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 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
Once we have a static geometry, we can of course directly apply all of the standard tricks  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 vector7 [376, 377, 378] since he implicitly took the speed of sound to be a position-independent constant. The fact that prefactor drops out of the final result for the surface gravity can be justified by appeal to the known conformal invariance of the surface gravity . Though derived in a totally different manner, this result is also compatible with the expression for “surface-gravity” obtained in the solid-state black holes of Reznik , wherein a position dependent (and singular) refractive index plays a role analogous to the acoustic metric. As a further consistency check, one can go to the spherically symmetric case and check that this reproduces the results for “dirty black holes” enunciated in . 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 [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 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.
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.
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 calculate 9 [376, 377, 378], 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 . (For a survey of the many analogies and similarities between the physics of superfluid and the Standard Electroweak Model see , 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  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 , but this approximation becomes progressively worse as the core is approached.)
The ergosphere 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 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 , and Corley . (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 ingoing () and outgoing () 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 - 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  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.
© Max Planck Society and the author(s)