Backreaction in Acoustic Black Holes

Abstract

The backreaction equations for the linearized quantum fluctuations in an acoustic black hole are given. The solution near the horizon, obtained within a dimensional reduction, indicates that acoustic black holes, unlike Schwarzschild ones, get cooler as they radiate phonons. They show remarkable analogies with near-extremal Reissner-Nordström black holes.PACS numbers: 04.62.+v, 04.70.Dy, 47.40.Ki One of the most surprising and far reaching result for its implications in modern theoretical physics is the prediction made by Hawking [1] that black holes emit thermal radiation at a temperature T H proportional to the surface gravity k of the horizon. For a Schwarzschild black hole of mass M , k = (4M ) −1 and T H = (8πM ) −1 (we have set the velocity of light and Boltzman constant equal to one). Hawking obtained this result using quantum field theory in curved space, a scheme for dealing with the matter-gravity system where matter is quantized according to quantum field theory whereas gravity is treated classically according to Einstein General Relativity. The scale at which this framework becomes unreliable is the Planck length where the description of spacetime as a continuous differentiable manifold probably breaks down. Coming back to black holes, because of the quantum emission, they are unstable. Extrapolating Hawking's result (which is strictly valid only for stationary or static black holes) one can conjecture that as the mass decreases, the hole gets hotter and hotter (being the temperature inversely proportional to the mass) and eventually disappears in a time scale of the order of the initial mass to the third power. A more quantitative analysis can be performed by looking at the first order (in ) corrections g (1) αβ to a classical black hole metric g (0) αβ induced by the quantum emission. These can be calculated using the semiclassical Einstein equations [2]Here G µν is the Einstein tensor evaluated for the quan- * Email addresses: balbinot@bo.infn.it, fagnocchi@bo.infn.it † Email address: fabbria@bo.infn.it ‡ Email address: gpp27@cam.ac.uk tum corrected metric g αβ = g (0) αβ + g (1) αβ and linearized in the perturbation g (1) αβ (of order ). The r.h.s. represents the expectation value of the stress tensor for the quantum matter field evaluated in the classical background g (0) αβ . In a very interesting paper, appeared in 1981, Unruh [3] showed that a thermal radiation similar to the one predicted by Hawking for black holes is expected in a completely (at first sight) different physical scenario, namely a fluid undergoing hypersonic motion. This opened the way for the study of condensed matter analogues of Hawking radiation [4], a rather promising field of research where the connections to the experimental side do not seem so remote, compared to gravity.The Eulerian equations of motion for an irrotational and homentropic fluid flow can be derived from the actionwhere ρ is the mass density, ψ the velocity potential, i.e. − → v = − → ∇ψ, u the internal energy density and a dot means time derivative....

Black-hole radiation in Bose-Einstein condensates

“…This is no longer necessarily true for acoustic black holes, because the dispersion limits the number of e-folding times during which Eq. (19) applies [43,44]. Thus, one should analyze each case to see if the vacuum state provides a reliable approximation.…”

Horizon in Random Matrix Theory, the Hawking Radiation, and Flow of Cold Atoms

“…Much less obvious [3] is the fact that the mere existence of a horizon kinematically implies the generation of Hawking radiation emanating from it [4]. Analogue gravitational models like this one are a topic of growing interest [5], since "table-top" experiments can be used to test predictions of quantum gravity and help in resolving some of its issues [11].…”

“…Unfortunately, direct experimental evidence of Hawking radiation is almost impossible (see, for instance, [3]). After the beautiful proposal by Unruh [4], there has been a growing interest in searching for similar phenomena in the so called analogous gravitational models [5].…”

Back-reaction in acoustic black holes

“…This behavior completely differs from what one finds in Schwarzschild black holes, where the back-reaction increases the temperature and the black hole lifetime is of order M −3 0 (M 0 is the initial mass of the hole). Acoustic black holes instead show a behavior similar to charged Reissner-Nordström black holes, whose temperature tends to zero as ∝ t −3 [13,14].…”

“…In the gravitational analogue equation is the flux which determines the mass correction. This is not surprising since one can not expect the very same back-reaction equations for the two settings, that, even if identical from the kinematical point of view, are completely different from the dynamical one [12,13,14].…”