In contrast, the acoustic metric is more constrained. Being specified completely by the three scalars , , and , 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 and .
Thus, the simple acoustic metric of this section can, at best, reproduce some subset of the generic metrics of interest in general relativity.
In the next two subsections we shall undertake to more fully explain some of the technical details underlying the acoustic analogy. Concepts and quantities such as horizons, ergo-regions and “surface gravity” are important features of standard general relativity, and analogies are useful only insofar as they adequately preserve these notions.
Let us start with the notion of an ergo-region: Consider integral curves of the vector5 [445, 275, 670].
A trapped surface in acoustics is defined as follows: Take any closed two-surface. If the fluid velocity is everywhere inward-pointing 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 outer-trapped. (For comparison with the usual situation in general relativity see [275, pp. 319–323] or [670, pp. 310–311].) Inner-trapped surfaces (anti-trapped surfaces) can be defined by demanding that the fluid flow is everywhere outward-pointing with supersonic normal component. It is only because of the fact that the background Minkowski metric provides a natural definition of “at rest” that we can adopt such a simple and straightforward definition. In ordinary general relativity we need to develop considerable additional technical machinery, such as the notion of the “expansion” of bundles of ingoing and outgoing null geodesics, before defining trapped surfaces. That the above definition for acoustic geometries is a specialization of the usual one can be seen from the discussion in [275, pp. 262–263]. The acoustic trapped region is now defined as the region containing outer trapped surfaces, and the acoustic (future) apparent horizon as the boundary of the trapped region. That is, the acoustic apparent horizon is the two-surface for which the normal component of the fluid velocity is everywhere equal to the local speed of sound. (We can also define anti-trapped regions and past apparent horizons but these notions are of limited utility in general relativity.)6
The fact that the apparent horizon seems to be fixed in a foliation-independent manner is only an illusion due to the way in which the analogies work. A particular fluid flow reproduces a specific “metric”, (a matrix of coefficients in a specific coordinate system, not a “geometry”), and, in particular, a specific foliation of spacetime. (Only “internal” observers see a “geometry”; see the discussion in Section 7.4). The same “geometry” written in different coordinates would give rise to a different fluid flow (if at all possible, as not all coordinate representations of a fixed geometry give rise to acoustic metrics) and, therefore, to a different apparent horizon.
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.
In all stationary geometries the apparent and event horizons coincide, and the distinction is immaterial. In time-dependent geometries the distinction is often important. When computing the surface gravity, we shall restrict attention to stationary geometries (steady flow). In fluid flows of high symmetry (spherical symmetry, plane symmetry), the ergosphere may coincide with the acoustic apparent horizon, or even the acoustic event horizon. This is the analogue of the result in general relativity that for static (as opposed to stationary) black holes the ergosphere and event horizon coincide. For many more details, including appropriate null coordinates and Carter–Penrose diagrams, both in stationary and time-dependent situations, see .
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 non-stationary (time-dependent) 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 time-coordinate unless there is a time-like Killing vector – this is why extending the notion of surface gravity to non-stationary 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.
To set up the appropriate framework, write the general stationary acoustic metric in the form
Since this is a static geometry, the relationship between the Hawking temperature and surface gravity may be verified in the usual fast-track manner – using the Wick rotation trick to analytically continue to Euclidean space . If you don’t like Euclidean signature techniques (which are in any case only applicable to equilibrium situations) you should go back to the original Hawking derivations [270, 271].8
We should emphasize that the formula for the Hawking temperature contains both the surface gravity and the speed of sound at the horizon . Specifically (which corresponds to being constant). Purely on dimensional grounds it is a spatial derivative of velocity (which has the same engineering dimension as frequency) that is the determining factor in specifying the physically-normalised Hawking temperature. (Since there is a strong tendency in classical general relativity to adopt units such that , and even in these analogue models it is common to adopt units such that , this has the potential to lead to some confusion. If you choose units to measure the surface gravity as a physical acceleration, then it is the quantity , which has the dimensions of frequency that governs the Hawking flux .)
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 de-synchronizing 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.
Recall that by construction the acoustic apparent horizon is, in general, defined to be a two-surface for which the normal component of the fluid velocity is everywhere equal to the local speed of sound, whereas the acoustic event horizon (absolute horizon) is characterised by the boundary of those null geodesics (phonons) that do not escape to infinity. In the stationary case these notions coincide, and it is still true that the horizon is a null surface, and that the horizon can be ruled by an appropriate set of null curves. Suppose we have somehow isolated the location of the acoustic horizon, then, in the vicinity of the horizon, we can split up the fluid flow into normal and tangential components
Consider the quantity and calculate9
This is in agreement with the previous calculation for static acoustic black holes, and insofar as there is overlap, is also consistent with results of Unruh [607, 608], Reznik , and the results for “dirty black holes” . 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.
As an example of a fluid flow where the distinction between ergosphere and acoustic event horizon is critical, consider the “draining bathtub” fluid flow. We shall model a draining bathtub by a (3+1) dimensional flow with a linear sink along the z-axis. Let us start with the simplifying assumption that the background density is a position-independent constant throughout the flow (which automatically implies that the background pressure and speed of sound are also constant throughout the fluid flow). The equation of continuity then implies that for the radial component of the fluid velocity we must have
(If these flow velocities are nonzero, then following the discussion of  there must be some external force present to set up and maintain the background flow. Fortunately it is easy to see that this external force affects only the background flow and does not influence the linearised fluctuations we are interested in.)
For the background velocity potential we must then have
Dropping a position-independent prefactor, the acoustic metric for a draining bathtub is explicitly given by, that metric being a model for the acoustic geometry surrounding physical vortices in superfluid 3He. (For a survey of the many analogies and similarities between the physics of superfluid 3He, see Section 4.2.2 and references therein. For issues specifically connected to the Standard Electroweak Model see .) Note that the metric given above is not identical to the metric of a spinning cosmic string, which would instead take the form 
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 time-like curves. At large distances it is possible to approximate the vortex geometry by a spinning cosmic string , but this approximation becomes progressively worse as the core is approached. Trying to force the existence of closed time-like curves leads to the existence of evanescent waves in what would be the achronal region, and therefore to the breakdown of the analogue model description .
The ergosphere (ergocircle) forms at
The acoustic event horizon forms once the radial component of the fluid velocity exceeds the speed of sound, that is, at
A popular model for the investigation of event horizons in the acoustic analogy is the one-dimensional slab geometry where the velocity is always along the direction and the velocity profile depends only on . The continuity equation then implies that is a constant, and the acoustic metric becomes
If we set and ignore the coordinates and conformal factor, we have the toy model acoustic geometry discussed in many papers. (See for instance the early papers by Unruh [608, p. 2828, Equation (8)], Jacobson [310, p. 7085, Equation (4)], Corley and Jacobson , and Corley .) Depending on the velocity profile one can simulate black holes, white holes or black hole-white hole pairs, interesting for analyzing the black hole laser effect  or aspects of the physics of warpdrives .
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 . 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.
To see how close the acoustic metric can get to reproducing the Schwarzschild geometry it is first useful to introduce one of the more exotic representations of the Schwarzschild geometry: the Painlevé–Gullstrand line element, which is simply an unusual choice of coordinates on the Schwarzschild spacetime.11 In modern notation the Schwarzschild geometry in outgoing (+) and ingoing (–) Painlevé–Gullstrand coordinates may be written as:, Gullstrand , Lemaître , the related discussion by Israel , and more recently, the paper by Kraus and Wilczek . The Painlevé–Gullstrand coordinates are related to the more usual Schwarzschild coordinates by
As emphasised by Kraus and Wilczek, the Painlevé–Gullstrand line element exhibits a number of features of pedagogical interest. In particular the constant-time spatial slices are completely flat. That is, the curvature of space is zero, and all the spacetime curvature of the Schwarzschild geometry has been pushed into the time–time and time–space components of the metric.
Given the Painlevé–Gullstrand line element, it might seem trivial to force the acoustic metric into this form: Simply take and to be constants, and set . While this certainly forces the acoustic metric into the Painlevé–Gullstrand form, the problem with this is that this assignment is incompatible with the continuity equation that was used in deriving the acoustic equations.
The best we can actually do is this: Pick the speed of sound to be a position-independent constant, which we normalise to unity (). Now set , and use the continuity equation plus spherical symmetry to deduce so that . Since the speed of sound is taken to be constant, we can integrate the relation to deduce that the equation of state must be and that the background pressure satisfies . Overall, the acoustic metric is now
So we see that the net result is conformal to the Painlevé–Gullstrand form of the Schwarzschild geometry but not identical to it. For many purposes this is 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  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 frequency-dependent 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.
We can now turn to another aspect of acoustic black holes, i.e., their global causal structure, which we shall illustrate making use of the Carter–Penrose conformal diagrams [445, 275]. A systematic study in this sense was performed in  for 1+1 geometries (viewed either as a dimensional reduction of a physical 3+1 system, or directly as geometrical acoustic metrics). The basic idea underlying the conformal diagram of any non-compact 1+1 manifold is that its metric can always be conformally mapped to the metric of a compact geometry, with a boundary added to represent events at infinity. Since compact spacetimes are in some sense “finite”, they can then properly be drawn on a sheet of paper, something that is sometimes very useful in capturing the essential features of the geometry at hand.
The basic steps in the acoustic case are the same as in standard general relativity: Starting from the coordinates () as in Equation (35), one has to introduce appropriate null coordinates (analogous to the Eddington–Finkelstein coordinates) (), then, by exponentiation, null Kruskal-like coordinates (), and finally compactify by means of a new coordinate pair () involving a suitable function mapping an infinite range to a finite one (typically the arctan function). We shall explicitly present only the conformal diagrams for an acoustic black hole, and a black hole-white hole pair, as these particular spacetimes will be of some relevance in what follows. We redirect the reader to  for other geometries and technical details.
In contradistinction to the Carter–Penrose diagram for the Schwarzschild black hole (which in the current context would have to be an eternal black hole, not one formed via astrophysical stellar collapse) there is no singularity. On reflection, this feature of the conformal diagram should be obvious, since the fluid flow underlying the acoustic geometry is nowhere singular.
Note that the event horizon is the boundary of the causal past of future right null infinity; that is, , with standard notations .
Living Rev. Relativity 14, (2011), 3
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