Acoustic
Metric
In
mathematical physics,
a metric
describes the arrangement of relative distances within a surface or
volume, usually measured by signals passing through the region. The
metric ("means of measurement") –
essentially describes the region's intrinsic geometry. An acoustic
metric will describe the signalcarrying properties
characteristic of a given particulate medium in acoustics,
or in fluid dynamics. Other descriptive
names such as sonic metric are also
sometimes used, interchangeably.
Since "acoustic"
behaviour is intuitively familiar from everyday
experience, many complex "acoustic" effects can be confidently
described without recourse to advanced mathematics. The rest of this
article contrasts the "everyday" properties of an acoustic metric with
the more intensely studied and betterdocumented "gravitational"
behaviour of general relativity.
Unusual properties of
an acoustic metric
Unlike
some other metrics, acoustic metrics can seem to show some very nonlinear
behaviour: where special relativity's Minkowski metric
is fixed and unchanging, and general relativity's
metric is more flexible (Wheeler: "spacetime tells
matter how to move,
matter tells spacetime how to bend"), acoustic
metrics take this a
stage further: in the most familiar example of an acoustic metric, the
behaviour of sound in air, the motion of a sound wavefront through a
region moves air, creating local variations and offset sin the average
speed of air molecules along the signal path, which in turn modifies
the local speed of sound at different points along that path. The
passage of a signal through an acoustic metric itself modifies the
metric and the notional speeds at which signals are transmitted.
This
can lead to definitional problems: we can't always start with
a clearlydefined acoustic metric, introduce a signal, and then assume
that the initial definitions will still be valid.
Acoustic
horizons
Under general relativity, absolute
gravitational horizons are sharply defined (at r=2M
for a spherical black hole), and once defined, this limit in the Schwarzchild
metric
is inviolable: events enclosed by the event horizon of a black hole
cannot modify the external properties of the object, because this would
require signals to move outward through the horizon, which is forbidden.
With
an acoustic horizon (a.k.a. "sonic
horizon"), this ordered set of definitions breaks down:
events behind an acoustic horizon can
modify the effective horizon position and allow information to escape
from a horizonbounded region. This results in acoustic horizons
following a different set of rules to gravitational horizons under
general relativity:
 Acoustic
horizons fluctuate and radiate. This effect is referred to as
acoustic Hawking radiation,
or sonic
Hawking radiation.
 Acoustic
horizons can be incomplete. If a jet aircraft is
stationary on a runway and firing its engines, a particle in the
supersonic exhaust stream cannot directly send signals "upstream" back
to the jet engine (except by weak indirect transmission). The particle
can be said to be separated from the engine by an acoustic horizon, and
from the particle's point of view, the engine is not directly
contactable due to the nominal existence of an antihorizon
surface intersecting the jet exhaust. However, the particle can
legally send a signal sideways out of the jetstream, and this signal
can then travel subsonically through the surrounding air to reach the
engine. The acoustic horizon does not completely enclose the particle,
and can be circumvented – the existence of an event horizon between two
points can said to be routedependent.
 Acoustic
horizons are "fuzzy". The precise position of a
nominal acoustic horizon surface can be difficult to locate at smaller
scales, since the process of measuring a horizon by probing it with
smallerwavelength signals itself alters the properties that we are
trying to measure. This property of "fuzziness" allows an incomplete
horizon surface to "peter out" gracefully at its limits without sharp
geometrical singularities or edges.
Acoustic metrics and
quantum mechanics
Although the underlying shape of
spacetime in an
acoustic metric is complete and continuous, if we project an acoustic
metric onto a more conventional observerspace
metric, parts of the
surface can be concealed behind curvature horizons, leading to an
apparent, projected surface that is discontinuous and incomplete.
These
projections can result in apparent acausalities
and apparent instances of reverse causality
... but these are artefacts of the projection method ... the underlying
physics still obeys the conventional rules of
causality. This behaviour is reminiscent of the "Hidden
Variable Interpretation" of quantum mechanics,
where smooth, classical mechanisms are assumed to underlie
apparently discontinuous quantum effects.
Acoustic metrics and
quantum gravity
As of 2005, work towards obtaining a theory
of quantum gravity
is still being complicated by the lack of a solid understanding of the
exact rules and principles that such a theory ought to follow.
Since
acoustic metrics share some statistical behaviours with the
way that we expect a future theory of quantum gravity to behave (such
as Hawking radiation), these metrics
are increasingly being used as intuitive toy models
for exploring aspects of statistical mechanics,
in a safer and more familiar context than quantum mechanics usually
allows. The use of "acoustic" effects as "analogs"
(/"analogues") of effects in advanced
gravitational physics has led to a number or
research papers whose titles refer to "analog", "analogue"
or
"analogous" Hawking radiation, horizons, and
gravitation.
References
 W.G. Unruh, "Experimental
black hole evaporation" Phys. Rev. Lett. 46
(1981), 1351–1353
– considers information
leakage through a transsonic horizon as an "analogue" of Hawking
radiation in black hole problems
 Matt
Visser "Acoustic black holes: Horizons, ergospheres,
and Hawking radiation" Class. Quant. Grav. 15
(1998), 1767–1791 grqc/9712010
– indirect radiation
effects in the physics of acoustic horizon explored as a case of
Hawking radiation
 Carlos
Barceló, Stefano Liberati, and Matt Visser, "Analogue
Gravity" grqc/0505065
–
huge review article of "toy models" of gravitation, 2005, currently on
v2, 152 pages, 435 references, alphabetical by author.
 M.
Novello, Matt Visser and G. E. Volovik, Artificial
Black Holes (2002)
 Kip
S Thorne, Richard H Price and Douglas A Macdonald
(eds.) Black Holes: The membrane paradigm
(1986)
 Eric
Baird, Relativity in Curved Spacetime
(2007), including Sections 9 "Moving bodies drag light",
11.6 "Dark stars and acoustic metrics",
11.7 "Acoustic metrics and nonlinearity", 11.16,
"Acoustic metrics, once again",
12.11 "Do cosmological horizons count as "acoustic"?",
19.9 "The "acoustics" analogue"
