Prof. Philippe Spindel

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    Université de Mons, Belgium

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    \title{ {\bf Hawking radiation}} \author{Renaud Parentani, Philippe Spindel } %\date{}  % Activate to display a given date or no date

    \begin{document} \maketitle %\section{} %\subsection{} When a sufficiently massive star has burned its nuclear combustible, %R3 , added its internal pressure will no longer be able to resist its own gravitational attraction. As a result, it will explode, becoming a supernova. During the explosion a lot of stellar matter is ejected, at a speed of the order of a few percent of the light velocity UNIQ525d9b9351881051-MathJax-1-QINU. If the residual material is not too massive it will contract, reach a new equilibrium state: a neutron star. But if its mass is greater than a few Solar masses, its own pressure will not be able to counterbalance its weight and it will ineluctably collapse onto itself and form a black hole. A black hole is a region of space-time characterised by a boundary surface called the {\em horizon} that separates the outer region from which light rays can escape and reach far distant observers from a trapped region from which neither matter nor light can possibly escape. The classical theory of gravitation predicts that during the last stage of this collapse, the temperature of the radiation emitted by the star, measured by the distant observers, decreases to zero in a universal manner. More precisely the temperature perceived by static observers sitting far from the star decreases following \begin{equation} T(t) \sim T_0\, e^{- (t-t_0)/\tau_\kappa}\, ,\end{equation} {\it i.e.} it goes to zero exponentially rapidly, with a characteristic lifetime given by UNIQ525d9b9351881051-MathJax-2-QINU. Interestingly, one finds that UNIQ525d9b9351881051-MathJax-3-QINU depends only on the mass of the black hole and its angular momentum. The properties of the collapsing star are thus erased. Roughly speaking, when the angular momentum vanishes, UNIQ525d9b9351881051-MathJax-4-QINU is given by the time it takes light to travel a distance of the order of the size of the black hole horizon. For a star of one Solar mass, one finds that UNIQ525d9b9351881051-MathJax-5-QINU which means that after one second, the collapsing star is actually black. If Nature were not fundamentally on quantum mechanical nature, this would be the end of the story: A collapsing star would stop radiating at the end of its collapse in a fraction of a second and would henceforth no longer radiate. However the collapsed star would still gravitationaly attract matter and light. As a result infalling matter will cross inwards the horizon and enter into the inside trapped region, thereby justifying why such an object has been called a black hole. It should be noticed that through this accretion, the mass of the black hole will necessarily increase. This resembles very much the second Law of thermodynamics which stipulates that the entropy, that measures the amount of info ..., of a system can never decrease. We shall return to this aspect below. However, to the greatest surprise of his colleagues, and to himself, Stephen Hawking discovered in 1974 that because of the quantum nature of light, which was neglected in the above reasoning, a newly formed black hole will spontaneously emit in a steady manner a thermal radiation with a temperature precisely given by \begin{equation} k_B T_{\rm Hawking} = { h \over (2 \pi)^2 }{1\over \tau_\kappa}, \end{equation} where UNIQ525d9b9351881051-MathJax-6-QINU is Boltzmann constant UNIQ525d9b9351881051-MathJax-7-QINU, and UNIQ525d9b9351881051-MathJax-8-QINU is the universal Planck constant. Using the famous Einstein relation UNIQ525d9b9351881051-MathJax-9-QINU, one sees that the typical frequency associated with this thermal radiation is, up to UNIQ525d9b9351881051-MathJax-10-QINU, equal to UNIQ525d9b9351881051-MathJax-11-QINU, i.e. it is fixed by the characteristic decay rate of Eq. (1). When considering a black hole with zero angular momentum, this temperature \begin{equation} T_{BH}=\frac {6.2\,10^{-8}\ \strut^\circ\! K}{m_{\odot}}\end{equation} where now UNIQ525d9b9351881051-MathJax-12-QINU denotes the mass of the black hole in solar mass UNIQ525d9b9351881051-MathJax-13-QINU. This evaporation process makes black holes loosing entirely their mass in a time of the order of \begin{equation} \tau_{BH}\approx m_{\odot}\ 6.6\,10^{74}\ s \end{equation} that for a solar mass black hole is much greater than the age of the Universe : UNIQ525d9b9351881051-MathJax-14-QINU. Actually, such a big black hole will absorb more interstellar radiation than it will emits radiation, making it mass increasing. It is only in a far future, when due to the expansion of the Universe the temperature of the microwave radiation in the Universe will be less than the black hole temperature, that the mass lost process actually will start. At first sight the thermal nature of the radiation can cause a surprise, but before Hawking's discovery there were already indications that BH behave as thermal objects. The first indications that black hole radiates are of thermodynamic nature. One one hand, a Kerr black hole, a rotating black hole, is characterised by only two parameters: its mass UNIQ525d9b9351881051-MathJax-15-QINU and its angular momentum UNIQ525d9b9351881051-MathJax-16-QINU. These parameters characterise completely the black hole. This is the meaning of Wheeler's claim :``{\em A black hole has no hair''}. Its area UNIQ525d9b9351881051-MathJax-17-QINU is equal to: \begin{equation} A_{BH}=8\,\pi\,\frac{G_N}{c^2} \left(M^2_{BH}+M_{BH}\,\sqrt{M^2_{BH}-a_{BH}^2 }\right)\end{equation} where UNIQ525d9b9351881051-MathJax-18-QINU is the Newton constant and UNIQ525d9b9351881051-MathJax-19-QINU the velocity of light: UNIQ525d9b9351881051-MathJax-20-QINU and UNIQ525d9b9351881051-MathJax-21-QINU is the specific angular momentum. Another important parameter, already introduced, is the surface gravity UNIQ525d9b9351881051-MathJax-22-QINU of the black hole: \begin{equation} \kappa = \frac12\frac{c^4}{G_N}\frac{\sqrt{M^2_{BH}-a^2_{BH}}}{2\,M_{BH} \left(M_{BH}+\sqrt{M^2_{BH}-a^2_{BH}}\right)}\end{equation} It corresponds to the gravitational acceleration that a test particle located near the black hole surface undergoes, when measured by the force exerted at infinity to maintain it with the help of an (infinite) unextensible rod. For a non rotating black hole it reduces to UNIQ525d9b9351881051-MathJax-23-QINU with UNIQ525d9b9351881051-MathJax-24-QINU the Schwardschild radius. Einstein general relativity equations predicts that in the gravitational field of a stationary black hole surrounded by matter, the following four laws of black hole mechanics are verified (\cite{Carter}) \begin{enumerate} \item The surface gravity of a stationary black hole is constant \item When two different black hole configurations are realised, their characteristic parameters cannot be varying arbitrarily but have to verified a variation relation of the type : \begin{equation} dM_{BH}=\frac \kappa{8\,\pi}dA_{BH}+\Omega_{BH}\,dJ_{BH}+\dots\end{equation} where UNIQ525d9b9351881051-MathJax-25-QINU is the angular velocity of the horizon and the dots represent extra term contributions describing the variation of the exterior energy--matter distribution. \item The area of a black hole could only increase : UNIQ525d9b9351881051-MathJax-26-QINU. \item It is impossible to decrease UNIQ525d9b9351881051-MathJax-27-QINU to zero \end{enumerate} These statements deserve some comments. \begin{itemize} \item The first one is obvious for a spherically symmetric black hole. That it is also true for a stationary, rotating, black hole, surrounded by matter is remarkable. Centrifugal force and gravitational attraction have to conspire to equilibrate each other. This uniformity is the analogous of the constancy of the temperature of bodies at equilibrium; the so-called zeroth law of thermodynamics. \item The second one related the parameters that describes slightly different black holes. It is the counterpart of the first law of thermodynamics UNIQ525d9b9351881051-MathJax-28-QINU. \item Once a black hole is created its horizon consists into endless light rays. The third law tell us that these light trajectories cannot disappears, but new ones can be added producing an increase in the horizon area. This is corresponds to the law of increase of the entropy in thermodynamics and leads us to identify UNIQ525d9b9351881051-MathJax-29-QINU with a multiple of the entropy and UNIQ525d9b9351881051-MathJax-30-QINU with the temperature. \item Finally the fourth law reinforces the previous identification as the corresponding thermodynamic law says that it is not possible to reach the absolute zero. \end{itemize} On the other hand, Bekenstein argued from purely thermodynamic considerations that black hole must posses an entropy. Its argument is basically as follows. Given that matter carrying entropy can fall into a black hole and never comes back, if no entropy were attributed to black holes, the second principel of thermodynamics, according to which the total entropy of an isolated system can never decrease would be violated. To estimate the entropy of the black hole, Bekenstein used an argument based on information theory. He compute the number of different way a black hole of a given mass can be obtain from elementary particles, taking into account that for an elementary particle to contribute to a black hole, its Compton wave length must be shorter than the black hole radius. This imply that the elementary particles constituting the black hole all have an energy bounded from below. Thus a black hole of a given mass can only be built from a finite number of such particles. The number UNIQ525d9b9351881051-MathJax-31-QINU of ways to built the black hole is proportional to the exponential of the mass of the black hole {\it i.e. } to its area. Then applying Boltzman formula, Bekenstein obtained the entropy formula: \begin{equation} S_{BH}\propto A_{BH}\qquad .\end{equation} These considerations have been made more precise by Hawking (\ref{Hw})quantum field theory calculations. To understand the spirit of Hawking calculation let us remind an aspect of the gravitational collapse. While for an observer sitting on the surface of a collapsing star, the time it takes to cross the horizon is finite, for an external observer the same phenomenon is diluted into an infinite period of time. Just as if he is observing Achilles and the tortoise race but in a scenario where it always takes (for him) the same amount of time for Achilles reaching the position the tortoise were staying. This time distorsion due to the increasing of the gravitational attraction at the surface of the star during its motion near the horizon is the source of an huge red-shift. It costs more and more energy to escape from the horizon vicinity. In quantum field theory, far from gravitational field, the particle description of a field is directly dictated by the sign (positive or negative) of the frequencies, i.e. energy, of the modes that solves its evolution equation. A quantum state associated to vacuum in the remote past will then be reinterpreted in the far future of a collapse process as containing radiation because the evolution of the modes in a time dependent gravitational field will lead to a mixing of both positive and negative frequencies. Physically the strong gravitational field can be seen as a distillation device that separate in quantum fluctuation high positive energy part from the negative one. The positive energy part escapes near infinity, and during the climbing of the gravitational field its energy is strongly reduced, to reach finally, with a Planckian distribution, the mean value of UNIQ525d9b9351881051-MathJax-32-QINU while the negative counterpart falls into the black hole and decrease its mass, as expected from energy conservation. The stationary aspect of this radiation process is due to the fact that to occur it needs a huge variation of the gravitational field met by the quantum mode between its entrance into the collapsing star and its exit. This can only be the case when the star surface is close to the horizon, during a short period of time for an observer following the star. Its is this short period of time that will be extended to a enormous time at interval for a far away observer that leads to a constant rate of emission, at least as long that the black hole didn't loose to much of its mass. The displaying of Hawking radiation by astrophysical observations is impossible. Fortunately, it exists physical processes that present a profound analogy with the propagation of a field in the neighbourhood of a black hole. One of them, first proposed by William Unruh, and susceptible to be observed in the laboratories, consists into the examination of perturbations travelling in a fluid that flows across a bottleneck where it reaches a supersonic velocity. Let us consider such a perturbation, a sonic wave, that propagates in the opposite direction of the fluid flow. There where the fluid velocity is subsonic, the perturbation may row up the stream, but where the fluid flows with the sound velocity, the perturbations become stationary. When the fluid velocity is higher than the speed of the sound, the perturbations are dragged with it. This kinematic is analogs to those of a black hole. A sonic horizon appears, just like a light horizon appears near a black hole, separating the fluid into two region by a one way boundary. The equations governing the evolution of sonic waves being, for high wave numbers, similar as those governing free quantum fields, we expect the occurrence of a Hawking radiation of phonons ( quanta of sound ). The temperature of this emission is of the order of the micro-Kelvin. Of course an emission at such a low temperature also is unobservable. But the kinematics of the fluid flow offers an other advantage over the black hole: the supersonic fluid may be slow down to subsonic velocities. This leads to the apparition of a second horizon: the analogs of the horizon of a white hole. Such horizons are always unobservable in the framework of gravity: they are always hidden by the external horizons, but of course not for a fluid in a pipe. This second horizon induces an amplification effect on the acoustic Hawking radiation that renders more optimistic our hope to observes it. The observation of this radiation will also bring some information about a conceptual puzzle that an elementary derivation of the Hawking radiation raises: those of the trans-Planckian energies. Indeed, if we consider a one Solar mass black hole, the Hawking radiation perceived by an asymptotic observer is almost constant during a period of time of the order of UNIQ525d9b9351881051-MathJax-33-QINU, while it appears to be emitted, for an observer falling with the collapsing star, only during a very short period of time of UNIQ525d9b9351881051-MathJax-34-QINU . It is the huge red shift factor between the neighbourhood of the black hole horizon and infinity that creates this difference. The same factor leads to think that a radiation quantum that arrives at infinity, with an energy of the order UNIQ525d9b9351881051-MathJax-35-QINU and a typical length of the order of the Schwardschild radius of the black hole, is issued from a fluctuation of energy much greater than the Planck energy : UNIQ525d9b9351881051-MathJax-36-QINU and concentrated on a scale less than the Planck length: UNIQ525d9b9351881051-MathJax-37-QINU. Here we touch at the TERRA INCOGNITA of the present day physics. The precise nature of the phenomenon involved raise to the quantum theory of gravity, a theory today always in elaboration. Someones have suggested that this difficulty is an indication that the Hawking radiation didn't exist! But such a reasoning raises numerous other difficulties. Indeed the prediction of the Hawking radiation bases itself on the attribution of a (finite) entropy and thus a (non zero) temperature to the black holes. The converse leads to violations of the second principle of thermodynamics! In the framework of the sonic horizons, it is the inter molecular distance of the fluid that will furnish a natural limit to the wave lengths of the sonic perturbations. Several theoretical works tend to show that this limitation, who modified at short wave lengths the relation between energy and momentum of the created quanta (the phonons) didn't prevent the occurrence of the radiation phenomenon. That is to said all the interest that such an experience, maybe feasible in a not to far future with Bose-Einstein condensates, presents for the physicists interested by such fundamental questions. \begin{thebibliography}{99} \bibitem{Carter} B. Carter B., {\em Black Hole equilibrium States}, in {\em Black Holes. Les Astres Occlus.}, Proceedings of Les Houches Summer School on Theoretical Physics, edited by C. DeWitt and B. S. DeWitt, Gordon and Breach (1972),\\ J.M. Bardeen, B. Carter, B. and S.W. Hawking, Commun. Math. Phys.{\bf 31}, 161--170 (1973) \bibitem{Be}J. D. Bekenstein, "Black holes and entropy", Phys. Rev. D {\bf 7}:2333-2346 (1973). \bibitem{Hw}S.W. Hawking, ,{\em Particle creation by black holes} Commun. Math. Phys. , {\bf 43}, 199--220 ( 1975),\\ S.W. Hawking, {\em Black hole explosions?}. Nature {\bf 248} (5443): 30–31(1974) . \bibitem{Unr} W. G. Unruh, {\em Experimental black hole evaporation}, Phys. rev. Lett. {\bf 46}: 1351-1353, 1981\\ W. G. Unruh, {\em Sonic analog of black holes and the effect of high frequencies on black hole evaporation}, Phys. rev. D {\bf 51}: 2827-2838, 1995 \bibitem{BMPS} R. Brout, S. Massar, , R. Parentani, , Ph. Spindel, {\em Hawking radiation without transPlanckian frequencies}, Phys.Rev.D {\bf 52}:4559-4568,1995. e-Print: hep-th/9506121 \end{thebibliography} \end{document} <review>message to curators</review>

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