- 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 itself. (This observation continues to hold in more general situations where it is often significantly easier to calculate the tensor density than it is to calculate the effective metric .)
- 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 is the speed of light in vacuum.) The fluid particles couple only to the physical metric . In fact the fluid motion is completely non-relativistic, so that , and it is quite sufficient to consider Galilean relativity for the underlying fluid mechanics.
- Sound waves on the other hand, do not “see” the physical metric at all. Acoustic perturbations couple only to the effective acoustic metric .

- It is quite remarkable that (to the best of our knowledge) a version of this acoustic metric was first derived and used in Moncrief’s studies of the relativistic hydrodynamics of accretion flows surrounding black holes [448]. Indeed, Moncrief was working in the more general case of a curved background “physical” metric, in addition to a curved “effective” metric. We shall come back to this work later on, in our historical section. (See also Section 4.1.2.)
- However, the geometry determined by the acoustic metric does inherit some key properties from the existence of the underlying flat physical metric. For instance, the topology of the manifold does not depend on the particular metric considered. The acoustic geometry inherits the underlying topology of the physical metric – ordinary – with possibly a few regions excised (due to whatever hard-wall boundary conditions one might wish to impose on the fluid). In systems constrained to have effectively less than 3 space-like dimensions one can reproduce more complicated topologies (consider, for example, an effectively one-dimensional flow in a tubular ring).
- Furthermore, the acoustic geometry automatically inherits from the underlying Newtonian time parameter, the important property of “stable causality” [275, 670]. Note that This precludes some of the more entertaining causality-related pathologies that sometimes arise in general relativity. (For a general discussion of causal pathologies in general relativity, see, for example, [275, 272, 273, 125, 274, 630]).
- Other concepts that translate immediately are those of “ergo-region”, “trapped surface”, “apparent horizon”, and “event horizon”. These notions will be developed more fully in the following subsection.
- The properly normalised four-velocity of the fluid is so that This four-velocity is related to the gradient of the natural time parameter by 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 and the integral curves are geodesics of the acoustic metric if and only if is position independent.
- Observe that in a completely general (3+1)-dimensional Lorentzian geometry the metric has 6 degrees
of freedom per point in spacetime. ( 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 , , 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.

- A point of notation: Where the general relativist uses the word “stationary” the fluid dynamicist uses the phrase “steady flow”. The general-relativistic 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 Einstein–Hilbert differential equations. In contrast, in the present context, the acoustic metric is related to the distribution of matter in a simple algebraic fashion.

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 vector

If the flow is steady, then this is the time translation Killing vector. Even if the flow is not steady the background Minkowski metric provides us with a natural definition of “at rest”. Then 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 [37].

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

The time translation Killing vector is simply , with The metric can also be written as

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 [245]. If you don’t like Euclidean signature techniques (which are in any
case only applicable to equilibrium situations) you should go back to the original Hawking
derivations [270, 271].^{8}

We should emphasize that the formula for the Hawking temperature contains both the surface gravity and the speed of sound at the horizon [624]. Specifically

In view of the explicit formula for above, this can also be written as which is closer to the original form provided by Unruh [607] (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 [624].)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

Here (and for the rest of this particular section) it is essential that we use the natural Newtonian time coordinate inherited from the background Newtonian physics of the fluid. In addition is a unit vector field that, at the horizon, is perpendicular to it, and away from the horizon is some suitable smooth extension. (For example, take the geodesic distance to the horizon and consider its gradient.) We only need this decomposition to hold in some open set encompassing the horizon and do not need to have a global decomposition of this type available. Furthermore, by definition we know that at the horizon. Now consider the vector field Since the spatial components of this vector field are by definition tangent to the horizon, the integral curves of this vector field will be generators for the horizon. Furthermore, the norm of this vector (in the acoustic metric) is In particular, on the acoustic horizon defines a null vector field, the integral curves of which are generators for the acoustic horizon. We shall now verify that these generators are geodesics, though the vector field is not normalised with an affine parameter, and in this way shall calculate the surface gravity.Consider the quantity and calculate

To calculate the first term note that Thus, And so: On the horizon, where , and additionally assuming so that the density is constant over the horizon, this simplifies tremendously Similarly, for the second term we have On the horizon this again simplifies There is partial cancellation between the two terms, and so while Comparing this with the standard definition of surface gravity [670] This is in agreement with the previous calculation for static acoustic black holes, and
insofar as there is overlap, is also consistent with results of Unruh [607, 608], Reznik [523],
and the results for “dirty black holes” [621]. From the construction it is clear that the
surface gravity is a measure of the extent to which the Newtonian time parameter inherited
from the underlying fluid dynamics fails to be an affine parameter for the null geodesics on the
horizon.^{10}

Again, the justification for going into so much detail on this specific model is that this style of argument can be viewed as a template – it will (with suitable modifications) easily generalise to more complicated analogue models.

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

In the tangential direction, the requirement that the flow be vorticity free (apart from a possible delta-function contribution at the vortex core) implies, via Stokes’ theorem, that(If these flow velocities are nonzero, then following the discussion of [641] there must be some external force present to set up and maintain the background flow. Fortunately it is easy to see that this external force affects only the background flow and does not influence the linearised fluctuations we are interested in.)

For the background velocity potential we must then have

Note that, as we have previously hinted, the velocity potential is not a true function (because it has a discontinuity on going through radians). The velocity potential must be interpreted as being defined patch-wise on overlapping regions surrounding the vortex core at . The velocity of the fluid flow isDropping a position-independent prefactor, the acoustic metric for a draining bathtub is explicitly given by

Equivalently A similar metric, restricted to (no radial flow), and generalised to an anisotropic speed of sound, has been exhibited by Volovik [646], that metric being a model for the acoustic geometry surrounding physical vortices in superfluidIn 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 [646], 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 [8].

The ergosphere (ergocircle) forms at

Note that the sign of is irrelevant in defining the ergosphere and ergo-region: It does not matter if the vortex core is a source or a sink.The acoustic event horizon forms once the radial component of the fluid velocity exceeds the speed of sound, that is, at

The sign of now makes a difference. For we are dealing with a future acoustic horizon (acoustic black hole), while for we are dealing with a past event horizon (acoustic white hole).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

That isIf 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 [148], and Corley [145].) 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 [150] or aspects of the physics of warpdrives [197].

In this situation one must again invoke an external force to set up and maintain the fluid flow. Since the conformal factor is regular at the event horizon, we know that the surface gravity and Hawking temperature are independent of this conformal factor [315]. In the general case it is important to realise that the flow can go supersonic for either of two reasons: The fluid could speed up, or the speed of sound could decrease. When it comes to calculating the “surface gravity” both of these effects will have to be taken into account.

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:

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 [315] 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 [37] for 1+1 geometries (viewed either as a dimensional reduction of a physical 3+1 system, or directly as geometrical acoustic metrics). The basic idea underlying the conformal diagram of any 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 [37] 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 [445].

Living Rev. Relativity 14, (2011), 3
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