Hawking radiation from acoustic black holes, short distance and backreaction effects
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hic,a and R. Parentanid
arXiv:gr-qc/0601079v1 19 Jan 2006
a) Dipartimento di Fisi a dell'Università di Bologna and INFN sezione di Bologna, via Irnerio, 46 - 40126 Bologna. E-mail: balbinotbo.infn.it b) Departamento de Físi a Teóri a and IFIC, Fa ultad de Físi a, Universidad de Valen ia, 46100 Burjassot-Valen ia (Spain). E-mail: afabbrii .uv.es
) Centro Enri o Fermi, Compendio Viminale, 00184 Roma (Italy). E-mail: fagno
hibo.infn.it d) Laboratoire de Physique Théorique, CNRS UMR 8627, Bâtiment 210, Université Paris XI, 91405 Orsay Cedex (Fran e). E-mail: renaud.parentanith.u-psud.fr
Abstra t Using the a tion prin iple we rst review how linear density perturbations (sound waves) in an Eulerian uid obey a relativisti equation: the d'Alembert equation. This analogy between propagation of sound and that of a massless s alar eld in a Lorentzian metri also applies to non-homogeneous ows. In these ases, sound waves ee tively propagate in a urved four-dimensional a ousti metri whose properties are determined by the ow.
Using this analogy, we onsider regular
ows whi h be ome supersoni , and show that the a ousti metri behaves like that of a bla k hole.
The analogy is so good that, when
onsidering quantum me hani s, a ousti bla k holes should produ e a thermal ux of Hawking phonons. We then fo us on two interesting questions related to Hawking radiation whi h are not fully understood in the ontext of gravitational bla k holes due to the la k of a theory of quantum gravity. The rst on erns the al ulation of the modi ations of Hawking radiation whi h are indu ed by dispersive ee ts at short distan es, i.e., approa hing the atomi s ale when onsidering sound.
We general-
ize existing treatments and al ulate the modi ations aused by the propagation near the bla k hole horizon. The se ond question on erns ba krea tion ee ts. We return to the Eulerian a tion, ompute se ond order ee ts, and show that the ba krea tion of sound waves on the uid's ow an be expressed in terms of their stress-energy tensor. Using this result in the ontext of Hawking radiation, we ompute the se ular ee t on the ba kground ow.
1
1 Introdu tion and summary A
ording to General Relativity bla k holes are spa etime regions where the gravitational eld is so strong that even light is unable to es ape. It is indeed against all expe tations in luding his own that Hawking dis overed in 1974 [1, 2℄ that bla k holes are no longer bla k when Quantum Me hani s is taken into a
ount.
More pre isely he showed that
they thermally radiate (photons, neutrinos, gravitons, ...) with a temperature inversely proportional to their mass. This dis overy had a major impa t on theoreti al physi s. On a general level, Hawking radiation (HR in the following) nds its origin in the
ombination of the three pillars of physi s as presently understood, namely Spe ial Relativity (with the speed of light
c
as fundamental onstant), Quantum Field Theory (the
relativisti version of Quantum Me hani s bringing
~), and General Relativity (the met
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