During the last twenty-five years evidence has been mounting that a blackhole surface area has a discrete spectrum. Moreover, it is widely believed that area eigenvalues are uniformally spaced. There is, however, no general agreement on the spacing of the levels. In this letter we use Bohr's correspondence principle to provide this missing link. We conclude that the area spacing of a black-hole is 4h ln 3. This is the unique spacing consistent both with the areaentropy thermodynamic relation for black holes, with Boltzmann-Einstein formula in statistical physics and with Bohr's correspondence principle.The necessity in a quantum theory of gravity was already recognized in the 1930s. However, despite the flurry of activity on this subject we still lack a complete theory of quantum gravity. It is believed that black holes may play a major role in our attempts to shed some light on the nature of a quantum theory of gravity (such as the role played by atoms in the early development of quantum mechanics).The quantization of black holes was proposed long ago in the pioneering work of Bekenstein [1]. The idea was based on the remarkable observation that the horizon area of nonextremal black holes behaves as a classical adiabatic invariant. In the spirit of Ehrenfest principle [2], any classical adiabatic invariant corresponds to a quantum entity with discrete
During the last ten years evidence has been mounting that generically the Cauchy horizon inside a charged or a spinning black hole becomes a null, weak singularity which is a precursor of a strong, spacelike singularity along the r 0 hypersurface. We present here the missing link in this picture: A complete calculation from a regular initial data to the formation of a black hole and its inner singularities. We follow the gravitational collapse of a self-gravitating charged massless scalar field and observe the formation of an apparent horizon, a null, weak, mass-inflation singularity along the Cauchy horizon, and a final, spacelike, central singularity. [S0031-9007(98)06951-8] PACS numbers: 04.70.Bw, 04.25.Dm, 04.40.NrThe no-hair conjecture, introduced by Wheeler [1] in the early 1970s, states that the external field of a black-hole relaxes to a Kerr-Newman field characterized solely by the black-hole's mass, charge and angular-momentum. This simple picture describing the exterior of a black hole is in dramatic contrast with its interior. The singularity theorems of Penrose and Hawking [2] predict the occurrence of inevitable spacetime singularities as a result of a gravitational collapse in which a black hole forms. According to the weak cosmic censorship conjecture [3] these spacetime singularities are hidden beneath the black hole's event horizon. However, these theorems tell us nothing about the nature of these spacetime singularities. In particular, the final outcome of a generic gravitational collapse is still an open question in general relativity.Until recently, our physical intuition regarding the nature of these inner singularities was largely based on the known static or stationary black-hole solutions: Schwarzschild (spacelike, strong, and unavoidable central singularity), Reissner-Nordström and Kerr (timelike, strong singularity). Further insight was gained from the work of Belinsky, Khalatnikov, and Lifshitz [4] who found a strong oscillatory spacelike singularity. However, a new and drastically different picture of these inner black-hole singularities has emerged in the last few years, according to which the Cauchy horizon (CH) inside charged or spinning black holes is transformed into a null, weak singularity [5][6][7][8][9]. The CH singularity is weak in the sense that an infalling observer which hits this null singularity experiences only a finite tidal deformation [7,8]. Nevertheless, curvature scalars (namely, the Newman-Penrose Weyl scalar C 2 ) diverge at the CH, a phenomena known as mass inflation [6]. The physical mechanism which underlies this CH singularity is actually quite simple: small perturbations, which are remnants of the gravitational collapse outside the collapsing object are gravitationally blueshifted as they propagate in the black hole's interior parallel to the CH [10] (the mass-inflation scenario itself includes in addition an outgoing radiation flux which irradiates the CH. This outgoing flux repre-sents a portion of the ingoing radiation which is scattered inside the blac...
From information theory and thermodynamic considerations a universal bound on the relaxation time τ of a perturbed system is inferred, τ ≥h/πT , where T is the system's temperature. We prove that black holes comply with the bound; in fact they actually saturate it. Thus, when judged by their relaxation properties, black holes are the most extreme objects in nature, having the maximum relaxation rate which is allowed by quantum theory.A fundamental problem in thermodynamic and statistical physics is to study the response of a system in thermal equilibrium to an outside perturbation [1][2][3]. In particular, one is typically interested in calculating the relaxation timescale at which the perturbed system returns to a stationary, equilibrium configuration. Can this relaxation time be made arbitrarily small? That the answer may be negative is suggested by the third-law of thermodynamics, according to which the relaxation time of a perturbed system is expected to go to infinity in the limit of absolute zero of temperature. Finite temperature systems are expected to have faster dynamics and shorter relaxation times-how small can these be made? In this Letter we use general results from quantum information theory in order to derive a fundamental bound on the maximal rate at which a perturbed system approaches thermal equilibrium.On another front, deep connections between the world of black-hole physics and the realms of thermodynamics and information theory were revealed by Hawking's theoretical discovery of black-hole radiation [4], and its corresponding temperature and black-hole entropy [5]. These discoveries imply that black holes behave as thermodynamic systems in many respects. Furthermore, black holes have been proven to be very useful in deriving fundamental, static bounds on information storage [6][7][8][9][10][11]. Can one use black holes to obtain deep insights into natural limitations on dynamical relaxation times? Indeed one can, as we shall show below.Quantum information theory and Thermodynamics.-A fundamental question in quantum information theory is what is the maximum rate,İ max , at which information may be transmitted by a signal of duration τ and energy ∆E. The answer to this question was already found in the 1960's (see e.g. [12,13]):(We use units in which k B = G = c = 1.) An outside perturbation to a thermodynamic system is characterized by energy and entropy changes in the system, ∆E and ∆S, respectively. By the complementary relation between entropy and information (entropy as a measure of one's uncertainty or lack of information about the actual internal state of the system [14][15][16]12]), the relation Eq. (1) sets an upper bound on the rate of entropy change [17],where τ is the characteristic timescale for this dynamical process (the relaxation time required for the perturbed system to return to a quiescent state). Taking cognizance of the second-law of thermodynamics, one obtains from Eq. (2)where T is the system's temperature. Thus, according to quantum theory, a thermodynamic system has...
The fundamental role played by black holes in many areas of physics makes it highly important to explore the nature of their stability. The stability of charged Reissner-Nordström black holes to neutral (gravitational and electromagnetic) perturbations was established almost four decades ago. However, the stability of these charged black holes under charged perturbations has remained an open question due to the complexity introduced by the well-known phenomena of superradiant scattering: A charged scalar field impinging on a charged Reissner-Nordström black hole can be amplified as it scatters off the hole. If the incident field has a non-zero rest mass, then the mass term effectively works as a mirror, preventing the energy extracted from the hole from escaping to infinity. One may suspect that the superradiant amplification of charged fields by the charged black hole may lead to an instability of the Reissner-Nordström spacetime (in as much the same way that rotating Kerr black holes are unstable under rotating scalar perturbations). However, in this Letter we show that, for charged Reissner-Nordström black holes in the regime (Q/M ) 2 ≤ 8/9, the two conditions which are required in order to trigger a possible superradiant instability [namely: (1) the existence of a trapping potential well outside the black hole, and (2) It is well-known that rotating Kerr black holes suffer from superradiant instability which may be triggered by massive scalar perturbations [1][2][3]. Superradiant scattering is a well-known phenomena in quantum systems [4,5] as well as in classical ones [6,7]. Considering a wave of the form e imφ e −iωt incident upon a rotating object whose angular velocity is Ω, one finds that if the frequency ω of the incident wave satisfies the relationthen the scattered wave is amplified. A bosonic field impinging upon a rotating Kerr black hole can be amplified if the superradiance condition (1) is satisfied, where in this case Ω is the angular velocity of the black-hole horizon. Press and Teukolsky suggested to use this mechanism to build a black-hole bomb [8]: If one surrounds the black hole by a reflecting mirror, the wave will bounce back and forth between the black hole and the mirror amplifying itself each time. It was later realized [9][10][11][12][13][14][15][16][17] that a natural mirror exists in a system composed of a rotating black hole and a massive scalar field: In this case the mass term effectively works as a mirror for modes in the regime ω < µ ≡ MG/hc, where M is the mass of the field. The gravitational force binds the massive field and keeps it from escaping to infinity. At the event horizon some of the field goes down the black hole, and if the frequency of the wave is in the superradiance regime (1) then the field is amplified. In this way the field is amplified at the horizon while being bound away from infinity. As a consequence, the rotational energy extracted from the black hole by the incident bosonic field grows exponentially over time [9][10][11][12][13][14][15][16][17].A sim...
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