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Hawking radiation and the boomerang behavior of massive modes near a horizon

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Niveau: Supérieur, Doctorat, Bac+8
Hawking radiation and the boomerang behavior of massive modes near a horizon G. Jannes,1,2 P. Maıssa,1 T. G. Philbin,3 and G. Rousseaux1,* 1Universite de Nice Sophia Antipolis, Laboratoire J.-A. Dieudonne, UMR CNRS-UNS 6621, Parc Valrose, 06108 Nice Cedex 02, France 2Low Temperature Laboratory, Aalto University School of Science, PO Box 15100, 00076 Aalto, Finland 3School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland, United Kingdom (Received 4 February 2011; published 17 May 2011) We discuss the behavior of massive modes near a horizon based on a study of the dispersion relation and wave packet simulations of the Klein-Gordon equation. We point out an apparent paradox between two (in principle equivalent) pictures of black-hole evaporation through Hawking radiation. In the picture in which the evaporation is due to the emission of positive-energy modes, one immediately obtains a threshold for the emission of massive particles. In the picture in which the evaporation is due to the absorption of negative-energy modes, such a threshold apparently does not exist. We resolve this paradox by tracing the evolution of the positive-energy massive modes with an energy below the threshold. These are seen to be emitted and move away from the black-hole horizon, but they bounce back at a ‘‘red horizon'' and are reabsorbed by the black hole, thus compensating exactly for the difference between the two pictures

  • rest mass

  • energy hawking

  • uk ? ck

  • energy massive

  • black hole

  • hawking radiation

  • negative

  • frequency root


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PHYSICAL REVIEW D83,104028 (2011) Hawking radiation and the boomerang behavior of massive modes near a horizon
1,2 1 3 1, * G.Jannes,P.Ma¨ıssa,T.G.Philbin,andG.Rousseaux 1 Universite´deNiceSophiaAntipolis,LaboratoireJ.A.Dieudonn´e,UMRCNRSUNS6621, Parc Valrose, 06108 Nice Cedex 02, France 2 Low Temperature Laboratory, Aalto University School of Science, PO Box 15100, 00076 Aalto, Finland 3 School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland, United Kingdom (Received 4 February 2011; published 17 May 2011) We discuss the behavior of massive modes near a horizon based on a study of the dispersion relation and wave packet simulations of the Klein-Gordon equation. We point out an apparent paradox between two (in principle equivalent) pictures of black-hole evaporation through Hawking radiation. In the picture in which the evaporation is due to the emission of positive-energy modes, one immediately obtains a threshold for the emission of massive particles. In the picture in which the evaporation is due to the absorption of negative-energy modes, such a threshold apparently does not exist. We resolve this paradox by tracing the evolution of the positive-energy massive modes with an energy below the threshold. These are seen to be emitted and move away from the black-hole horizon, but they bounce back at a ‘‘red horizon’’ and are reabsorbed by the black hole, thus compensating exactly for the difference between the two pictures. For astrophysical black holes, the consequences are curious but do not affect the terrestrial constraints on observing Hawking radiation. For analogue-gravity systems with massive modes, however, the consequences are crucial and rather surprising.
DOI:10.1103/PhysRevD.83.104028
I. INTRODUCTION A. Black hole evaporation through the emission of positiveenergy Hawking modes The evaporation of black holes through Hawking radia-tion is one of the cornerstones of post-classical gravity [13]. In Hawking’s original derivation [1] for the case of a gravitational collapse, the emphasis lay on the positive-energy modes that cross the horizon just before it is actually formed, escape from the black hole space-time, and can in principle be detected by an asymptotic observer. Hawking already observes that the decrease of the black-hole mass, and the accompanying decrease of the area of the event horizon, ‘‘must, presumably, be caused by a flux of negative energy across the event horizon which balances the positive energy flux emitted to infinity. One might picture this negative energy flux in the following way. Just outside the event horizon there will be virtual pairs of particles, one with negative energy and one with positive energy. The negative particle. . .can tunnel through the event horizon to the region inside the black hole. . .In this region the particle can exist as a real particle with a timelike momentum vector even though its energy relative to infinity as measured by the time translation Killing vector is negative. The other particle of the pair, having a positive energy, can escape to infinity where it constitutes a part of the thermal emission described above.’’ Hawking warns, however, that it ‘‘should be em-phasized that these pictures of the mechanism responsible
* Germain.Rousseaux@unice.fr
1550-7998=2011=83(10)=104028(11)
PACS numbers: 04.70.s, 04.62.+v, 04.70.Dy
for the thermal emission and area decrease are heuristic only and should not be taken too literally.’’ With regard to massive particles, Hawking notes that ‘‘As [the black holes] got smaller, they would get hotter and so would radiate faster. As the temperature rose, it would exceed the rest mass of particles such as the electron and the muon and the black hole would begin to emit them also.’’ Therefore, ‘‘the rate of particle emission in the asymptotic future. . .will again be that of a body with temperature=2. The only difference from the zero rest mass case is that the frequency!in the thermal factor 11 ðexpð2!Þ 1Þnow includes the rest mass energy of the particle. Thus there will not be much emission of particles of rest massmunless the temperature=2is greater thanm.’’ A similar conclusion was reached in [4]. Indeed, for the black hole to emit, for example, an elec-19 tron, its temperature must be on the order ofT¼10 2 Kelvin. Because of emission of massless particles the black hole will eventually become small enough for the 9 temperature to reach10 K, and then the radiation will contain electrons and positrons. But for most of the life-time of the black hole, the mass cutoff prevents any (significant) radiation of electrons. Implicit in the above reasoning is that the black-hole emission corresponds (by definition) to what can be de-tected by an asymptotic observer. This definition makes
1 We neglect complications [5] due to the electrical charge. 2231 Frommc¼kBTand usingme¼9:1110 kg, one ob-9 tainsT¼5:9310Kelvin. This rough estimate is in agree-ment with [2] (see p. 261). The associated frequency, from 2 20 !¼mc, is!c¼7:7610 Hz.
104028-1
2011 American Physical Society