Analogue Gravity
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Abstract
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
And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of Nature, and they ought least to be neglected in Geometry.
— Johannes Kepler
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 focussing 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.
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 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.

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 (hopefully complete) catalogue of extant models.

Discuss the main physics results obtained to date.

Outline the many possible directions for future research.

Summarise the current state of affairs.
By that stage the interested reader will have had a quite thorough introduction to the ideas, techniques, and hopes of the analogue gravity programme.
1.1 Going further

The book “Artificial Black Holes”, edited by Mario Novello, Matt Visser, and Grigori Volovik [284].
 The websites for the “Analogue models” workshop:

The book “The Universe in a Helium droplet”, by Grigori Volovik [418].

The Physics Reports article, “Superfluid analogies of cosmological phenomena”, by Grigori Volovik [413].
2 The Simplest Example of an Analogue Model
Acoustics in a moving fluid is the simplest and cleanest example of an analogue model [376, 387, 391, 389]. 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.
The advantage of geometrical acoustics is that the derivation of the precise mathematical form of the analogy is so simple as to be almost trivial, and that the derivation is extremely general. The disadvantage is that in the geometrical acoustics limit one can deduce only the causal structure of the spacetime, and does not obtain a unique effective metric. The advantage of physical acoustics is that while the derivation of the analogy holds in a more restricted regime, the analogy can do more for you in that it can now specify a specific effective metric and accommodate a wave equation for the sound waves.
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 space plus time, 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 [389], 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:(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.$${\eta _{\mu \nu}} \equiv {({\rm{diag}}[  c_{{\rm{light}}}^2,1,1,1])_{\mu \nu}}.$$(37)

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) the acoustic metric was first derived and used in Moncrief’s studies of the relativistic hydrodynamics of accretion flows surrounding black holes [268]. 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.

The geometry determined by the acoustic metric does however 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 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” [164, 422]. Note thatThis 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 [164, 161, 162, 72, 163, 396]).$${g^{\mu \nu}}({\nabla _\mu}t)\,({\nabla _\nu}t) =  {1 \over {{\rho _0}\,c}} < 0.$$(38)

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$$\begin{array}{*{20}c} {{\nabla _\mu}t = (1,0,0,0);} & {{\nabla ^\mu}t =  {{(1;\;v _0^i)} \over {{\rho _0}\;c}} =  {{{V^\mu}} \over {\sqrt {{\rho _0}\;c}}}.} \\ \end{array}$$(41)and the integral curves are geodesics of the acoustic metric if and only if α_{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), μ_{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.
2.5 Dumb holes — ergoregions, horizons, and surface gravity
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 [164, pages 319–323] or [422, pages 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 on pages 262–263 of Hawking and Ellis [164]. 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 event horizon (absolute horizon) is defined, as in general relativity, by demanding that it be the boundary of the region from which null geodesics (phonons) cannot escape. This is actually the future event horizon. A past event horizon can be defined in terms of the boundary of the region that cannot be reached by incoming phonons — strictly speaking this requires us to define notions of past and future null infinities, but we will simply take all relevant incantations as understood. In particular the event horizon is a null surface, the generators of which are null geodesics.
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.
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 [147]. 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 [159, 160].^{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.
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 [376, 377, 378], Reznik [319], and the results for “dirty black holes” [386]. 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.
2.5.1 Example: vortex geometry
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 [404], but this approximation becomes progressively worse as the core is approached.)
2.5.2 Example: slab geometry
If we set c = 1 and ignore the conformal factor we have the toy model acoustic geometry discussed by Unruh [378, page 2828, equation (8)], Jacobson [188, page 7085, equation (4)], Corley and Jacobson [88], and Corley [86]. (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 [192].) 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.
2.5.3 Example: PainlevéGullstrand geometry
As emphasised by Kraus and Wilczek, the PainlevéGullstrand line element exhibits a number of features of pedagogical interest. In particular the constant time spatial slices are completely flat — 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 \(v = \sqrt {2GM/r}\) 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.
So we see that the net result is conformal to the PainlevéGullstrand form of the Schwarzschild geometry but not identical to it. For many purposes this is quite good enough: We have an event horizon, we can define surface gravity, we can analyse Hawking radiation.^{12} Since surface gravity and Hawking temperature are conformal invariants [192] this is sufficient for analysing basic features of the Hawking radiation process. The only way in which the conformal factor can influence the Hawking radiation is through backscattering off the acoustic metric. (The phonons are minimally coupled scalars, not conformally coupled scalars so there will in general be effects on the frequencydependent greybody factors.)
If we focus attention on the region near the event horizon, the conformal factor can simply be taken to be a constant, and we can ignore all these complications.
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 (1+1) or (2+1) dimensions.

Adding vorticity, to go beyond the irrotational constraint.
2.7.1 External forces
Adding external forces is easy, an early discussion can be found in [389] and more details are available in [401]. The key point is that with an external force one can to some extent shape the background flow (see for example the discussion on [149]). Upon linearization, the fluctuations are however 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.
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 well behaved 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 [84], 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, vorticity free 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.^{13} An approach similar to the spirit of the present discussion, but in terms of Clebsch potentials, can be found in [307]. 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.)^{14}
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. [15]
2.9 Going further
 Working with specific fluids.

Superfluids.

BoseEinstein condensates.

 Abstract generalizations.

Normal modes in generic systems.

Multiple signal speeds.

3 History and Motivation
From the point of view of the general relativity community the history of analogue models can reasonably neatly (but superficially) be divided into a “historical” period (essentially pre1981) and a “modern” period (essentially post1981).
3.1 Modern period
3.1.1 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” [376], 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” general relativity 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.^{15} 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” [185] 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 [186] and [377, 378]. (This period also saw the independent rediscovery of the fluid analogue model by one of the present authors [387], and the first explicit consideration of superfluids in this regard [84].)
The later 1990’s then saw continued work by Jacobson and his group [187, 188, 88, 90, 198], with new and rather different contributions coming in the form of the solid state models considered by Reznik [319, 318].^{16} This period also saw the introduction of the more general class of superfluid models considered by Volovik and his collaborators [402, 403, 213, 110, 407, 405, 406, 199, 409, 410], more precise formulations of the notions of horizon, ergosphere, and surface gravity in analogue models [389, 391], and discussions of the implications of analogue models regarding BekensteinHawking entropy [390, 391]. Finally, analogue spacetimes based on special relativistic acoustics were considered in [33].
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.
3.1.2 The year 2000
Key developments in 2000 were the introduction, by Garay and collaborators, of the use of BoseEinstein condensates as a working fluid [136, 137], and the extension of those ideas by the present authors [14]. 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 [205, 273].
That year also marked the appearance of a review article on superfluid analogues [413], more work on “nearhorizon” physics [123], 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 [28]. Models based on nonlinear electrodynamics were investigated in [11], ^{3}He — A based models were reconsidered in [193, 411], and “slow light” models in quantum dielectrics were considered in [235, 236, 231].
The most radical proposal to appear in 2000 was that of Laughlin et al. [76]. Based on taking a superfluid analogy rather literally they mooted an actual physical breakdown of general relativity at the horizon of a black hole [76].
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 the book [284] in 2002.
3.1.3 The year 2000
This year saw more applications of analogueinspired ideas to cosmological inflation [107, 263, 262, 207, 275], to neutron star cores [66], and to the cosmological constant [414, 416].
Closer to the heart of the analogue programme were the development of a “normal mode” analysis in [15, 16, 398], the development of dielectric analogues in [342], speculations regarding the possibly emergent nature of Einstein gravity [20, 398], and further developments regarding the use of ^{3}He − A [106] as an analogue for electromagnetism. Experimental proposals were considered in [19, 398, 331].
Vorticity was discussed in [307], and the use of BECs as a model for the breakdown of Lorentz invariance in [397]. Analogue models based on nonlinear electrodynamics were discussed in [101]. Acoustics in an irrotational vortex were investigated in [120].
The excitation spectrum in superfluids, specifically the fermion zero modes, were investigated in [412, 182], while the relationship between rotational friction in superfluids and superradiance in rotating spacetimes was discussed in [57]. More work on “slow light” appeared in [48]. The possible role of Lorentz violations at ultrahigh energy was emphasised in [190].
3.1.4 The year 2002
“What did we learn from studying acoustic black holes?” was the title and theme of Parentani’s article in 2002 [300], while Schutzhold and Unruh developed a rather different fluidbased analogy based on gravity waves in shallow water [344, 345]. Superradiance was investigated in [27], while the propagation of phonons and quasiparticles was discussed in [122, 121]. More work on “slow light” appeared in [124, 311].
The stability of an acoustic white hole was investigated in [234], while further developments regarding analogue models based on nonlinear electrodynamics were presented by Novello and collaborators in [102, 103, 282, 278, 126]. Analogue spacetimes relevant to braneworld cosmologies were considered in [12].
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 [195, 194].
3.1.5 The year 2003
That year saw further discussion of analogueinspired models for black hole entropy and the cosmological constant [419, 421], and the development of analogue models for FRW geometries [115, 114, 17, 105, 242]. There were several further developments regarding the foundations of BECbased models in [18, 116], while analogue spacetimes in superfluid neutron stars were further investigated in [67].
Effective geometry was the theme in [280], while applications of nonlinear electrodynamics (and its effective metric) to cosmology were presented in [281]. Superradiance was further investigated in [26, 24], while the limitations of the “slow light” analogue were explained in [379]. Vachaspati argued for an analogy between phase boundaries and acoustic horizons in [381]. Emergent relativity was again addressed in [227].
The review article by Burgess [53], 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 [191] give a nice introduction to Hawking radiation and its connection to analogue spacetimes.
3.1.6 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 [306], while the physical realizability of acoustic Hawking radiation was addressed in [95, 382]. More cosmological issues were raised in [382, 424], while a specifically astrophysical use of the acoustic analogy was invoked in [96, 97, 98].
BECbased horizons were again considered in [149, 148], while backreaction effects were the focus of attention in [10, 9, 208]. More issues relating to the simulation of FRW cosmologies were raised in [118, 119].
Unruh and Schützhold discussed the universality of the Hawking effect [380], and a new proposal for possibly detecting Hawking radiation in a electromagnetic wave guide [347]. The causal structure of analogue spacetimes was considered in [13], while quasinormal modes attracted attention in [31, 237, 64, 269]. Two dimensional analogue models were considered in [55].
There were attempts at modelling the Kerr geometry [401], and generic “rotating” spacetimes [77], a proposal for using analogue models to generate massive phonon modes in BECs [400], and an extension of the usual formalism for representing weakfield gravitational lensing in terms of an analogue refractive index [38].
Finally we mention the development of yet more strong observational bounds on possible ultra high energy Lorentz violation [196, 197].
3.1.7 The year 2005
The first few months of 2005 have seen continued and vigourous activity on the analogue model front.
More studies of the superresonance phenomenon have appeared [25, 113, 209, 354], and a minisurvey was presented in [63]. Quasinormal modes have again received attention in [78], while the Magnus force is reanalysed in terms of the acoustic geometry in [432]. Singularities in the acoustic geometry are considered in [56], while backreaction has received more attention in [343].
Interest in analogue models is intense and shows no signs of abating.
We shall in the next subsection 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 15 years.
3.2 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.^{17}
3.2.1 Optics
After that, there was sporadic interest in effective metric techniques. One historically important contribution was one of the problems in the wellknown book “The classical theory of fields” by Landau and Lifshitz [222]. 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.
Three articles that directly used the dielectric analogy to analyse specific physics problems are those of Skrotskii [352], Balazs [8], and Winterberg [427]. The general formalism was more fully developed in articles such as those by Peblanski [304, 303], and good summary of this classical period can be found in the article by de Felice [100].
If an optical medium does not satisfy this constraint (with a position independent proportionality constant) then it is not completely equivalent to a gravitational field. For a position dependent proportionality constant complete equivalence can be established in the geometric optics limit, but for wave optics the equivalence is not complete.
3.2.2 Acoustics
There were several papers in the 1980’s using an acoustic analogy to investigate the propagation of shockwaves in astrophysical situations, most notably those of Moncrief [268] and Matarrese [259, 260, 258]. In particular in Moncrief’s work [268] the linear perturbations of a relativistic perfect fluid on an arbitrary general relativistic metric 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 [268] can be seen as a precursor to the later works on acoustic geometries and acoustic horizons.^{18}
3.2.3 Electromechanical analogy
The socalled “electromechanical analogy” has also had a long history within the engineering community. It is sometimes extended to obtain an “electromechanicalacoustic” analogy, or even an “electrothermal” analogy. Unfortunately the issues of interest to the engineering community rarely resonate within the relativity community, and these engineering analogies (though powerful in their own right) have no immediate impact for our purposes.^{19}
3.3 Motivation

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 (much more tenuous) to gain insight into new and radically different ways of dealing with “quantum gravity”.
3.4 Going further

Analoguebased “geometrical” interpretations of pseudomomentum, Iordanskii forces, Magnus forces, and the acoustic AharanovBohm effect [133, 365, 366, 367, 368, 408].

An analogueinspired interpretation of the Kerr spacetime [157].

The use of analogies to clarify the Newtonian limit of general relativity [373], to provide heuristics for motivating interest in specific spacetimes [320, 395], and to discuss a simple interpretation of the notion of a horizon [287].

Discrete [359] and noncommutative [81] spacetimes partially influenced and flavoured by analogue ideas.

Analoguebased hints on how to implement “double special relativity” (DSR) [215, 216, 217, 370], and a cautionary analysis of why this might be difficult [346].

Possible blackhole phase transitions placed in an analogue context [364].

Cosmological structure formation viewed as noise amplification [351].

Discussions of unusual topology, “acoustic wormholes”, and unusual temporal structure [270, 272, 313, 357, 358, 433].

Abstract quantum field theoretic considerations of the Unruh effect [428].

Numerous suggestions regarding possible transPlanckian physics [7, 29, 69, 74, 75, 171, 262, 322, 371].

Numerous suggestions regarding a minimum length in quantum gravity [32, 35, 49, 85, 135, 175, 176, 215, 216, 217, 243, 245, 244, 355].

Standard quantum field theory physics reformulated in the light of analogue models [4, 5, 117, 127, 238, 239, 240, 241, 248, 285, 286, 295, 294, 296, 301, 314, 428].

Standard general relativity supplemented with analogue viewpoints and insights [212, 225, 248].

The discussion of, and argument for, a possible reassessment of fundamental features of quantum physics and general relativity [6, 152, 206, 226, 241, 297, 328, 335].

Nonstandard viewpoints on quantum physics and general relativity [93, 174, 290, 324, 323, 336, 337, 338, 339].

Soliton physics [302], defect physics [246], and the Fizeau effect [271], presented with an analogue flavour.

Analogueinspired models of black hole accretion [315, 316].

Cosmological horizons from an analogue spacetime perspective [146].

Analogueinspired insights into renormalization group flow [60].

An analysis of “wave catastrophes” inspired by analogue models [210].

Improved numerical techniques for handling wave equations [426], and analytic techniques for handling wave tails [37], partially based on analogue ideas.
From the above the reader can easily appreciate the broad interest in, and wide applicability of, analogue spacetime models.
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.

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 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.
4.1.2 Shallow water waves (gravity waves)
Here ρ is the density of the fluid, ρ its pressure, g the gravitational acceleration and V_{∥} a potential associated with some external force necessary to establish an horizontal flow in the fluid. We denote that flow by \({\bf{v}}_{\rm{B}}^{}\). We must also impose the boundary conditions that the pressure at the surface, and the vertical velocity at the bottom, both vanish: p(z = h_{B}) = 0, v_{⊥}(z = 0) = 0.
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 Hawking evaporation and other quantum phenomena, one would need to use ultra cold quantum fluids.) However, the gravity wave analogue can certainly serve to investigate the classical phenomena of mode mixing that underlies the quantum processes.
4.1.3 Classical refractive index
4.1.3.1 Eikonal approximation
The behaviour of this dispersion relation now depends critically on the way that the eigenvalues of \({\bf{\tilde \epsilon}}\) are distributed.
4.1.3.2 3 degenerate eigenvalues
4.1.3.3 2 degenerate eigenvalues
If \({\bf{\tilde \epsilon}}\) 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 [39]. From the point of view of analogue models this corresponds to a twometric theory.
4.1.3.4 3 distinct eigenvalues
If \({\bf{\tilde \epsilon}}\) 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 [39, 399], and it seems that no meaningful notion of 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 [184].)
4.1.3.5 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 [288, 165, 220, 166].
4.1.3.6 Nonlinear electrodynamics
4.1.3.7 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 (155), 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 PainleveGullstrand type, and therefore geometries with ergoregions. If the proportionality constant relating ϵ ∞ μ is position independent, one can make the stronger statement (144) which holds true at the level of physical optics.
4.1.4 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.
As was the case for the Fresnel Equation (170), the determinant is to be taken on the field indices AB. (Remember to eliminate spurious and gauge degrees of freedom so that this determinant is not identically zero.) We emphasise that the algebraic equation defining the normal cone is the leading term in the Fresnel equation encountered in discussing the eikonal approximation. If there are N fields in total then this “normal cone” will generically consist of N nested sheets each with the topology (not necessarily the geometry) of a cone. Often several of these cones will coincide, which is not particularly troublesome, but unfortunately it is also common for some of these cones to be degenerate, which is more problematic.
 Suppose that \({f^{\mu v}}_{AB}\) factorisesThen$${f^{\mu \nu}}_{AB} = {h_{AB}}\,{f^{\mu \nu}}.$$(178)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 \({f^{\mu v}}_{AB}\).$$Q(x,\,k) = \det ({h_{AB}})\,{\left[ {{f^{\mu \nu}}\,{k_\mu}\,{k_\nu}} \right]^N}$$(179)
 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 \({f^{\mu v}}_{AB}\) can be simultaneously diagonalised in field space leading to$${f^{\mu \nu}}_{AB}\,{f^{\alpha \beta}}_{BC} = {f^{\alpha \beta}}_{AB}\,{f^{\mu \nu}}_{BC};\quad {\rm{that}}\,{\rm{is}}\quad \left[ {{f^{\mu \nu}},\,{f^{\alpha \beta}}} \right] = 0.$$(180)and then$${\bar f^{\mu \nu}}{\,_{AB}} = {\rm{diag}}\{\bar f_1^{\mu \nu},\,\bar f_2^{\mu \nu},\,\bar f_3^{\mu \nu}, \ldots, \bar f_N^{\mu \nu}\}$$(181)This case corresponds to an Nmetric theory, where up to an unspecified conformal factor \(g_A^{\mu v} \propto \bar f_A^{\mu v}\). This is the natural generalization of the two metric situation in biaxial crystals.$$Q(x,\,k) = \prod\limits_{A = 1}^N {[\bar f_A^{\mu \nu}\,{k_\mu}\,{k_\nu}].}$$(182)

If \({f^{\mu v}}_{AB}\) 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 [184].)
 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. 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 [136, 137, 16, 19, 18, 115, 114].
4.2.2 BEC models in the eikonal approximation
 1.
It is interesting to recognise that the dispersion relation (228) is exactly in agreement with that found in 1947 by Bogoliubov [36] (reprinted in [310]; see also [223]) 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.
It is easy to see that (228) 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 5.1.3, this is the reason why sonic horizons in a BEC can exhibit different features from those in standard general relativity.$$\delta \equiv {{\sqrt {1 + {{\lambda _c^2} \over {4{\lambda ^2}}}}  1} \over {(1  {v_0}/{c_{\rm{s}}})}} \simeq {1 \over {(1  {v_0}/{c_{\rm{s}}})}}{{\lambda _c^2} \over {8{\lambda ^2}}}.$$(229)  3.The dispersion relation (228) exhibits a contribution due to the background flow \(v _0^i{k_i}\), plus a quartic dispersion at high momenta. The group velocity isDispersion relations of this type (but in most cases with the sign of the quartic term reversed) have been used by Corley and Jacobson in analysing the issue of transPlanckian modes in the Hawking radiation from general relativistic black holes [185, 186, 88]. The existence of modified dispersion relations (MDR), that is, dispersion relations that break Lorentz invariance, can be taken as a manifestation of new physics showing up at high energies/short wavelengths. In their analysis, the group velocity reverses its sign for large momenta. (Unruh’s analysis of this problem used a slightly different toy model in which the dispersion relation saturated at high momentum [377].) In our case, however, the group velocity grows without bound allowing highmomentum modes to escape from behind the horizon. Thus the acoustic horizon is not absolute in these models, but is instead frequency dependent, a phenomenon that is common once nontrivial dispersions are included.$$v_g^i = {{\partial \omega} \over {\partial {k_i}}} = v_0^i \pm {{\left({{c^2} + {{{\hbar ^2}} \over {2{m^2}}}{k^2}} \right)} \over {\sqrt {{c^2}{k^2} + {{\left({{\hbar \over {2m}}{k^2}} \right)}^2}}}}{k^i}.$$(230)
Indeed, 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 well behaved 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.
This type of superluminal dispersion relation has also been analysed by Corley and Jacobson [90]. They found that this escape of modes from behind the horizon often leads to selfamplified instabilities in systems possessing both an inner horizon as well as an outer horizon, possibly causing them to disappear in an explosion of phonons. This is also in partial agreement with the stability analysis performed by Garay et al. [136, 137] using the whole Bogoliubov equations. Let us however leave further discussion regarding these developments to the Section 5.1.3 on horizon stability.
4.2.3 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 [418] 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 the 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 of 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 very 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 socalled ^{3}He — A 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_{ 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 [415, 417].
4.2.4 Slow light
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, centred 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 [284, 379], such a conclusion would turn out to be overenthusiastic. In order to obtain particle creation an inescapable requirement is to have socalled “mode mixing”, that is, mixing between the positive and negative frequency modes of the incoming and outgoing states. This is tantamount to saying that there must be regions where the frequency of the quanta as seen by a stationary observer at infinity (laboratory frame) becomes negative beyond the ergosphere at g_{00} = 0.
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 (238), but this also tells us that the condition ω_{0} − u · k < 0 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. 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 [379] that a different setup for slow light might deal with this and other issues (see [379] for a detailed summary). In the setup suggested by these authors there are two strong background 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.
4.3 Going further
We feel that the catalogue we have just presented is reasonably complete and covers the key items. For additional background on any of these topics, we would suggest sources such as the books “Artificial Black Holes” [284] and “The universe in a Helium droplet” [418]. For more specific detail, check this review’s bibliography, and use Spires to check for recent developments.
5 Lessons from 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 condensed matter systems, one immediately realises that these features only appear in the lowenergy regime of the analogue systems. When the 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 well understood 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 [379]).
 2.
To configure the analogue geometry such that it includes an apparent horizon. That is, within an appropriate quantum analogue model, the formation of an apparent horizon for the propagation of the quantum fields should excite the fields as to result in the emission of a thermal distribution of field particles.^{22}.
 1.
The effective description of the quantum analogue systems as fields over a background geometry breaks down when probed at sufficiently short length scales. This could badly influence the main features of Hawking radiation. In fact, immediately after the inception of the idea that black holes radiate, it was realised that there was a potential problem with the calculation [375]. It strongly relies on the validity of quantum field theory on curved backgrounds up to arbitrary high energies. Following a wave packet with a certain frequency at future infinity backwards in time, we can see that it had to contain arbitrarily large frequency components with respect to a local free fall observer (well beyond the Planck scale) when it was close to the horizon. In principle any unknown physics at the Planck scale could strongly influence the Hawking process so that one should view it with suspicion. This is the socalled 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 of the order of the critical Landau speed, which in these cases coincides with the sound speed [213]) with the surface of the container (an electromagnetic potential in the BECs case) might cause the production of rotons and quantised vortices, respectively. Thus, in order to produce an analogue model of Hawking radiation, one has to be somewhat ingenious. For example in the liquid Helium case, instead of taking acoustic waves in a supersonic flow stream as the analogue model, it is preferable to use as analogue model ripplons in the interface between two different phases, A and B phases, of Helium3 [415]. Another option is to start from a moving domain wall configuration. Then, the topological stability of the configuration prevents its destruction when creating a horizon [199, 200]. In the case of BECs a way to suppress the formation of quantised vortices is to take effectively 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 [19]. In either liquid Helium or BECs, there is also the possibility of creating an apparent horizon by rapidly approaching a critical velocity profile (see Figure 10), but without actually crossing into the supersonic regime [13], softening in this way the appearance of dynamical instabilities.
 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 [380].
 4.
Another important issue is the need to characterise “how quantum” a specific analogue model is. Even though, strictly speaking, one could say that any system undergoes quantum fluctuations, the point is how important they are in its description. In trying to build an analogue model of Hawking’s quantum effect, the relative value of Hawking temperature with respect to the environment is going to tell us whether the system can be really thought as a quantum analogue model or as effectively classical. For example, in our standard cosmological scenario, for a black hole to radiate at temperatures higher than that of the Cosmic Microwave Background, ≈ 3 K, the black hole should have a diameter of the order of micrometers or less. We would have to say that such black holes are no longer classical, but semiclassical. The black holes for which we have some observational evidence are of much higher mass and size, so their behaviour can be thought of as completely classical. Estimates of the Hawking temperature reachable in BECs yield T ∼ 100 nK [19]. This has the same order of magnitude of the temperature as the BECs themselves. This is telling us that regarding the Hawking process, BECs can be considered to be 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 ([167, 168, 169], 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 TransPlanckian problem
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.
As we have said, the requirement of a reservoir of ultrahigh frequency modes nearby the horizon seems to indicate a possible (and worrisome) sensitivity of the black hole radiation to the microscopic structure of spacetime. Actually by assuming exact Lorentz invariance one could in principle always locally transform the problematic ultra high frequency modes to low energy ones via some appropriate Lorentz transformation [185]. However in doing so it would have to rely on the physics of reference frames moving ultra fast with respect to us, as the reference frame needed would move arbitrarily close to the speed of light. Hence we would have to apply Lorentz invariance in a regime of arbitrary large boosts yet untested and in principle never completely testable given the noncompactness of the boost subgroup. The assumption of an exact boost symmetry is linked to a 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 idea on the nature of spacetime at ultrashort distances.
It was this type of reasoning that led in the nineties to a careful reconsideration of the crucial ingredients required for the derivation of Hawking radiation [185, 186, 377]. In particular investigators explored the possibility that spacetime microphysics could provide a short distance, 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 [376]. However such ideas were running into serious difficulties given that a naive short distance cutoff posed on the available modes of a free field theory results in a complete removal of the evaporation process (see e.g., Jacobson’s article [185] and references therein, and the comments in [167, 168, 169]). Indeed there are alternative ways through which the effect of the short scales physics could be taken into account, and analogue models provide a physical framework where these ideas could be put to the test. In fact analogue models provide explicit examples of emergent spacetime symmetries, they can be used to simulate black hole backgrounds, they may be endowed with quantizable perturbations and, in most of the cases, they have a well known microscopic structure. Given that Hawking radiation can be, at least in principle, simulated in such systems one might ask how and if the 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 lowest order terms), although in some special models (like BEC) the series is automatically finite due to intrinsic properties of the system. (In any case one can see that most of the analogue models so far considered lead to modifications of the form ±k^{3}/K^{2} 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 in the literature as “superluminal” and “subluminal” dispersion relations.
Most of the work on the transPlanckian problem in the nineties focussed on studying the effect on Hawking radiation due to such modifications of the dispersion relations at high energies in the case of acoustic analogues [185, 186, 377, 378, 88], and the question of whether such phenomenology could be applied to the case of real black holes (see e.g., [50, 188, 88, 299]).^{24} In all the aforementioned works Hawking radiation can be recovered under some suitable assumptions, as long as neither the black hole temperature nor the frequency at which the spectrum is considered are too close to the scale of microphysics K. However, the applicability of these assumptions to the real case of black hole evaporation is an open question. Also, in the case of the analogue models the mechanism by which the Hawking radiation is recovered is not always the same. We concisely summarise here the main results (but see e.g., [380] for further details).
5.1.2.2 Subluminal dispersion relations
The key feature is that in the presence of a subluminal modification the group velocity of the modes increases with k only up to some turning point (which is equivalent to saying that the group velocity does not asymptote to c, which could be the speed of sound, but instead is upper bounded). For values of k beyond the turning point the group velocity decreases to zero (for (248)) or becomes imaginary (for (249)). In the latter case this can be interpreted as signifying the breakdown of the regime where the dispersion relation (249) can be trusted. The picture that emerged from these analyses concerning the origin of the outgoing Hawking modes at infinity is quite surprising. In fact, if one traces back in time an outgoing mode, as it approaches the horizon it decreases its group velocity below the speed of sound. At some point before reaching the horizon, the outgoing mode will appear as an ingoing one dragged into the black hole by the flow. Stepping further back in time it is seen that such a mode was located at larger and larger distances from the horizon, and tends to a very high energy mode far away at early times. In this case one finds what might be called a “mode conversion”, where the origin of the outgoing Hawking quanta seems to originate from ingoing modes which have “bounced off” the horizon without reaching transPlanckian frequencies in its vicinities. Several detailed analytical and numerical calculations have shown that such a conversion indeed happens [378, 50, 88, 87, 170, 330, 380] and that the Hawking result can be recovered for k ≪ K where k is the black hole surface gravity.
5.1.2.3 Superluminal dispersion relations
The case of a superluminal dispersion relation is quite different and, as we have seen, has some experimental interest given that this is the kind of dispersion relation that arises in some promising analogue models (e.g., BECs). In this situation, the outgoing modes are actually seen as originating from behind the horizon. This implies that these modes somehow originate from the singularity (which can be a region of high turbulence in acoustic black hole analogues), and hence it would seem that not much can be said in this case. However it is possible to show that if one imposes vacuum boundary conditions on these modes near the singularity, then it is still possible to recover the Hawking result, i.e., thermal radiation outside the hole [87]. It is particularly interesting to note that this recovering of the standard result is not always guaranteed in the presence of superluminal dispersion relations. Corley and Jacobson [90] 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 [90] is that the compact ergoregion characterizing such configurations is unstable to selfamplifying Hawking radiation. The presence of such an instability seems to be confirmed by the analysis carried on in [136, 137] where a BoseEinstein condensate analogue black hole was considered.
5.1.2.4 General conditions for Hawking radiation
Is it possible to reduce the rather complex phenomenology just described to a few basic assumptions that must be satisfied in order to recover Hawking radiation in the presence of Lorentz violating dispersion relations? A tentative answer is given in [380], where the robustness of the Hawking result is considered for general modified (subluminal as well as superluminal) dispersion relations. The authors of [380] assume that the geometrical optics approximation breaks down only in the proximity of the event horizon (which is equivalent to saying that the particle production happens only in such a region). Here, the wouldbe transPlanckian modes are converted in subPlanckian ones. Then, they try to identify the minimal set of assumptions that guarantees that such “converted modes” are generated in their ground states (with respect to a freely falling observer), as this is a well known condition in order to recover Hawking’s result. They end up identifying three basic assumptions that guarantee such emergence of modes in the ground state at the horizon. First, the preferred frame selected by the breakdown of Lorentz invariance must be the freely falling one instead of the rest frame of the static observer at infinity (which coincides in this limit with the laboratory observer). Second, the Planckian excitations must start off in the ground state with respect to freely falling observers. Finally, they must evolve in an adiabatic way (i.e., the Planck dynamics must be much faster than the external subPlanckian dynamics). Of course, although several systems can be found in which such conditions hold, it is also possible to show [380] that realistic situations in which at least one of these assumptions is violated can be imagined. It is hence still an open question whether real black hole physics does indeed satisfy such conditions, and whether it is hence robust against modifications induced by the violation of Lorentz invariance.
5.1.2.5 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 [89, 189] 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 (248), 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 (249) 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 (249) (which anyway in realistic analogue models is emergent and not fundamental), and hence the model has no predictive power regarding the ultimate origin of the relevant incoming modes.
These conclusions regarding the impossibility of clearly predicting the origin at early times of the modes ultimately to be converted into Hawking radiation are not specific to the particular dispersion relations (248) or (249) one is using. The Killing frequency is in fact conserved on a static background, thus the incoming modes must have the same frequency as the outgoing ones, hence there can be no modemixing and particle creation. This is why one has actually to assume that the WKB approximation fails in the proximity of the horizon and that the modes are there in the vacuum state for the comoving observer. In this sense the need for these assumptions can be interpreted as evidence that these models are not fully capable of solving the transPlanckian problem.
5.1.2.6 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 [318, 319, 89]. In particular in [319] the transparency of the refractive medium at high frequencies has been used to introduce an effective cutoff for the modes involved in Hawking radiation in a classical refractive index analogue model (see Section 4.1.3 of this review). In this model an event horizon for the electromagnetic field modes can be simulated by a surface of singular electric and magnetic permeabilities. This would be enough to recover Hawking radiation but it would imply the unphysical assumption of a refractive index which is valid at any frequency. However it was shown in [319] that the Hawking result can be recovered even in the case of a dispersive medium which becomes transparent above some fixed frequency K (which we can imagine as the plasma frequency of the medium), the only (crucial) assumption being again that the “transPlanckian” modes with k > K are in their ground state near the horizon.
An alternative avenue was considered in [89]. There a lattice description of the background was used for imposing a cutoff in a more physical way with respect to the continuum dispersive models previously considered. In such a discretised spacetime the field takes values only at the lattice points, and wavevectors are identified modulo 2π/ℓ where ℓ is the lattice characteristic spacing, correspondingly one obtains a sinusoidal dispersion relation for the propagating modes. Hence the problem of recovering a smooth evolution of incoming modes to outgoing ones is resolved by the intrinsically regularised behaviour of the wave vectors field. In [89] the authors explicitly considered the Hawking process for a discretised version of a scalar field, where the lattice is associated to 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 short distance cutoff leads to the choice of a lattice spacing constant at infinity, and that the lattice points are at rest at infinity and fall freely into the black hole.^{25} In this case the lattice spacing grows in time and the lattice points spread in space as they fall toward the horizon. However this time dependence of the lattice points is found to be of order 1/κ, and hence unnoticeable to long wavelength modes and relevant only for those with wavelengths of the order of the lattice spacing. The net result is that on such a lattice long wavelength outgoing modes are seen to originate from short wavelength incoming modes via a process analogous to the Bloch oscillations of accelerated electrons in crystals [89, 189].
5.1.3 Horizon stability
Although closely related, as we will soon see, we have to distinguish carefully between the mode analysis of a linear field theory (with or without modified dispersion relations — MDR) over a fixed background and the stability analysis of the background itself.
In a normal mode analysis one requires boundary conditions such that the field is regular everywhere, even at infinity. However, if one is analysing the solutions of the linear field theory as a way of probing the stability of the background configuration, one can consider less restrictive boundary conditions. For instance, one can consider the typical boundary conditions that lead to quasinormal modes: These modes have to be purely 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 to this sink configuration have been analysed in [31]. The results found are qualitatively similar to those in the classical linear stability analysis of the Schwarzschild black hole in general relativity [384, 385, 317, 431, 267]. Of course, the gravitational field in general relativity has two dynamical degrees of freedom — those associated with gravitational waves — that have to be analysed separately; these are the “axial” and “polar” perturbations. In contrast, in the present situation we only have scalar perturbations. Nevertheless, the potentials associated with “axial” and “polar” perturbations of Schwarzschild spacetime, and that associated with scalar perturbations of the canonical acoustic black hole, produce qualitatively the same behaviour: There is a series of damped quasinormal modes — proving the linear stability of the system — with higher and higher damping rates.
An important point we have to highlight in here is that although in the linear regime the dynamical behaviour of the acoustic system is similar to general relativity, this is no longer true once one enters the 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, independently of how small they are initially, can lead to the development of shocks, a situation completely absent in vacuum general relativity.
Now, given an approximately stationary, and at the very least metastable, classical 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. Moreover, in an analogue system with quantum fluctuations that maintain strict adherence to the equivalence principle (no MDR) it must then be impossible to create an isolated truly stationary horizon by external means — any truly stationary horizon must be provided with an external power source to stabilise it against Hawking emission. That is, in an analogue system one could in principle, by manipulating external forces, compensate for the backreaction effects that in a physical general relativity scenario cause the horizon to shrink (or evaporate) and thus become nonstationary.
Let us describe what happens when one takes into account the existence of MDR. A wonderful physical system that has MDR explicitly incorporated in its description is the 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 of 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 [136, 137] but taking the flow to be effectively onedimensional. What they found is that these configurations are dynamically unstable: There are modes satisfying the appropriate boundary conditions such that the imaginary parts of their associated frequencies are positive. These instabilities are associated basically with the bound states inside the black hole. The dynamical tendency of the system to evolve is suggestively similar to that in the standard evaporation process of a black hole in semiclassical general relativity. This observation alone could make us question whether the first signatures of a quantum theory underlying general relativity might show up in precisely this manner. Interest in this question is reinforced by a specific analysis in the “loop quantum gravity” approach to quantizing gravity that points towards the existence of fundamental MDR at high energies [134]. The formulation of effective gravitational theories that incorporate some sort of MDR at high energies is currently under investigation (see for example [247, 2]); these exciting developments are however beyond the scope of this review.

The “classical” or macroscopic wave function of the BEC represents the classical spacetime of GR, but only when probed at long enough wavelengths.

The “classical” 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 its underlying quantum origin.

The Bogoliubov quantum quasiparticles over the “classical” wave function correspond to a further step away from semiclassical gravity in that they are analogous to the existence of quantum gravitons over a (semiclassical) background spacetime.
At this point we would like to remark, once again, that the analysis based on the evolution of a BEC has to be used with care. For example, they cannot directly serve to shed light on what happens in the final stages of the evaporation of a black hole, as the BEC does not fulfil, at any regime, the Einstein equations.
Now continuing the discussion, what happens when treating the perturbations to the background BEC flow as quantum excitations (Bogoliubov quasiparticles)? What we certainly know is that the analysis of modes in a collapsingtoformablackhole background spacetime leads to the existence of radiation emission very much like Hawking emission [50, 87, 91]. (This is why it is said that Hawking’s process is robust against modifications of the physics at high energies.) The comparison of these calculations with that of Hawking (without MDR), tells us that the main modification to Hawking’s result is that now the Hawking flux of particles would not last forever but would vanish after a long enough time (this is why, in principle, we can dynamically create a configuration with a sonic horizon in a BEC). The emission of quantum particles reinforces the idea that the supersonic sink configurations are unstable.

If the perturbation 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.
A quantum mode analysis of the black holewhite hole configuration in a straight line, taking into account the existence of superluminal dispersion relations (similar to those in a BEC), led to the conclusion that the emission of particles in this configuration proceeds in a selfamplified (or runaway) manner [90]. We can understand this effect as follows: At the black horizon a virtual pair of phonons are converted into real phonons, the positive energy phonon goes towards infinity while the negative energy pair falls beyond the black horizon. However, the white horizon makes this negative energy phonon bounce back towards the black horizon (thanks to superluminal motion) stimulating the emission of additional phonons. Although related to Hawking’s process this phenomenon has a quite different nature: For example, there is no temperature associated with it. A stability analysis of a configuration like this in a BEC would lead to strong instabilities. This same configuration, but compactified into a ring configuration, has been dynamically analysed in [136, 137]. What they found is that there are regions of stability and instability depending on the parameters characterizing the configuration. We suspect that the stability regions appear because of specific periodic arrangements of the modes around the ring. Among other reasons, these arrangements are interesting because they could be easier to create in laboratory with current technology and their instabilities easier to detect than Hawking radiation itself.
To conclude this subsection, we would like to highlight that there is still much to be learned by studying the different levels of description of an analogue system, and how they influence the stability or instability of configurations with horizons.
5.1.4 Analogue spacetimes as background gestalt

“Top down” calculations of Hawking radiation starting from some idealised model of quantum gravity [1, 201, 276, 277].

“Bottom up” calculations of Hawking radiation starting from curved space quantum field theory [22, 23, 61, 62, 71, 111, 112, 128, 253, 254, 255, 256, 257, 298, 321].

TransHawking versions of Hawking radiation, either as reformulations of the physics, or as alternative scenarios [30, 68, 130, 131, 150, 156, 167, 168, 169, 233, 289, 291, 325, 329, 333, 334, 340, 341, 350, 364].

Black hole entropy viewed in the light of analogue spacetimes [94].

Hawking radiation interpreted as a statement about particles travelling along complex spacetime trajectories [292, 350, 360].
5.2 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 [401]. 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 [345]. Equivalently to what happens with Kerr black holes, this configuration is classically stable (in the linear regime) [31]. 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 [415]. 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 [25, 27, 26, 64, 113, 237].
A related phenomenon one can consider is the blackhole bomb mechanism [312]. 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 of the order of the healing length, so one cannot directly associate an effective Lorentzian geometry to this portion of the configuration. Any analysis that neglects the high energy terms is not going to give any sensible result in these cases.
5.3 Cosmological geometries
Analogue model techniques have also been applied to cosmology. 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 [17, 18, 116, 115, 114, 242, 58, 59, 424] with a specific view to enhancing our understanding of “cosmological particle production” driven by the expansion of the universe.

Fedichev and Fischer [115, 114] have investigated WKB estimates of the cosmological particle production rate and (1+1) dimensional cosmologies, both in expanding BECs.

Lidsey [242], and Fedichev and Fischer [116] have focussed on the behaviour of cigarlike condensates in grossly asymmetric traps.

Barcelo et al. [17, 18] have focussed on the central region of the BEC and thereby tried (at least locally) to mimic FLRW behaviour as closely as possible.

Fischer and Schützhold [119] propose the use of twocomponent BECs to simulate cosmic inflation.

Weinfurtner [425, 424] has concentrated on the approximate simulation of de Sitter spacetimes.
In all of these models the general expectations of the relativity community have been borne out — theory definitely predicts particle production, and the very interesting question is the extent to which the formal predictions are going to be modified when working with real systems experimentally [18]. We expect that these analogue models provide us with new insights as to how their inherent modified dispersion relations affect cosmological processes such as the generation of a primordial spectrum of perturbations (see for example [42, 41, 43, 44, 45, 46, 47, 70, 107, 158, 177, 178, 207], [229, 230, 244, 252, 249, 250, 251, 274, 275, 296, 349, 361, 362, 363, 371] where analoguelike ideas are applied to cosmological inflation).
An interesting sideeffect of the original investigation, is that birefringence can now be used to model “variable speed of light” (VSL) geometries [28, 108]. 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 [28, 108].
5.4 Bose novae: an example of the reverse flow of information?
As we have seen in the previous sections, analogue models have in the past been very useful in providing new, condensed matter physics inspired, 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 well known concepts of semiclassical gravity into condensed matter frameworks.
The phenomenon we are referring to is the so called “Bose nova” [104]. 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 sizeable 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 t_{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 [58, 59], based on the well known semiclassical gravity analysis of particle creation in an expanding universe. In fact the dynamics of quantum excitations over the collapsing BEC was 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 [58, 59] 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 [58, 59] 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.
5.5 Going further
For more details on the transPlanckian problem some of the key papers are the relatively early papers of Unruh [377, 378], and Jacobson [185, 186]. For superradiance and cosmological issues there seems to be considerable ongoing interest, and one should carefully check Spires for the most recent articles.
6 Future Directions

Back reaction,

Equivalence principle,

Emergent gravity,

Quantum gravity phenomenology,

Quantum gravity,

Experimental analogue gravity.
6.1 Back reaction
There are important phenomena in gravitational physics whose understanding needs analysis well beyond classical general relativity and field theory on curved background spacetimes. The black hole evaporation process can be considered as paradigmatic among these phenomena. Here, we particularise our discussion to this case. Since we are currently unable to analyse the entire process of black hole evaporation within a complete quantum theory of gravity, a way of proceeding is to analyse the simpler (but still extremely difficult) problem of semiclassical back reaction (see for example [99, 79, 34, 132, 51, 257]). One takes a background black hole 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 back reaction 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 [9, 10]. 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 back reaction 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 back reaction 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 back reaction (which incorporates deviations from standard general relativity as well). In other words, one would have to understand the differences between the standard back reaction scheme in general relativity, and that based on Equations (186) and (187).
To end this subsection, we would like to comment that one can go beyond the semiclassical backreaction scheme by using the socalled stochastic semiclassical gravity programme [179, 180, 181]. 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 [299], 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.
6.2 Equivalence principle
Analogue models are of particular interest in that the analogue spacetimes that emerge often violate, to some extent, the Einstein equivalence principle [16, 399]. Since the Einstein equivalence principle (or more precisely the universality of free fall) is experimentally tested to the accuracy of about 1 part in 10^{13}, it is important to build this principle into realistic models of gravitation — most likely at a fundamental level.
One way of interpreting the Einstein equivalence principle is as 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.
Now it is this feature that is relatively difficult to arrange in analogue models. If one is dealing with a single degree of freedom then monometricity is no great constraint. But with multiple degrees of freedom, analogue spacetimes generally lead to refringence — that is the occurrence of Fresnel equations that often imply multiple propagation speeds for distinct normal modes. To even obtain a bimetric model (or more generally, a multimetric model), requires an algebraic constraint on the Fresnel equation that it completely factorises into a product of quadratics in frequency and wavenumber. Only if this algebraic constraint is satisfied can one assign a “metric” to each of the quadratic factors. If one further wishes to impose monometricity then the Fresnel equation must be some integer power of some single quadratic expression, an even stronger algebraic statement [16, 399].
 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.

6.3 Emergent gravity
One of the more fascinating approaches to “quantum gravity” is the suggestion, typically attributed to Sakharov [332, 393], 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. Sakharov had in mind a specific model in which gravity could be viewed as an “elasticity” of the spacetime medium, and was “induced” via oneloop physics in the matter sector [332, 393]. 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.
6.4 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 [261]. The key point is that at low energies (well below the Planck energy) one expects the locally Minkowskian structure of the spacetime manifold to guarantee that one sees only special relativistic effects; general relativistic effects are negligible at short distances. However as ultra high energies are approached (although still below Planck scale energies) several quantum gravity 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 [214, 109] to brane worlds [54] and loop quantum gravity [134]. 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 [3, 82, 195, 194, 196, 197, 261, 355] 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 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 new ideas about other effects of physical quantum gravity that might be observable at subPlanckian energies.
6.5 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 [65, 356, 129].
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) [65, 356, 129, 327, 326, 40]. 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 [356].
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 [376, 389, 284]. 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 [389, 284]. 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.
6.6 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.
While the consensus in the theoretical community is that BoseEinstein condensates are likely to provide the best working model for analogue gravity, it is possible that we might still be surprised by experimental developments. We leave this as an open challenge to the experimental community.
7 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
7.1 Going further

http://www.slac.stanford.edu/spires/ — the bibliographic database for keeping track of (almost all of) the relevant literature.

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

http://www.livingreviews.org/ — the Living Reviews portal.
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 [80]. 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 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 [239].
 11.
The PainlevéGullstrand line element is sometimes called the Lemaître line element.
 12.
 13.
Vorticity is automatically generated, for instance, whenever the background fluid is nonbarotropic, and in particular when ∇_{ ρ } × ∇_{ p } ≠ 0. Furthermore, it has been argued in [343] that quantum backreaction can also act as a source for vorticity.
 14.
In references [141, 143, 139, 140, 145, 144, 138, 142] 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.
 15.
We emphasise: To get Hawking radiation you need an effective geometry, a horizon, and a suitable quantum field theory on that geometry.
 16.
 17.
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.
 18.
Indeed the results of Moncrief [268] are more general than those considered in the standard acoustic gravity papers that followed because they additionally permit a general relativistic curved background.
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 reference [268] predates Unruh’s 1981 paper by one year, and [259, 260, 258] postdate Unruh’s 1981 paper by a few years, there seems to have not been any crossconnection.
 19.
A recent attempt at connecting the electromechanical analogy back to relativity can be found in [434].
 20.
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.
 21.
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.
 22.
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 dissappear 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 deserve future analysis. Certainly systems of this type lie well outside the class of usual analogue models.
 23.
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.
 24.
 25.
Reference [89] 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 short distance cutoff.
 26.
It is however important to keep in mind that not all the above cited 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. Carlos Barceló has been supported by a Marie Curie European Reintegration Grant, and by the Education and Science Council of Junta de Andalucía (Spain). We also wish to thank Enrique Arilla for Figures 1 and 2 of the art work.
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