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Encyclopedia > Acoustic horizon

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. Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... Acoustics is a branch of physics and is the study of sound (mechanical waves in gases, liquids, and solids). ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ...


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. General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

Contents

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. To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...


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.


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. General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... The Schwarzschild radius (sometimes inappropriately referred to as the gravitational radius[1]) is a characteristic radius associated with every mass. ...


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.

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. 2005 is a common year starting on Saturday of the Gregorian calendar. ... This article or section does not adequately cite its references or sources. ...


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. In physics, Hawking radiation is thermal radiation thought to be emitted by black holes due to quantum effects. ... In physics, a toy model is a simplified set of objects and equations relating them that can nevertheless be used to understand a mechanism that is also useful in the full, non-simplified theory. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...


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. In physics, decoupling is the general phenomenon in which the interactions between some physical objects (such as elementary particles) disappear. ...


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 Galileans (or Galilæans) were members of a fanatical sect (Zealots), followers of Judas of Galilee, who fiercely resented the taxation of the Romans, and whose violence contributed to induce the latter to vow the extermination of the whole race. ... Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... Helium II will creep along surfaces in order to find its own level - after a short while, the levels in the two containers will equalize. ... Fluid Dynamics Compressibility (physics) is a measure of the relative volume change of fluid or solid as a response to a pressure (or mean stress) change: . For a gas the magnitude of the compressibility depends strongly on whether the process is adiabatic or isothermal, while this difference is small in... Galilean invariance is a principle which states that the fundamental laws of physics are the same in all inertial (uniform-velocity) frames of reference. ...

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. In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...


The acoustic metric



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

g00 + 2g0iui + gijuiuj = 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.


See also

In physics, Hawking radiation is thermal radiation thought to be emitted by black holes due to quantum effects. ... In astrophysics, the Gravastar theory is a proposal by Emil Mottola and Pawel Mazur to replace the black hole. ... Acoustics is a branch of physics and is the study of sound (mechanical waves in gases, liquids, and solids). ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... Fig. ... This article or section does not adequately cite its references or sources. ...

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

  Results from FactBites:
 
Acoustic Hawking radiation - Wikipedia, the free encyclopedia (1818 words)
For an observer near r=2M, watching the particle's journey, the particle seems to originate in the vicinity of the horizon, and, for a common coordinate system for all hovering observers outside the horizon, the particle is described as originating at the horizon itself (or at least, at a Planck scale distance from the horizon).
Hawking suggests that since GR and QM seem to disagree about horizon radiation (fl hole information paradox), that QM should be altered to remove the conflict.
The apparent generality of Hawking radiation effects prompts John Preskill, to say that this would face us with "the daunting task of finding a new conceptual basis for all of physics" (1992), and to draw a historical comparison with the ultraviolet catastrophe as to the theoretical importance of the situation.
  More results at FactBites »

 
 

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