Acoustic metric

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In mathematical physics, a metric describes the arrangement of relative distances within a surface or volume, usually measured by signals passing through the region – essentially describing the intrinsic geometry of the region. An acoustic metric will describe the signal-carrying 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 better-documented "gravitational" behaviour of general relativity.

Contents 1 Unusual properties of an acoustic metric 2 Acoustic horizons 3 Acoustic metrics and quantum gravity 4 A simple fluid example 5 See also 6 References

[edit] 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 offsets in 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: one cannot always start with a clearly-defined acoustic metric, introduce a signal, and then assume that the initial definitions will still be valid.

[edit] 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 (also known as "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 horizon-bounded 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 route dependent.

• 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 smaller-wavelength 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.

[edit] 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 "analog(ue)s" 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.

Some people have suggested that analog models are more than just an analogy and that the actual gravity that we observe is actually an analog theory. But in order for this to hold, since a generic analog model depends upon BOTH the acoustic metric AND the underlying background geometry, the low energy large wavelength limit of the theory has to decouple from the background geometry.

[edit] A simple fluid example

For simplicity, we will assume that the underlying background geometry is Galilean. This is absolutely unnecessary in analog models in general (even isotropy is unnecessary) but this is only an expository toy example and Galilean symmetry will simplify some of the results. We will also assume that this Galilean spacetime is filled with an isotropic inviscid fluid at zero temperature (e.g. a superfluid). This fluid is described by a density field ρ and a velocity field . The speed of sound at any given point depends upon the compressibility which in turn depends upon the density at that point. This can be specified by the "speed of sound field" c. Now, the combination of both isotropy and Galilean covariance tells us that the permissible velocities of the sound waves at a given point x, has to satisfy

This restriction can also arise if we imagine that sound is like "light" moving though a spacetime described by an effective metric tensor called the acoustic metric.

The acoustic metric

"Light" moving with a velocity of (NOT the 4-velocity) has to satisfy

g₀₀ + 2g_0iuⁱ + g_ijuⁱu^j = 0

If

where α is some conformal factor which is yet to be determined (see Weyl rescaling), we get the desired velocity restriction. α may be some function of the density, for example.

[edit] See also

• Analog models of gravity
• Hawking radiation
• gravastar
• acoustics
• metric (mathematics)
• quantum mechanics
• quantum gravity

[edit] 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. gr-qc/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" gr-qc/0505065

– huge review article of "toy models" of gravitation, 2005, currently on v2, 152 pages, 435 references, alphabetical by author.

• Acoustic black holes on arxiv.org