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2.5 Dumb holes - ergo-regions, horizons, and surface gravity

Let us start with the notion of an ergo-region: Consider integral curves of the vector
Km =_ (@/@t)m = (1,0,0,0)m. (43)
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”. Then5
g (@/@t)m (@/@t)n = g = - [c2- v2]. (44) mn tt
This quantity changes sign when ||v ||> c. Thus any region of supersonic flow is an ergo-region. (And the boundary of the ergo-region may be deemed to be the ergo-surface.) The analogue of this behaviour in general relativity is the ergosphere surrounding any spinning black hole - it is a region where space “moves” with superluminal velocity relative to the fixed stars [265Jump To The Next Citation Point164Jump To The Next Citation Point422Jump To The Next Citation Point]. 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 [164Jump To The Next Citation Point, pages 319-323] or [422Jump To The Next Citation Point, pages 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 on pages 262-263 of Hawking and Ellis [164Jump To The Next Citation Point]. 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
View Image

Figure 5: A moving fluid can form “trapped surfaces” when supersonic flow tips the sound cones past the vertical.
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 [13Jump To The Next Citation Point].

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

2 r-[ 2 2 2] ds = c -c dt + (dx - v dt) . (45)
The time translation Killing vector is simply m K = (1;0), with
2 m n 2 r- 2 2 K =_ gmnK K =_ - ||K || = - c [c - v ]. (46)
The metric can also be written as
2 r[ 2 2 2 2] ds = c -(c - v ) dt - 2v .dx dt + (dx ) . (47)
Now suppose that the vector 2 2 v/(c - v ) is integrable, then we can define a new time coordinate by
-v-.dx- dt = dt + c2 - v2. (48)
Substituting this back into the acoustic line element gives
[ { i j } ] ds2 = r-- (c2- v2) dt 2 + dij + -v-v--- dxi dxj . (49) c c2- v2
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
{ } ---v----- \~/ × (c2- v2) = 0, (50)
that is, (since in deriving the existence of the effective metric we have already assumed the fluid to be irrotational),
2 2 v × \~/ (c - v ) = 0. (51)
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, 1 \~/ v2 2 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

K K VFIDO =_ ----- = V~ --------------. (52) ||K || (r/c) [c2- v2]
The four-acceleration of the FIDOS is defined as
AFIDO =_ (VFIDO . \~/ )VFIDO, (53)
and using the fact that K is a Killing vector, it may be computed in the standard manner
1 \~/ ||K ||2 AFIDO = + ---------. (54) 2 ||K ||2
That is
1 [\ ~/ (c2 - v2) \~/ (r/c) ] AFIDO = -- -----------+ -------- . (55) 2 (c2 - v2) (r/c)
The surface gravity is now defined by taking the norm ||A || FIDO, multiplying by the lapse function, V~ -------2----2- ||K ||= (r/c) [c - v ], and taking the limit as one approaches the horizon: |v|--> c (remember that we are currently dealing with the static case). The net result is
||A ||||K ||= 1v . \~/ (c2 - v2) + O(c2 - v2), (56) FIDO 2
so that the surface gravity is given in terms of a normal derivative by7
2 2 g = 1-@(c----v-) = c @(c---v). (57) H 2 @n @n
This is not quite Unruh’s result [376Jump To The Next Citation Point377Jump To The Next Citation Point378Jump To The Next Citation Point] since he implicitly took the speed of sound to be a position-independent constant. The fact that prefactor r/c drops out of the final result for the surface gravity can be justified by appeal to the known conformal invariance of the surface gravity [192Jump To The Next Citation Point]. 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 [319Jump To The Next Citation Point], 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 [386Jump To The Next Citation Point]. 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 [147]. 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 [159Jump To The Next Citation Point160Jump To The Next Citation Point].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

v = v _L + v||; where v _L = v _L ^n. (58)
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 ^n 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 v _L = c at the horizon. Now consider the vector field
m i L = (1; v|| ). (59)
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
2 r [ 2 2 ] r 2 2 ||L || = - c- - (c - v ) - 2v|| .v + v|| .v|| = c-(c - v _L ). (60)
In particular, on the acoustic horizon Lm 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 L is not normalised with an affine parameter, and in this way shall calculate the surface gravity.

Consider the quantity (L . \~/ )L and calculate

La\ ~/ aLm = La(\ ~/ aLb - \~/ bLa)gbm + 1 \~/ b(L2)gbm. (61) 2
To calculate the first term note that
r Lm = --(-[c2- v2 _L ];v _L ). (62) c
Thus
|_ .. [r- 2 2 ] _| 0 . - \~/ i c(c - v _L ) L = - ............ . ...... . (63) [a,b] |_ +\ ~/ [r(c2- v2 )]... (r v _L ) _| j c _L c [i,j]
And so:
( ) a [r 2 2 ] [r 2 2 ] i(r _L ) L L[b,a] = v|| . \~/ -(c - v _L ) ; \~/ j -(c - v _L ) + v|| --v . (64) c c c [j,i]
On the horizon, where c = v _L, this simplifies tremendously
a r- ( 2 2 ) r- @(c2---v2 _L )- (L L[b,a])|horizon = - c 0; \~/ j(c - v _L ) = - c @n (0;^nj) . (65)
Similarly, for the second term we have
( [ ]) \~/ (L2) = 0; \~/ r-(c2 - v2 ) . (66) b j c _L
On the horizon this again simplifies
2 r ( 2 2 ) r @(c2- v2 _L ) \~/ b(L )|horizon = + -- 0; \~/ j(c - v _L ) = + -- -----------(0;^nj). (67) c c @n
There is partial cancellation between the two terms, and so
2 2 (La\ ~/ aLm)horizon = + 1-r-@(c---v _L )-(0;^nj) , (68) 2 c @n
while
(L ) = r-(0;c ^n ). (69) m horizon c j
Comparing this with the standard definition of surface gravity [422Jump To The Next Citation Point] 9
gH (La\ ~/ aLm)horizon = +---(Lm)horizon, (70) c
we finally have
1-@(c2---v2 _L )- @(c---v _L ) gH = 2 @n = c @n . (71)
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 [376Jump To The Next Citation Point377Jump To The Next Citation Point378Jump To The Next Citation Point], Reznik [319Jump To The Next Citation Point], and the results for “dirty black holes” [386]. 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.

2.5.1 Example: vortex geometry

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 r is a position-independent constant throughout the flow (which automatically implies that the background pressure p and speed of sound c 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

1 v^r oc -. (72) r
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
v^t oc 1-. (73) r

(If these flow velocities are nonzero, then following the discussion of  [401Jump To The Next Citation Point] 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

f(r,h) = - A ln(r/a) - B h. (74)
Note that, as we have previously hinted, the velocity potential is not a true function (because it has a discontinuity on going through 2p radians). The velocity potential must be interpreted as being defined patch-wise on overlapping regions surrounding the vortex core at r = 0. The velocity of the fluid flow is
(A ^r + B ^h) v = - \~/ f = ------------. (75) r

Dropping a position-independent prefactor, the acoustic metric for a draining bathtub is explicitly given by

( )2 ( )2 ds2 = - c2 dt2 + dr - A-dt + rdh - B-dt + dz2. (76) r r
Equivalently
( A2 + B2 ) A ds2 = - c2- -----2--- dt2- 2--dr dt- 2B dh dt + dr2 + r2dh2 + dz2. (77) r r
A similar metric, restricted to A=0 (no radial flow), and generalised to an anisotropic speed of sound, has been exhibited by Volovik [404Jump To The Next Citation Point], that metric being a model for the acoustic geometry surrounding physical vortices in superfluid 3 He. (For a survey of the many analogies and similarities between the physics of superfluid 3 He and the Standard Electroweak Model see [420], this reference is also useful as background to understanding the Lorentzian geometric aspects of 3He 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]
ds2 = - c2(dt- ~A dh)2 + dr2 + (1 - B~)r2dh2 + dz2. (78)
View Image

Figure 6: A collapsing vortex geometry (draining bathtub): The green spirals denote streamlines of the fluid flow. The outer circle represents the ergo-surface while the inner circle represents the [outer] event horizon.
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

V~ -------- A2 + B2 rergosphere = ----------. (79) c
Note that the sign of A 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

|A| rhorizon = ---. (80) c
The sign of A now makes a difference. For A < 0 we are dealing with a future acoustic horizon (acoustic black hole), while for A > 0 we are dealing with a past event horizon (acoustic white hole).

2.5.2 Example: slab geometry

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 z direction and the velocity profile depends only on z. The continuity equation then implies that r(z) v(z) is a constant, and the acoustic metric becomes

ds2 oc ----1----[-c(z)2 dt2 + {dz - v(z) dt}2 + dx2 + dy2]. (81) v(z) c(z)
That is
1 [ { } ] ds2 oc --------- - c(z)2 - v(z)2 dt2- 2v(z) dz dt + dx2 + dy2 + dz2 . (82) v(z) c(z)

If we set c = 1 and ignore the conformal factor we have the toy model acoustic geometry discussed by Unruh [378Jump To The Next Citation Point, page 2828, equation (8)], Jacobson [188Jump To The Next Citation Point, page 7085, equation (4)], Corley and Jacobson [88Jump To The Next Citation Point], 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 [192Jump To The Next Citation Point].) 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.

2.5.3 Example: Painlevé-Gullstrand geometry

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:

( V~ ------ )2 2 2 2GM 2( 2 2 2) ds = - dt + dr ± -----dt + r dh + sin h df . (83) r
Equivalently
V~ ------ ( 2GM ) 2GM ( ) ds2 = - 1 - ------ dt2± ------dr dt + dr2 + r2 dh2 + sin2 h df2 . (84) r r
This representation of the Schwarzschild geometry was not (until the advent of the analogue models) particularly well-known, and it has been independently rediscovered several times during the 20th century. See for instance Painlevé [293], Gullstrand [154], Lemaître [228], the related discussion by Israel [183], and more recently, the paper by Kraus and Wilczek [218]. The Painlevé-Gullstrand coordinates are related to the more usual Schwarzschild coordinates by
[ (V ~ ------) ] 2GM V~ ------- tPG = tS ± 4M arctanh --r--- - 2 2GM r . (85)
Or equivalently
V~ -------- 2GM/r dtPG = dtS ± ------------dr. (86) 1 - 2GM/r
With these explicit forms in hand, it becomes an easy exercise to check the equivalence between the Painlevé-Gullstrand line element and the more usual Schwarzschild form of the line element. It should be noted that the + sign corresponds to a coordinate patch that covers the usual asymptotic region plus the region containing the future singularity of the maximally extended Schwarzschild spacetime. It thus covers the future horizon and the black hole singularity. On the other hand the - sign corresponds to a coordinate patch that covers the usual asymptotic region plus the region containing the past singularity. It thus covers the past horizon and the white hole singularity.

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 r and c to be constants, and set V~ -------- v = 2GM/r? 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 \~/ .(rv) /= 0 that was used in deriving the acoustic equations.

The best we can actually do is this: Pick the speed of sound c to be a position independent constant, which we normalise to unity (c = 1). Now set v = V~ 2GM/r---, and use the continuity equation \~/ .(rv) = 0 to deduce 2 r| v | oc 1/r so that -3/2 r oc r. Since the speed of sound is taken to be constant we can integrate the relation c2 = dp/dr to deduce the equation of state must be p = p oo + c2r and that the background pressure satisfies p - p oo oc c2r-3/2. Overall the acoustic metric is now

|_ _| ( V~ ------ )2 ( ) ds2 oc r-3/2 |_ - dt2 + dr ± 2GM---dt + r2 dh2 + sin2 h df2 _| . (87) r

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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