List of 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 Navier–Stokes equation. See External Link 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 first-order quasi-linear symmetric hyperbolic form, then it is an exact non-perturbative result that the matrix of coefficients for the first-derivative terms can be used to construct a conformal class of metrics that encodes the causal structure of the system of PDEs. For barotropic hydrodynamics this is briefly discussed in [138]. This analysis is related to the behaviour of characteristics of the PDEs, and ultimately can be linked back to the Fresnel equation that appears in the eikonal limit.
4 It is straightforward to add external forces, at least conservative body forces such as Newtonian gravity.
5 Henceforth, in the interests of notational simplicity, we shall drop the explicit subscript 0 on background field quantities unless there is specific risk of confusion.
6 This discussion naturally leads us to what is perhaps the central question of analogue models – just how much of the standard “laws of black hole mechanics” [51, 671] carry over into these analogue models? Quite a lot but not everything – that is our main topic for the rest of the review.
7 Because of the background Minkowski metric there can be no possible confusion as to the definition of this normal derivative.
8 There are a few potential subtleties in the derivation of the existence of Hawking radiation, which we are, for the time being, glossing over; see Section 5.1 for details.
9 There is an issue of normalization here. On the one hand we want to be as close as possible to general relativistic conventions. On the other hand, we would like the surface gravity to really have the dimensions of an acceleration. The convention adopted here, with one explicit factor of c, is the best compromise we have come up with. (Note that in an acoustic setting, where the speed of sound is not necessarily a constant, we cannot simply set c → 1 by a choice of units.)
10 There are situations in which this surface gravity is a lot larger than one might naively expect [398Jump To The Next Citation Point].
11 The Painlevé–Gullstrand line element is sometimes called the Lemaître line element.
12 Similar constructions work for the Reissner–Nordström geometry [398Jump To The Next Citation Point], as long as one does not get too close to the singularity. (With c = 1 one needs r > Q2∕(2m ) to avoid an imaginary fluid velocity.) Likewise, certain aspects of the Kerr geometry can be emulated in this way [641Jump To The Next Citation Point]. (One needs r > 0 in the Doran coordinates [176, 633Jump To The Next Citation Point] to avoid closed timelike curves.) As a final remark, let us note that de Sitter space corresponds to v ∝ r and ρ ∝ 1∕r3. For further details see Section 2.5.
13 Mathematically, one can view the time taken to traverse such a path as a particular instance of Finsler distance – it is, in fact, the distance function associated with a Randers metric. See [246Jump To The Next Citation Point], and brief discussion in Section 3.2.10.
14 Vorticity is automatically generated, for instance, whenever the background fluid is non-barotropic, and, in particular, when ∇ρ × ∇p ⁄= 0. Furthermore, it has been argued in [559Jump To The Next Citation Point] that quantum backreaction can also act as a source for vorticity.
15 In [233, 234, 235, 236, 237, 238, 239, 240] the author has attempted to argue that vorticity can be related to the concept of torsion in a general affine connexion. We disagree. Although deriving a wave equation in the presence of vorticity very definitely moves one beyond the realm of a simple Riemannian spacetime, adding torsion to the connexion is not sufficient to capture the relevant physics.
16 Indeed, historically, though not of direct relevance to general relativity, analogue models played a key role in the development of electromagnetism – Maxwell’s derivation of his equations for the electromagnetic field was guided by a rather complicated “analogue model” in terms of spinning vortices of aether. Of course, once you have the equations in hand you can treat them in their own right and forget the model that guided you – which is exactly what happened in this particular case.
17 We emphasise: To get Hawking radiation you need an effective geometry, a horizon, and a suitable quantum field theory on that geometry.
18 Of course, we now mean “gravity wave” in the traditional fluid mechanics sense of a water wave whose restoring force is given by ordinary Newtonian gravity. Waves in the fabric of spacetime are more properly called “gravitational waves”, though this usage seems to be in decline within the general relativity community. Be very careful in any situation where there is even a possibility of confusing the two concepts.
19 The existence of this constraint has been independently re-derived several times in the literature. In contrast, other segments of the literature seem blithely unaware of this important restriction on just when permittivity and permeability are truly equivalent to an effective metric.
20 One could also imagine systems in which the effective metric fails to exist on one side of the horizon (or even more radically, on both sides). The existence of particle production in this kind of system will then depend on the specific interactions between the sub-systems characterizing each side of the horizon. For example, in stationary configurations it will be necessary that these interactions allow negative energy modes to disappear beyond the horizon, propagating forward in time (as happens in an ergoregion). Whether these systems will provide adequate analogue models of Hawking radiation or not is an interesting question that deserves future analysis. Certainly systems of this type lie well outside the class of usual analogue models.
21 Actually, even relativistic behaviour at low energy can be non-generic, but we assume in this discussion that an analogue model by definition is a system for which all the linearised perturbations do propagate on the same Lorentzian background at low energies.
22 However, see also [522Jump To The Next Citation Point, 528] for a radically different alternative approach based on the idea of “superoscillations” where ultra-high frequency modes near the horizon can be mimicked (to arbitrary accuracy) by the exponential tail of an exponentially-large amplitude mostly hidden behind the horizon.
23 Note however, that for a real black hole and Lorentz-invariant physics, the spectrum observed at infinity is indistinguishable from thermal given that no correlation measurement is allowed, since the Hawking partners are space-like separated across the horizon. This fact is indeed the root of the information paradox.
24 [149Jump To The Next Citation Point] 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.
25 Personal communication by R. Parentani
26 However, it is 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.