeit the degree of mode conversion appears to be significantly higher than that expected from theory. The same group has now set up a more compact experiment based on the hydraulic jump, wherein measurements of the “Froude cones” convincingly demonstrate the presence of a surfacewave white hole [334], (as described in Sections 4.1.3 and 4.1.4 and possibly implicit in the results of Badulin et al. [17]).
6.2 BoseEinstein condensate experiments
We have already extensively discussed the theoretical aspects of BoseEinsteincondensatebased analogue models. Regarding the actual experimental possibility of generating BECbased acoustic black holes, several options have been envisaged in the literature. In particular, commonly proposed settings are long thin condensates in a linear or circular trap [231, 232] as well as Lavalnozzleshaped traps [48, 539, 228, 476]. However, it is only very recently that an experiment aimed at the formation of a sonic black hole in a BEC has been set up, and the creation of a sonic horizon has been convincingly argued for [369]. In this case, a sonic horizon was achieved by a counterintuitive effect of “density inversion”, in which a deep potential minimum creates a region of low density, as would a potential maximum. This low density region corresponds to a slowerthannormal speed of sound, and hence to the possibility for the flow speed to exceed the speed of sound and generate sonic horizons at the crossing points (where the speed of the flow and that of sound coincide). The density inversion is achieved by overlapping a lowfrequency (broad) harmonic potential and a highfrequency (narrow) Gaussian potential generated via an elongated laser. In this manner a sonic black hole was generated, and kept stable for about 8 ms. The Hawking radiation predicted for the system as realised has a temperature of about 0.20–0.35 nK; unfortunately, one order of magnitude smaller than the lowest temperature allowed by the size of the system. (Lower temperatures for the condensate permit longerwavelength characteristic Hawking quanta, which must still fit into the condensate. So there is a tradeoff between Hawking temperature and physical size of the condensate.) However, higher densities could allow one to increase the Hawking temperature, and T_{ H } ≈ a few nK seems within experimental reach. Given that 8 ms would correspond to one cycle of 6 nK Hawking radiation, it appears that increases in T_{ H } together with amelioration of the lifetime of the sonic black hole might put the detection of the analogue spontaneous quantum Hawking effect within experimental reach (via correlation experiments) in the near future.
6.3 Differentiallyrotating flows in superfluid helium
At the end of Section 4.2.2 we briefly described an analogue model based on the ripplons in the surface separating two differentiallymoving superfluids, in particular an ABinterphase in ^{3}He. These interphases are being produced in Helsinki’s Low Temperature Lab [76, 77, 201, 202]. The ABinterphase is prepared in a small quartz cylinder (3 mm radius times 11 cm long) inside a rotating cryostat. The ^{3}HeA is rotating with the cryostat while the ^{3}HeB remains at rest with respect to the lab. Among other things, in this setting the critical values at which instabilities appear as functions of the temperature, and the nature of these instabilities, are being investigated. These instabilities are related to the appearance of an ergoregion in the analogue metric for ripplons and to the KelvinHelmholtz instability [657]. In particular, this has been the first time that the KelvinHelmholtz instability has been observed in superfluids [76]. The nature of the instability in these experiments is controlled by the difference in velocities between the normal and superfluid components of the flow. It still remains to further lower the ambient temperature so as to probe the nature of the instabilities in the absence of any normal fluid component.
6.4 Fibreoptic models
A recent implementation of analogue models based on electrodynamics is that based on fibreoptic engineering [504, 505]. The basic idea in this case is to use long dispersive light pulses (solitons), generated with a suitable laser, to create a propagating front at which the refractive index of the fibre changes suddenly (albeit by a small amount). Basically, the refractive index of the fibre, n_{0}, acquires a time and positiondependent correction δn, which is proportional to the instantaneous pulse intensity I at a give spacetime position, δn ∝ I(t, x). The wavefront at which this change in the refractive index occurs will move naturally at a speed close to the speed of light (and fibre optic engineering allows one to control this feature). If one now sends a continuous wave of light, what we might call a probe, along the fibre in such a manner that the probe group velocity in the fibre is arranged to be slightly larger than the pulse group velocity, then it will be possible to obtain horizonlike effects. In fact, as the probe wave reaches the back of the pulse, the increase in the refractive index will slow it down, until the probe group velocity will match the pulse one. Effectively, the rear end of the pulse will act as a white hole for the probe wave. Similarly, there will be a point on the front side of the pulse where the two group velocities will match. This will be the equivalent of a blackhole horizon for the probe wave. In [504, 505] the behaviour of the probe waves at the pulse was investigated, and it was shown for the white hole case that the expected classical behaviour is theoretically reproduced. Since this behaviour lies at the core of the mechanism responsible for the mode conversion underlying the Hawking effect, it is then expected that the quantum counterpart should also be reproducible in this manner. Indeed, very recently Belgiorno et al. have reported experimental detection of photons from a blackholewhitehole configuration possessing a “phase velocity horizon” [66]. The underlying theory behind their specific experiment is considered in [63, 64].
6.5 Going further
There seems to be considerable ongoing interest in experimental probes of analogue spacetimes, and quantum effects in analogue spacetimes, and one should carefully check SPIRES (or the beta version of INSPIRE) for the most recent articles.
7 Towards a Theory of Quantum Gravity?

Backreaction.

Equivalence principle.

Diffeomorphism invariance.

Effective spintwo excitations.

WeinbergWitten theorem.

Emergent gravity.

The cosmological constant problem.

Quantum gravity phenomenology.

Quantum gravity.
7.1 Backreaction
There are important phenomena in gravitational physics whose understanding needs analysis well beyond classical general relativity and field theory on (fixed) curved background spacetimes. The blackhole evaporation process can be considered as paradigmatic among these phenomena. Here, we confine our discussion to this case. Since we are currently unable to analyse the entire process of blackhole evaporation within a complete quantum theory of gravity, a way of proceeding is to analyse the simpler (but still extremely difficult) problem of semiclassical backreaction (see, for example, [166, 137, 75, 224, 95, 431]). One takes a background blackhole spacetime, calculates the expectation value of the quantum energymomentum tensor of matter fields in the appropriate quantum state (the Unruh vacuum state for a radiating black hole), and then takes this expectation value as a source for the perturbed Einstein equations. This calculation gives us information about the tendency of spacetime to evolve under vacuum polarization effects.
A nice feature of analogue models of general relativity is that, although the underlying classical equations of motion have nothing to do with Einstein equations, the tendency of the analogue geometry to evolve due to quantum effects is formally equivalent (approximately, of course) to that in semiclassical general relativity. Therefore, the onset of the backreaction effects (if not their precise details) can be simulated within the class of analogue models. An example of the type of backreaction calculations one can perform are those in [23, 25]. These authors started from an effectively onedimensional acoustic analogue model, configured to have an acoustic horizon by using a Laval nozzle to control the flow’s speed. They then considered the effect of quantizing the acoustic waves over the background flow. To calculate the appropriate backreaction terms they took advantage of the classical conformal invariance of the (1+1)dimensional reduction of the system. In this case, we know explicitly the form of the expectation value of the energymomentum tensor trace (via the trace anomaly). The other two independent components of the energymomentum tensor were approximated by the Polyakov stress tensor. In this way, what they found is that the tendency of a leftmoving flow with one horizon is for it to evolve in such a manner as to push the horizon downstream at the same time that its surface gravity is decreased. This is a behaviour similar to what is found for nearextremal ReissnerNordström black holes. (However, we should not conclude that acoustic black holes are, in general, closely related to nearextremal ReissnerNordström black holes, rather than to Schwarzschild black holes. This result is quite specific to the particular onedimensional configuration analysed.)
Can we expect to learn something new about gravitational physics by analysing the problem of backreaction in different analogue models? As we have repeatedly commented, the analyses based on analogue models force us to consider the effects of modified highenergy dispersion relations. For example, in BECs, they affect the “classical” behaviour of the background geometry as much as the behaviour of the quantum fields living on the background. In seeking a semiclassical description for the evolution of the geometry, one would have to compare the effects caused by the modified dispersion relations to those caused by pure semiclassical backreaction (which incorporates deviations from standard general relativity as well). In other words, one would have to understand the differences between the standard backreaction scheme in general relativity, and that based on Equations (229) and (230).
To end this subsection, we would like to comment that one can go beyond the semiclassical backreaction scheme by using the stochastic semiclassical gravity programme [298, 301, 302]. This programme aims to pave the way from semiclassical gravity toward a complete quantumgravitational description of gravitational phenomena. This stochastic gravity approach not only considers the expectation value of the energymomentum tensor but also its fluctuations, encoded in the semiclassical EinsteinLangevin equation. In a very interesting paper [490], Parentani showed that the effects of the fluctuations of the metric (due to the ingoing flux of energy at the horizon) on the outgoing radiation led to a description of Hawking radiation similar to that obtained with analogue models. It would be interesting to develop the equivalent formalism for quantum analogue models and to investigate the different emerging approximate regimes.
7.2 Equivalence principle
Analogue models are of particular interest in that the analogue spacetimes that emerge often violate, to some extent, the Einstein equivalence principle [45, 638]. This is the heart and soul of any metric theory of gravity and is basically the requirement of the universality of free fall, plus local Lorentz invariance and local position invariance of nongravitational experiments.
As such, the Einstein equivalence principle is a “principle of universality” for the geometrical structure of spacetime. Whatever the spacetime geometrical structure is, if all excitations “see” the same geometry, one is well on the way to satisfying the observational and experimental constraints. In a metric theory, this amounts to the demand of monometricity: A single universal metric must govern the propagation of all excitations.
 1.
Try to find a broad class of analogue models (either physically based or mathematically idealised) that naturally lead to monometricity. Little work along these lines has yet been done; at least partially because it is not clear what features such a model should have in order to be “clean” and “compelling”.
 2.Accept refringence as a common feature of the analogue models and attempt to use refringence to ones benefit in one or more ways:

There are real physical phenomena in nongravitational settings that definitely do exhibit refringence and sometimes multimetricity. Though situations of this type are not directly relevant to the gravity community, there is significant hope that the mathematical and geometrical tools used by the general relativity community might in these situations, shed light on other branches of physics.

Use the refringence that occurs in many analogue models as a way of “breaking” the Einstein equivalence principle, and indeed as a way of “breaking” even more fundamental symmetries and features of standard general relativity, with a view to exploring possible extensions of general relativity. While the analogue models are not themselves primary physics, they can nevertheless be used as a way of providing hints as to how more fundamental physics might work.

7.3 Nontrivial dispersion as Einsteinaether theory
Of course, in standard analogue models such an aether field does not come with its own dynamics: It is a background structure which breaks the physicallyrelevant content of what is usually called diffeomorphism invariance (see next Section 7.4). However, in a gravitation theory context one might still want to require background independence taking it as a fundamental property of any gravity theory, even a Lorentz breaking one. In this case one has to provide the aether field with a suitable dynamics; we can then rephrase much of the analogue gravity discussion in the presence of nontrivial dispersion relations in terms of a variant of the Einsteinaether models [323, 180, 219, 314].
7.4 Diffeomorphism invariance
When looking at the analogue metrics one problem immediately comes to mind. The laboratory in which the condensedmatter system is set up provides a privileged coordinate system. Thus, one is not really reproducing a geometrical configuration but only a specific metrical representation of it. This naturally raises the question of whether or not diffeomorphism invariance is lost in the analogue spacetime construction. Indeed, if all the degrees of freedom contained in the metric had a physical role, as opposed to what happens in a general relativistic context in which only the geometrical degrees of freedom (metric modulo diffeomorphism gauge) are physical, then diffeomorphism invariance would be violated. Here we are thinking of “active” diffeomorphisms, not “passive” diffeomorphisms (coordinate changes). As is well known, any theory can be made invariant under passive diffeomorphisms (coordinate changes) by adding a sufficient number of external/background/nondynamical fields (prior structure). See, for instance, [256]. Invariance under active diffeomorphisms is equivalent to the assertion that there is no “prior geometry” (or that the prior geometry is undetectable). Many readers may prefer to rephrase the current discussion in terms of the undetectability of prior structure.
The answer to this question is that active diffeomorphism invariance is maintained but only for (lowenergy) internal observers, i.e., those observers who can only perform (lowenergy) experiments involving the propagation of the relativistic collective fields. By revisiting classic LorentzFitzGerald ideas on length contraction, and analyzing the MichelsonMorley experiment in this context, it has been explicitly shown in [36] that (lowenergy) Lorentz invariance is not broken, i.e., that an internal observer cannot detect his absolute state of motion. (For earlier suggestions along these lines, see, for example, [400] and [660].)
The argument is the same for curved spacetimes; the internal observer would have no way to detect the “absolute” or fixed background. So the apparent background dependence provided by the (nonrelativistic) condensedmatter system will not violate active diffeomorphism invariance, at least not for these internal inhabitants. These internal observers will then have no way to collect any metric information beyond what is coded into the intrinsic geometry (i.e., they only get metric information up to a gauge or diffeomorphism equivalence factor). Internal observers would be able to write down diffeomorphism invariant Lagrangians for relativistic matter fields in a curved geometry. However, the dynamics of this geometry is a different issue. It is a wellknown issue that the expected relativistic dynamics, i.e., the Einstein equations, have to date not been reproduced in any known condensedmatter system.
7.5 Effective spintwo particles
Related to the previous point is the possibility of having quantum systems with no pregeometric notions whatsoever (i.e., a condensedmatterlike system) that still exhibit in their lowenergy spectrum effective massless spintwo excitations. This precise question has been investigated in [262, 263, 693, 694] for abstract quantum systems based on the underlying notion of qubits. Although not fully conclusive, these works indicate the possible existence of systems exhibiting purely helicity ±2 excitations. One crucial ingredient in these constructions is the existence of a specific vacuum state with the characteristics of a stringnet condensate.
These authors also show that it is not easy to have just helicity ±2 excitations — typically one would also generate helicity ±1 and massless scalar excitations. This is what would happen, for instance, in the emergent gravity scenario inspired in the phenomenology of ^{3}He, which will be described below.
7.6 WeinbergWitten theorem
Furthermore, when it comes to Sakharovstyle induced gravity those authors explicitly state [673]:Of course, there are acceptable theories that have massless charged particles with spin j > 1/2 (such as the massless version of the original YangMills theory), and also theories that have massless particles with spin j > 1 (such as supersymmetry theories or general relativity). Our theorem does not apply to these theories because they do not have Lorentzcovariant conserved currents or energymomentum tensors, respectively.
That is: The WeinbergWitten theorem has no direct application to analogue spacetimes — at the kinematic level it has nothing to say, at the dynamic level its applicability is rather limited by the stringent technical assumptions invoked — specifically exact Lorentz invariance at all scales — and the fact that these technical assumptions are not applicable in the current context. For careful discussions of the technical assumptions see [596, 366, 212, 404]. Note particularly the comment by Kubo [366]However, the theorem dearly does not apply to theories in which the gravitational field is a basic degree of freedom but the Einstein action is induced by quantum effects.
Finally we mention that, though motivated by quite different concerns, the review article [61] gives a good overview of the WeinbergWitten theorem, and the ways in which it may be evaded.… the powerful second part of the theorem becomes empty in the presence of gravity …
7.7 Emergent gravity
One of the more fascinating approaches to “quantum gravity” is the suggestion, typically attributed to Sakharov [540, 628], that gravity itself may not be “fundamental physics”. Indeed it is now a relatively common opinion, maybe not mainstream but definitely a strong minority opinion, that gravity (and in particular the whole notion of spacetime and spacetime geometry) might be no more “fundamental” than is fluid dynamics. The word “fundamental” is here used in a rather technical sense — fluid mechanics is not fundamental because there is a known underlying microphysics, that of molecular dynamics, of which fluid mechanics is only the lowenergy lowmomentum limit. Indeed, the very concepts of density and velocity field, which are so central to the Euler and continuity equations, make no sense at the microphysical level and emerge only as one averages over timescales and distancescales larger than the mean free time and mean free path.
In the same way, it is plausible (even though no specific and compelling model of the relevant microphysics has yet emerged) that the spacetime manifold and spacetime metric might arise only once one averages over suitable microphysical degrees of freedom.
7.8 One specific route to the Einstein equations?
In fact, there is a specific route to reproduce Einstein equations within a Fermiliquidlike system advocated by G. Volovik [660]. In a Fermi liquid like ^{3}HeA phase, there exist two important energy scales. One is the energy scale E_{ B } at which bosonization in the system start to develop. This scale marks the onset of the superfluid behaviour of helium three. At energies below E_{ B }, the different bosons appearing in the systems condense so that they can exhibit collective behaviour. The other energy scale, E_{ L }, is the Lorentz scale at which the quasiparticles of the system start to behave relativistically (as Weyl spinors). As we mention in Section 4.2.2 discussing the “heliocentric universe”, this occurs in the ^{3}HeA phase because the vacuum has Fermi points. It is in the immediate surroundings of these Fermi points where the relativistic behaviour shows up. There are additional relevant scales in these systems, but in this section we are going to talk exclusively of these two.
When one is below both energy scales, one can describe the system as a set of Weyl spinors coupled to background electromagnetic and gravitational fields. For a particular Fermi point, the electromagnetic and gravitational fields encode, respectively, its position and its “lightcone” structure through space and time. Both electromagnetic and gravitational fields are built from bosonic degrees of freedom, which have condensed. Apart from any predetermined dynamics, these bosonic fields will acquire additional dynamical properties through the Sakharovinduced gravity mechanism. Integrating out the effect of quantum fluctuations in the Fermionic fields à la Sakharov, one obtains a oneloop effective action for the geometric field, to be added to the treelevel contribution (if any). This integration cannot be extended beyond E_{b}, as at that energy scale the geometrical picture based on the bosonic condensate disappears. Thus, E_{b} will be the cutoff of the integration.
 1.
E_{ l } > E_{ b }: For the induction mechanics to give rise to an EinsteinHilbert term, \(\sqrt { g}R\), in the effective Lagrangian we have to be sure that the fluctuating Fermionic field “feels” the geometry (fulfilling a locallyLorentzinvariant equation) at all scales up to the cutoff. The term \(\sqrt { g}R\) will appear multiplied by a constant proportional to \(E_B^2\). That is why from now on we can called E_{ b } alternatively the Planck energy scale E_{ P } ≡ E_{ b }.
 2.
Special relativity dominance or E_{ l } ≫ E_{ P }: The \(E_P^2\) dependence of the gravitational coupling constant tells us that the fluctuations that are more relevant in producing the EinsteinHilbert term are those with energies close to the cutoff, that is, around the Planck scale. Therefore, to assure the induction of an EinsteinHilbert term one needs the Fermionic fluctuation with energies close to the Planck scale to be perfectly Lorentzian to a high degree. This can only be assured if E_{ L } ≫ E_{ B }.
 3.
Sakharov oneloop dominance: Finally, one also needs the induced dynamical term to dominate over the preexisting treelevel contribution (if any).
In counterpoint, in Hořava gravity the graviton appears to be fundamental, and need not be emergent [287, 288, 289]. Additionally, the Lorentz breaking scale and the Planck scale are in this class of models distinct and unconnected, with the possibility of driving the Lorentz breaking scale arbitrarily high [585, 584, 635, 681]. In this sense the Horava models are a useful antidote to the usual feeling that Lorentz violation is typically Planckscale.
7.9 The cosmological constant problem
The condensed matter analogies offer us an important lesson concerning the cosmological constant problem [660]. Sakharov’s induced gravity not only can give rise to an EinsteinHilbert term under certain conditions, it also gives rise to a cosmological term. This contribution would depend on the cutoff as \(E_P^4\), so if there were not additional contributions counterbalancing this term, emergent gravity in condensedmatterlike systems will always give place to an enormous cosmological constant inducing a stronglyrepulsive force between quasiparticles.
However, we know that at low temperatures, depending on the microscopic characteristics of the system we can have quite different situations. Remarkably, liquid systems (as opposed to gases) can remain stable on their own, without requiring any external pressure. Their total internal pressure at equilibrium is (modulo finitesize effects) always zero. This implies that, at zero temperature, if gravity emerges from a liquidlike system, the total vacuum energy Λ ∝ ρ_{ V } = − P_{ V } will be automatically forced to be (relatively) small, and not a large number. The \(E_P^4\), contribution coming from quasiparticle fluctuations will be exactly balanced by contributions from the microphysics or “transPlanckian” contributions.
If the temperature is not zero there will be a pressure p_{ M } associated with the thermal distribution of quasiparticles, which constitute the matter field of the system. Then, at equilibrium one will have p_{ m } + p_{ V } = 0, so that there will be a small vacuum energy Λ ∝ p_{ M }. This value is not expected to match exactly the preferred value of Λ obtained in the standard cosmological model (ACDM) as we are certain to be out of thermodynamic equilibrium. However, it is remarkable that it matches its order of magnitude, (at least for the current epoch), albeit any dynamical model implementing this idea will probably have to do so only at late times to avoid possible tension with the observational data. Guided by these lessons, there are already a number of heuristic investigations about how a cosmological term could dynamically adapt to the evolution of the matter content, and which implications it could have for the evolution of the universe [27, 350, 349, 351].
As final cautionary remark let us add that consideration of an explicit toy model for emergent gravity [252] shows that the quantity that actually gravitates cannot be so easily predicted without an explicit derivation of the analogue gravitational equations. In particular in [198] it was shown that the relevant quantity entering the analogue of the cosmological constant is a contribution coming only from the excitations above the condensate.
7.10 Other pieces of the puzzle
Sakharov had in mind a specific model in which gravity could be viewed as an “elasticity” of the spacetime medium, and where gravity was “induced” via oneloop physics in the matter sector [540, 628]. In this way, Sakharov had hoped to relate the observed value of Newton’s constant (and the cosmological constant) to the spectrum of particle masses.
More generally, the phrase “emergent gravity” is now used to describe the whole class of theories in which the spacetime metric arises as a lowenergy approximation, and in which the microphysical degrees of freedom might be radically different. Analogue models, and in particular analogue models based on fluid mechanics or the fluid dynamic approximation to BECs, are specific examples of “emergent physics” in which the microphysics is well understood. As such, they are useful for providing hints as to how such a procedure might work in a more fundamental theory of quantum gravity.
7.11 Quantum gravity — phenomenology
Over the last few years a widespread consensus has emerged that observational tests of quantum gravity are for the foreseeable future likely to be limited to precision tests of dispersion relations and their possible deviations from Lorentz invariance [435, 321]. The key point is that at low energies (well below the Planck energy) one expects the locallyMinkowskian structure of the spacetime manifold to guarantee that one sees only special relativistic effects; general relativistic effects are negligible at short distances. However, as ultrahigh energies are approached (although still below Planckscale energies) several quantumgravity models seem to predict that the locally Euclidean geometry of the spacetime manifold will break down. There are several scenarios for the origin of this breakdown ranging from string theory [360, 182] to brane worlds [99] and loop quantum gravity [229]. Common to all such scenarios is that the microscopic structure of spacetime is likely to show up in the form of a violation of Lorentz invariance leading to modified dispersion relations for elementary particles. Such dispersion relations are characterised by extra terms (with respect to the standard relativistic form), which are generally expected to be suppressed by powers of the Planck energy. Remarkably, the last years have seen a large wealth of work in testing the effects of such dispersion relations and in particular strong constraints have been cast by making use of high energy astrophysics observations (see, for example, [6, 141, 318, 317, 319, 320, 321, 435, 579, 396] and references therein).
Several of the analogue models are known to exhibit similar behaviour, with a lowmomentum effective Lorentz invariance eventually breaking down at high momentum once the microphysics is explored.^{26} Thus, some of the analogue models provide controlled theoretical laboratories in which at least some forms of subtle highmomentum breakdown of Lorentz invariance can be explored. As such, the analogue models provide us with hints as to what sort of modified dispersion relation might be natural to expect given some general characteristics of the microscopic physics. Hopefully, an investigation of appropriate analogue models might be able to illuminate possible mechanisms leading to this kind of quantum gravity phenomenology, and so might be able to provide us with new ideas about other effects of physical quantum gravity that might be observable at subPlanckian energies.
7.12 Quantum gravity — fundamental models
When it comes to dealing with “fundamental” theories of quantum gravity, the analogue models play an interesting role which is complementary to the more standard approaches. The search for a quantum theory of gravity is fundamentally a search for an appropriate mathematical structure in which to simultaneously phrase both quantum questions and gravitational questions. More precisely, one is searching for a mathematical framework in which to develop an abstract quantum theory which then itself encompasses classical Einstein gravity (the general relativity), and reduces to it in an appropriate limit [113, 580, 221].
The three main approaches to quantum gravity currently in vogue, “string models” (also known as “Mmodels”), “loop space” (and the related “spin foams”), and “lattice models” (Euclidean or Lorentzian) all share one feature: They attempt to develop a “pregeometry” as a replacement for classical differential geometry (which is the natural and very successful mathematical language used to describe Einstein gravity) [113, 580, 221, 534, 533, 83, 507]. The basic idea is that the mooted replacement for differential geometry would be relevant at extremely small distances (where the quantum aspects of the theory would be expected to dominate), while at larger distances (where the classical aspects dominate) one would hope to recover both ordinary differential geometry and specifically Einstein gravity or possibly some generalization of it. The “string”, “loop”, and “lattice” approaches to quantum gravity differ in detail in that they emphasise different features of the longdistance model, and so obtain rather different shortdistance replacements for classical differential geometry. Because the relevant mathematics is extremely difficult, and by and large not particularly well understood, it is far from clear which, if any, of these three standard approaches will be preferable in the long run [580].
A recent (Jan. 2009) development is the appearance of Hořava gravity [287, 288, 289]. This model is partially motivated by condensed matter notions such as (deeply nonperturbative) anomalous scaling and the existence of a “Lifshitz point”, and additionally shares with most of the analogue spacetimes the presence of modified dispersion relations and highenergy deviations from Lorentz invariance [287, 288, 289, 634, 635, 585, 584, 681]. Though Hořava gravity is not directly an analogue model per se, there are deep connections — with some steps toward an explicit connection being presented in [694].
We feel it likely that analogue models can shed new light on this very confusing field by providing a concrete specific situation in which the transition from the shortdistance “discrete” or “quantum” theory to the longdistance “continuum” theory is both well understood and noncontroversial. Here we are specifically referring to fluid mechanics, where, at short distances, the system must be treated using discrete atoms or molecules as the basic building blocks, while, at large distances, there is a welldefined continuum limit that leads to the Euler and continuity equations. Furthermore, once one is in the continuum limit, there is a welldefined manner in which a notion of “Lorentzian differential geometry”, and in particular a “Lorentzian effective spacetime metric” can be assigned to any particular fluid flow [607, 624, 470]. Indeed, the “analogue gravity programme” is extremely successful in this regard, providing a specific and explicit example of a “discrete” → “continuum” → “differential geometry” chain of development. What the “analogue gravity programme” does not seem to do as easily is to provide a natural direct route to the Einstein equations of general relativity, but that merely indicates that current analogies have their limits and therefore, one should not take them too literally [624, 470]. Fluid mechanics is a guide to the mathematical possibilities, not an end in itself. The parts of the analogy that do work well are precisely the steps where the standard approaches to quantum gravity have the most difficulty, and so it would seem useful to develop an abstract mathematical theory of the “discrete” → “continuum” → “differential geometry” chain using this fluid mechanical analogy (and related analogies) as inspiration.
7.13 Going further

Identify a particular analogue model easily amenable to laboratory investigation, and double check the extent to which the model provides a theoretically robust and clean analogue to general relativistic curved spacetime.

Identify the technical issues involved in actually setting up a laboratory experiment.
8 Conclusions
In this review article we have seen the interplay between standard general relativity and various analogies that can be used to capture aspects of its behaviour. These analogies have ranged from rather general but very physical analogue models based on fluidacoustics, geometrical optics, and wave optics, to highly specific models based on BECs, liquid helium, slow light, etc. Additionally, we have seen several rather abstract mathematical toy models that bring us to such exotic structures and ideas as birefringence, bimetricity, Finsler spaces, and Sakharov’s induced gravity.
The primary reason that these analogies were developed within the general relativity community was to help in the understanding of general relativity by providing very downtoearth models of otherwise subtle behaviour in general relativity. Secondary reasons include the rather speculative suggestion that there may be more going on than just analogy — it is conceivable (though perhaps unlikely) that one or more of these analogue models could suggest a relatively simple and useful way of quantizing gravity that sidesteps much of the technical machinery currently employed in such efforts. A tertiary concern (at least as far as the general relativity community is concerned) is the use of relativity and differential geometric techniques to improve understanding of various aspects of condensed matter physics.
The authors expect interest in analogue models to continue unabated, and suspect that there are several key but unexpected issues whose resolution would be greatly aided by the analysis of appropriate analogue models.
8.1 Going further

http://www.slac.stanford.edu/spires/ — the SPIRES bibliographic database for keeping track of (almost all of) the general relativity and particle physics aspects of the relevant literature.

http://inspirebeta.net/ — the new INSPIRE bibliographic database (currently beta version) for keeping track of (almost all of) the general relativity and particle physics aspects of the relevant literature.

http://ads.harvard.edu/ — the SAO/NASA ADS bibliographic database for keeping track of (almost all of) the astrophysical aspects of the relevant literature.

http://www.arXiv.org — the electronicpreprint (eprint) database for accessing the text of (almost all, post 1992) relevant articles.

http://relativity.livingreviews.org/ — the Living Reviews in Relativity journal.
Footnotes
 1.
The need for a certain degree of caution regarding the allegedly straightforward physics of simple fluids might be inferred from the fact that the Clay Mathematics Institute is currently offering a US$ 1,000,000 Millennium Prize for significant progress on the question of existence and uniqueness of solutions to the NavierStokes equation. See http://www.claymath.org/millennium/ for details.
 2.
In correct English, the word “dumb” means “mute”, as in “unable to speak”. The word “dumb” does not mean “stupid”, though even many native English speakers get this wrong.
 3.
For instance, whenever one has a system of PDEs that can be written in firstorder quasilinear symmetric hyperbolic form, then it is an exact nonperturbative result that the matrix of coefficients for the firstderivative terms can be used to construct a conformal class of metrics that encodes the causal structure of the system of PDEs. For barotropic hydrodynamics this is briefly discussed in [138]. This analysis is related to the behaviour of characteristics of the PDEs, and ultimately can be linked back to the Fresnel equation that appears in the eikonal limit.
 4.
It is straightforward to add external forces, at least conservative body forces such as Newtonian gravity.
 5.
Henceforth, in the interests of notational simplicity, we shall drop the explicit subscript 0 on background field quantities unless there is specific risk of confusion.
 6.
 7.
Because of the background Minkowski metric there can be no possible confusion as to the definition of this normal derivative.
 8.
There are a few potential subtleties in the derivation of the existence of Hawking radiation, which we are, for the time being, glossing over; see Section 5.1 for details.
 9.
There is an issue of normalization here. On the one hand we want to be as close as possible to general relativistic conventions. On the other hand, we would like the surface gravity to really have the dimensions of an acceleration. The convention adopted here, with one explicit factor of c, is the best compromise we have come up with. (Note that in an acoustic setting, where the speed of sound is not necessarily a constant, we cannot simply set c → 1 by a choice of units.)
 10.
There are situations in which this surface gravity is a lot larger than one might naively expect [398].
 11.
The Painlevé—Gullstrand line element is sometimes called the Lemaître line element.
 12.
Similar constructions work for the ReissnerNordström geometry [398], as long as one does not get too close to the singularity. (With c = 1 one needs r > Q^{2}/(2m) to avoid an imaginary fluid velocity.) Likewise, certain aspects of the Kerr geometry can be emulated in this way [641]. (One needs r > 0 in the Doran coordinates [176, 633] to avoid closed timelike curves.) As a final remark, let us note that de Sitter space corresponds to v ∝ r and ρ ∝ 1/r^{3}. For further details see Section 2.5.
 13.
 14.
Vorticity is automatically generated, for instance, whenever the background fluid is nonbarotropic, and, in particular, when ∇π × ∇p ≠ 0. Furthermore, it has been argued in [559] that quantum backreaction can also act as a source for vorticity.
 15.
In [233, 234, 235, 236, 237, 238, 239, 240] the author has attempted to argue that vorticity can be related to the concept of torsion in a general affine connexion. We disagree. Although deriving a wave equation in the presence of vorticity very definitely moves one beyond the realm of a simple Riemannian spacetime, adding torsion to the connexion is not sufficient to capture the relevant physics.
 16.
Indeed, historically, though not of direct relevance to general relativity, analogue models played a key role in the development of electromagnetism — Maxwell’s derivation of his equations for the electromagnetic field was guided by a rather complicated “analogue model” in terms of spinning vortices of aether. Of course, once you have the equations in hand you can treat them in their own right and forget the model that guided you — which is exactly what happened in this particular case.
 17.
We emphasise: To get Hawking radiation you need an effective geometry, a horizon, and a suitable quantum field theory on that geometry.
 18.
Of course, we now mean “gravity wave” in the traditional fluid mechanics sense of a water wave whose restoring force is given by ordinary Newtonian gravity. Waves in the fabric of spacetime are more properly called “gravitational waves”, though this usage seems to be in decline within the general relativity community. Be very careful in any situation where there is even a possibility of confusing the two concepts.
 19.
The existence of this constraint has been independently rederived several times in the literature. In contrast, other segments of the literature seem blithely unaware of this important restriction on just when permittivity and permeability are truly equivalent to an effective metric.
 20.
One could also imagine systems in which the effective metric fails to exist on one side of the horizon (or even more radically, on both sides). The existence of particle production in this kind of system will then depend on the specific interactions between the subsystems characterizing each side of the horizon. For example, in stationary configurations it will be necessary that these interactions allow negative energy modes to disappear beyond the horizon, propagating forward in time (as happens in an ergoregion). Whether these systems will provide adequate analogue models of Hawking radiation or not is an interesting question that deserves future analysis. Certainly systems of this type lie well outside the class of usual analogue models.
 21.
Actually, even relativistic behaviour at low energy can be nongeneric, but we assume in this discussion that an analogue model by definition is a system for which all the linearised perturbations do propagate on the same Lorentzian background at low energies.
 22.
However, see also [522, 528] for a radically different alternative approach based on the idea of “superoscillations” where ultrahigh frequency modes near the horizon can be mimicked (to arbitrary accuracy) by the exponential tail of an exponentiallylarge amplitude mostly hidden behind the horizon.
 23.
Note however, that for a real black hole and Lorentzinvariant physics, the spectrum observed at infinity is indistinguishable from thermal given that no correlation measurement is allowed, since the Hawking partners are spacelike separated across the horizon. This fact is indeed the root of the information paradox.
 24.
[149] also considered the case of a lattice with proper distance spacing constant in time but this has the problem that the proper spacing of the lattice goes to zero at spatial infinity, and hence there is no fixed shortdistance cutoff.
 25.
Personal communication by R. Parentani
 26.
However, it is important to keep in mind that not all the abovecited quantum gravity models violate the Lorentz symmetry in the same manner. The discreteness of spacetime at short scales is not the only way of breaking Lorentz invariance.
Notes
Acknowledgements
The work of Matt Visser was supported by the Marsden fund administered by the Royal Society of New Zealand. MV also wishes to thank both SISSA (Trieste) and the IAACSIC (Granada) for hospitality during various stages of this work. Carlos Barceló has been supported by the Spanish MICINN through the project FIS200806078C0301, and by the Junta de Andalucía through the projects FQM2288 and fQm219.
The authors also wish to specifically thank Enrique Arilla for providing Figures 1 and 2, and Silke Weinfurtner for providing Figure 6 of the artwork. Additionally, the authors wish to thank Renaud Parentani for helpful comments, specifically with respect to the question of which notion of surface gravity is the most important for Hawking radiation. Finally, the authors also wish to thank Germain Rousseaux for bringing several historicallyimportant references to our attention. —
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