A graduate student summary of the public lecture
Black Holes/Dumb Holes: Condensed Matter Analogues
by Bill Unruh
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In this talk Bill Unruh draws analogies between black holes (singularities in space-time) and a fluid-mechanical model of "dumb holes." It is shown that the horizon of a black-hole is analogous to the point in a fluid system where the background fluid flow exceeds the velocity of sound (and thus sound waves travelling against the flow cannot propagate across this boundary).
Unruh begins by outlining a problem with Stephen Hawking's famous derivation of the thermal radiation produced near a black hole's event horizon. If the modes of the emitted radiation are naively propagated backwards in time, however, they blue-shift to absurdly large frequencies where a single photon would have more energy than the mass of the universe.
Despite this problem the Hawking radiation seems to give much insight into black hole physics, in particular by providing a framework for black hole thermodynamics. The thermodynamical justification for Hawking radiation is so strong that Unruh feels that "this thermodynamic analogy makes one believe that the result is right even though the derivation is nonsense."
To get a better sense of the physics, Unruh has examined a fluid mechanical model of "dumb holes." The fluid mechanical action for the system can be rewritten to show the mathematical similarity between sound waves in a flowing fluid and scalar fields on a background space-time. The terms in the metric depend on the fluid velocity such that when the radial velocity of the fluid is equal to the speed of sound there is an "horizon" analogous to the black hole event horizon.
Unruh outlines three uses of this mathematical similarity. The first is that many calculations made by general relativists may be applied to problems in fluid dynamics. For example, the idea of considering a flowing fluid as a bi-metric theory in which both the background fluid flow and the ordinary space-time metric have associated stress-energy tensors and conservation laws has provided a new framework for discussing certain aspects of these systems.
One may also obtain insight into black holes through fluid experiments. Since surface waves in an incompressible fluid look like scalar fields on a metric, we can do "black hole physics in a bath-tub." The effective metric is controlled by the variation in the depth of the tub, while a drain in the system provides a net inward radial flow. Analogues of wave scattering and the Penrose process (where reflected waves have more energy than the incident waves) can be observed.
One can even obtain insight into quantum mechanical properties of the black hole by considering phonons in the fluid. Hawking's method can be applied to the fluid system to show that there is radiation from the hole, and that it has thermal spectrum as in the black hole case. This would give the same exponential blue-shift problem, except that we know that in acoustic systems the dispersion relation is linear only at low frequencies, and may have a variety of behaviours at high frequencies.
Unruh then goes on to investigate the behaviour of a simple system in which the group velocity of sound waves falls of at high frequency. A wave packet scattering from the black hole will experience a blue-shift wherein the higher frequency modes are swept over the horizon (since they have lower group velocity) while the lower frequency modes escape and form the thermal radiation. The analysis shows that, despite this modified dispersion relation, the radiation still has a thermal spectrum. So the theory reproduces the thermal radiation spectrum of black holes without the extreme photon energies.
Furthermore, by examining the spectrum of system with a dispersion relation with two characteristic temperatures (i.e. a dispersion that is linear at low frequencies, and linear at higher frequencies, but with different group velocities in the two regimes) Unruh claims that the thermal radiation is in fact determined by the low energy behaviour of the fields. The result is that Planck scale physics does not seem to determine the characteristics of the thermal radiation (though it certainly essential to other aspects of black hole physics).
Finally, Unruh comments on the possibility of performing experiments on fluid systems to observe the acoustic analogue of Hawking radiation. This would require measuring power on the order of ~10^-16 Watts in liquid helium, or ~10^-33 Watts in a Bose-Einstein condensate. There is some hope that these miniscule powers might be accessible to experimenters.
Unruh began the talk by claiming that "theoretical physics is puns," that theoretical physicists use the same mathematics to describe a variety of different situations. In the question period, Unruh warned of taking this particular analogy too seriously. We should not, for example, reintroduce the ether in order to provide a moving fluid in which light waves propagate like sound waves. "Some parts of the analogy are good and some parts are just awful."