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 [13].

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 timelike 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 Now suppose that the vector is integrable, then we can define a new time coordinate by Substituting this back into the acoustic line element gives In this coordinate system the absence of the time-space cross-terms makes manifest that the acoustic geometry is in fact static (there exists a family of spacelike hypersurfaces orthogonal to the timelike Killing vector). The condition that an acoustic geometry be static, rather than merely stationary, is thus seen to be that is, (since in deriving the existence of the effective metric we have already assumed the fluid to be irrotational), This requires the fluid flow to be parallel to another vector that is not quite the acceleration but is closely related to it. (Note that, because of the vorticity free assumption, is just the three-acceleration of the fluid, it is the occurrence of a possibly position dependent speed of sound that complicates the above.)Once we have a static geometry, we can of course directly apply all of the standard tricks [372] for calculating the surface gravity developed in general relativity. We set up a system of fiducial observers (FIDOS) by properly normalizing the time-translation Killing vector

The four-acceleration of the FIDOS is defined as and using the fact that is a Killing vector, it may be computed in the standard manner That is The surface gravity is now defined by taking the norm , multiplying by the lapse function, , and taking the limit as one approaches the horizon: (remember that we are currently dealing with the static case). The net result is so that the surface gravity is given in terms of a normal derivative byIf 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.

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 , 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 [422]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 [401] 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 A=0 (no radial flow), and generalised to an anisotropic speed of sound, has been exhibited by Volovik [404], that metric being a model for the acoustic geometry surrounding physical vortices in superfluid . (For a survey of the many analogies and similarities between the physics of superfluid and the Standard Electroweak Model see [420], this reference is also useful as background to understanding the Lorentzian geometric aspects of fluid flow.) Note that the metric given above is not identical to the metric of a spinning cosmic string, which would instead take the form [388] 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.)The ergosphere 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 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.

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 ingoing () and outgoing () 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 - 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 to deduce so that . Since the speed of sound is taken to be constant we can integrate the relation to deduce 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 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
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.

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