# Analog Hawking radiation

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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 $T_H << T$?

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Great question!

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@Thomas, I went ahead and merged both accounts, to avoid confusion, hope that is OK with you.

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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

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@Tim. I don't think John's post on liquid light is related to Unruh's dumb hole.

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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)

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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.

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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.

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Maybe, but the temperature is proportional to $\hbar$, so I would call it quantum.

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Doesn't matter what you call it --- it's still perfectly described as classical...

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If you have radiation at a rate proportional to $\hbar$, how can it not be a quantum effect?

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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

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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.

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