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I am confused by most discussions of analog Hawking radiation in fluids (see, for example, the recent experimental result of Weinfurtner et al. Phys. Rev. Lett. 106, 021302 (2011), arXiv:1008.1911). The starting point of these discussions is the observation that the equation of motion for fluctuations around stationary solutions of the Euler equation have the same mathematical structure as the wave equation in curved space (there is a fluid metric $g_{ij}$ determined by the background flow). This background metric can have sonic horizons. The sonic horizons can be characterized by an associated surface gravity $\kappa$, and analog Hawking temperature $T_H \sim \kappa\hbar/c_s$.

My main questions is this: Why would $T_H$ be relevant? Corrections to the Euler flow are not determined by quantizing small oscillations around the classical flow. Instead, hydrodynamics is an effective theory, and corrections arise from higher order terms in the derivative expansion (the Navier-Stokes, Burnett, super-Burnett terms), and from thermal fluctuations. Thermal fluctuations are governed by a linearized hydro theory with Langevin forces, but the strength of the noise terms is governed by the physical temperature, not by Planck's constant.

A practical question is: In practice $T_H$ is very small (because it is proportional to $\hbar$). How can you claim to measure thermal radiation at a temperature $T_H << T$?

Qmechanic
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    Great question! –  Jan 06 '12 at 22:08
  • @Thomas, I went ahead and merged both accounts, to avoid confusion, hope that is OK with you. –  Jan 06 '12 at 23:12
  • Just a few remarks. Firstly, GR is also an effective theory, with higher order corrections in the string length expansion. Secondly, I also don't understand how quantum effects can be measured in such a setting. Maybe it's possible for very low temperatures but then we need something like liquid Helium –  Jan 06 '12 at 23:16
  • FWIF John Baez had a blog post over at Azimuth about this topic: http://johncarlosbaez.wordpress.com/2011/11/28/liquid-light – Tim van Beek Jan 07 '12 at 14:40
  • @Tim. I don't think John's post on liquid light is related to Unruh's dumb hole. – Demian Cho Jan 07 '12 at 17:08
  • John indeed mentions a possible analogue of Hawking radiation in polariton liquid. However in this case it might actually make sense since the polariton liquid is quantum (i.e. the wavelength isn't small w.r.t. the mean free path) –  Jan 07 '12 at 19:49
  • Indeed, I refer to the Weinfurtner et al paper because I find this particularly puzzling (they have a water tank). But I am confused even in the case of quantum fluids: 1) Ordinary (non superfluid) quantum fluids are described by standard hydro, the only thing that is quantum is the microscopic kinetic description. 2) In superfluids, it is indeed the case that the normal components can be described (in some limit) as a gas of quantized sound waves (phonons). But even in this situation you would think that the main effect is ordinary thermal fluctuations at temperature T. –  Jan 09 '12 at 04:04
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    @Thomas: I used to be puzzled by this a bit when I was doing stuff on it. I think I eventually convinced myself that it's because the thermal state you get is actually "classical", in that e.g. as a Wigner distribution it's always positive. That turns out to be the weird thing about Unruh/Hawking radiation --- the form is actually not very quantum at all. – genneth Jan 10 '12 at 13:28
  • Maybe, but the temperature is proportional to $\hbar$, so I would call it quantum. –  Jan 10 '12 at 15:35
  • Doesn't matter what you call it --- it's still perfectly described as classical... – genneth Jan 10 '12 at 18:14
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    If you have radiation at a rate proportional to $\hbar$, how can it not be a quantum effect? –  Jan 12 '12 at 04:09
  • Thomas I think this effect should be noticeable when the external temperature is not very high with respect to the Hawking temperature, that is, the external temperature has to be very low. This is why I don't think it makes sense for water which cannot remain liquid at low temperatures –  Jan 14 '12 at 19:09
  • The sonic analogue of Unruh effect is fully classical. You can read more about it in G.E. Volovik's book, "The universe in a helium droplet". The $\hbar$ you mention is merely metaphorical. – Javier Rodriguez Laguna Jan 15 '12 at 15:31
  • @Thomas : Stefan's constant has some $\hbar$ in it, so any ember in your fireplace glows throug a quantum effect, according to your reasoning. – Frédéric Grosshans May 23 '12 at 10:07
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    @FrédéricGrosshans: and, of course, blackbody radiation was how quantum mechanics was originally discovered in the first place. – Zo the Relativist May 23 '12 at 14:07
  • We are moving away from the initial question ... Regarding black body radiation Jeremy is of course correct in stating that this is a quantum effect (the classical calculation of the energy density of thermal EM radiation is UV divergent, and this divergence is regularized in QM). There is an interesting question whether there is an analogous divergence in fluid dynamics. I'm not completely sure, since high momentum sound waves are strongly damped. – Thomas May 24 '12 at 01:14
  • The important point is this: 1) The dominant contribution to the energy are the kinetic and internal energy, as well as gradient corrections (dissipative terms), 2) Fluctuations are a higher order effect, and in general UV (cutoff) sensitive. This is expected for an effective field theory. 3) The UV completion is not a quantum theory of sound waves, but a kinetic theory (classical or quantum, depending on the fluid) of the microscopic degrees of freedom (typically atoms). – Thomas May 24 '12 at 01:14
  • The analogy is almost ridiculous, like comparing the moon to a supernova. Although it may seem a little too simple, a 2019 alternative to HR is suggested at https://arxiv.org/pdf/1910.10819.pdf . – Edouard Feb 18 '20 at 19:22

2 Answers2

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The conditions for the existence of the Hawking effect are described in classical terms, i.e you need

1) A Lorentz signature metric

2) A horizon (given, for example, by space flowing into a BH faster than the speed of light, or fluid flowing downstream faster than the speed of sound)

3) Surface gravity at the horizon

Those conditions are then applied to a quantum field which satisfies a wave equation (e.g. Klein Gordon field on spacetime). The standard analysis proceeds by treating photon paths as null geodesics (eikonal approximation). The acoustic analogue of this is that sound waves (the quantum excitations of which are phonons) follow null geodesics in the Lorentzian acoustic metric. So my guess is that the reason for the presence of Planck's constant in the Hawking temperature expression in the acoustic case is that they're treating Hawking radiation in the phonon (quantum) field.

Indeed, Visser says

an acoustic event horizon will emit Hawking radiation in the form of a thermal bath of phonons at a temperature

$$kT_H=\frac{\hbar g_H}{2\pi c} $$

Here $g_H$ is the acoustic surface gravity and $c$ is the speed of sound

twistor59
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  • Can you please suggest a reference for the three conditions you have stated above? Thank you. – Dee May 11 '15 at 14:34
  • I realized that you already gave a reference. Sorry and thanks again. – Dee May 11 '15 at 14:35
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I remember attending a seminar by Unruh a few months ago and the same question arised. As far as I remember, he enfasized that in these hydrodynamic analogs of black holes, the flow is not quantized, it is a classical fluid, and everything is classical and that the dumb hole behaves like a quantum amplifier emitting quantum noise from the Horizon. Calculations indicate that the spectrum of these outgoing phonons must be "thermal" (like in black hole case) and a small temperature can be associated to it. I would not call this temperature "quantum", even if it is proportional to $\hbar$ since it can be completely determined from classical attributes of the fluid (speed of flow, adiabatic properties of the fluid, etc) which can be measured classically. The appearance of $\hbar$ is supposed to be a consequence of quantum amplification of the "phonon field" near the Horizon.

Looking at the experimental paper I don't see anywhere a direct measurement of temperature. They don't stick a thermometer in the water stream (for obvious reasons). They just verify that the dumb hole seems to have a "thermal character" while acting on surface waves. They just measure the amplitude of ingoing and outgoing waves and find that their squared ratios follows the Boltzmann distribution in the frequency which agrees with Hawking radiation.

Dar
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