Quantized black holes, their spectrum and radiation
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ELEMENTARY PARTICLES AND FIELDS Theory
Quantized Black Holes, Their Spectrum and Radiation* I. B. Khriplovich** Budker Institute of Nuclear Physics, Novosibirsk, Russia Received August 13, 2007
Abstract—Under quite natural general assumptions, the following results are obtained. The maximum entropy of a quantized surface is demonstrated to be proportional to the surface area in the classical limit. The general structure of the horizon spectrum is found. In the special case of loop quantum gravity, the value of the Barbero–Immirzi parameter is found. The discrete spectrum of thermal radiation of a black hole fits the Wien profile. The natural widths of the lines are much smaller than the distances between them. The total intensity of the thermal radiation is estimated. If the density of quantized primordial black holes is close to the present upper limit on the dark-matter density in our Solar system, the sensitivity of modern detectors is close to that necessary for detecting this radiation. PACS numbers: 97.60.-s DOI: 10.1134/S1063778808040078
1. INTRODUCTION At the school together with A. Vainstein, I was awarded the Pomeranchuk Prize. The Pomeranchuk Prize is a great honor to me. Isaak Yakovlevich was a remarkable physicist; his fundamental results in various fields of theoretical physics are widely known and will remain forever. His life and work serve as an example of complete devotion to science. For Pomeranchuk, physics was a temple, where there was no place for money changers. The idea of quantizing the horizon area of black holes was put forward by Bekenstein in the pioneering article [1]. He pointed out that reversible transformations of the horizon area of a nonextremal black hole found by Christodoulou and Ruffini [2, 3] have an adiabatic nature. Of course, the quantization of an adiabatic invariant is perfectly natural, in accordance with the correspondence principle. Later, the quantization of black holes was discussed by Mukhanov [4] and Kogan [5]. In particular, Kogan was the first to investigate this problem within the string approach. Once the idea of area quantization is accepted, the general structure of the quantization condition for large quantum numbers N becomes obvious, up to an overall numerical constant (written usually as 8πγ). It should be [6] A = 8πγlP2 N. ∗ **
The text was submitted by the authors in English. E-mail: [email protected]
(1)
Indeed, the presence of the Planck length squared lP2 = k/c3 is only natural in this quantization rule. Then, for the horizon area A to be finite in the classical limit, the power of N should be the same as that of in lP2 . This argument can be checked by considering any expectation value in quantum mechanics, nonvanishing in the classical limit. It is worth mentioning that, contrary to wide spread beliefs, there are no compelling reasons to believe that the black-hole spectrum (1) is equidistant. A quite popular argument in favor of its equidistance is as follows [7] (see also [8, 9]). On the one hand, the entropy S of the horizon is related to
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