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
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
This quantity changes sign when
. 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 [265
, 164
, 422
].
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 [164
, pages 319-323] or [422
, 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 [164
]. 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.)
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 [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
by
This is not quite Unruh’s result [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 [192
]. 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 [319
], 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 [386
].
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 [159
, 160
].
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
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
]
we finally have
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 [376
, 377
, 378
], Reznik [319
],
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.
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
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 is
Dropping 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).
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
direction and the velocity profile depends only on
.
The continuity equation then implies that
is a constant, and the acoustic metric becomes
That is
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 [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.
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.
In modern notation the Schwarzschild geometry in ingoing (
) and outgoing (
) Painlevé-Gullstrand
coordinates may be written as:
Equivalently
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
Or equivalently
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
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.
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.