On Stimulated Radiation of Black Holes

The Schwarzschild radius and the event horizon are the notions of classical physics and general relativity. The quantum aspects of black holes appeared first in the papers of Hawking [1,2] which changed radically our understanding of the phenomena connected with them and introduced their thermal radiation. Together with the work of Unruh [3,4] this conceptual framework provides a basis for better understanding of phenomena occurring close to the event horizons of black holes. It was shown in [3] that the fiducial observer located outside the Schwarzschild radius will see the vacuum near the event horizon as a many-particle thermal ensemble. In standard interpretation the free falling observer will perceive this as Heisenberg fluctuations while the accelerated observer will detect these thermal particles as also shown in [3]. Thus orbiting particle with perimelasma 1 gradually shrinking to the value of Schwarzschild radius will ultimately interact with the thermal particles.

The quantum phenomena taking place around the event horizon of a black hole pose a formidable problem of quantization in a curved space. Actually [1] tackles the problem by describing asymptotic states of vacuum in flat space before and after collapse of a star. The relation between accelerated and nonaccelerated systems is studied in [3]. At first the quantization of massive scalar field in Rindler coordinate system is compared to that in Minkowski space as in the case of large black hole the use of flat space-time is locally valid approximation of the horizon. The quantization in Rindler coordinates based on [5] leads to the result inequivalent with the standard quantization in Minkowski space, namely the respective vacua differ. The vacuum in Minkowski space can be expressed as thermal many-particle ensemble of the Unruh-Fulling-Rindler (UFR) state. The procedure is examined also in the Schwarzschild metric, in this case the problem with definition of positive frequency waves has to be addressed, but the result remains valid. The temperature of the UFR state increases to infinity as the Rindler variable

dr ′ approaches to zero, i.e. the observer approaches the Schwarzschild radius R = 2M, where M is the mass of the black hole and we put 1 = G = = c = k (Bolzmann ′ s constant). This holds for any black hole regardless of its mass while the temperature of the Hawking radiation changes with its mass as 8π/M. Indeed, a black hole can radiate a particle of mas m if it drags down another particle from the UFR state beyond the event horizon so that the conservation laws are satisfied. The intensity of gravitational field close to the Schwarzschild radius behaves as m/M. Thus the probability of a particle with mass m to be radiated decreases with the mass M of the black hole. The black hole radiation is entirely determined by the surface gravity κ introduced in [6] and it plays the role of temperature, in the case of a black hole with zero angular momentum κ ∼ 1/M.

As mentioned above we can observe the UFR vacuum state. This is examined in detail in [3]. It is shown that accelerated detector -testing particle or field, interacts with particles of the vacuum state, provided these two fields have nonzero coupling. Interesting consequences arise if we implement this abstract example to the Standard Model or its extension. The UFR state is then occupied by all SM particle species and the orbiting particle can interact with them. The stretched horizon is introduced as the horizon several Planck lengths away from the mathematical horizon in [7] and it is shown that it has properties of an electric conductor. Thus the dynamics of a charged particle in the vicinity of the stretched horizon might be quite complex. The most simple case will be that of a massive neutral particle X 0 with gravitational interaction only. To conciliate this assumption with extension of SM we can consider this particle as an example of cold dark matter. Then besides gravitation this particle interacts with Higgs bosons H 0 provided it is sufficiently close to the horizon, i.e. the UFR state has a temperature corresponding to the occurrence of Higgs bosons. With the increasing proximity to the stretched horizon and with increasing temperature the symmetries are gradually restored. Thus the particles will be generally off mass shell. The most simple process is then X 0 H 0 * → H 0 X 0 * . The proximity of the event horizon and the environment of the UFR state allow that the mass m(X 0 * ) = m(X 0 ) -∆ and ∆ > 0, i.e. X 0 gets off mass shell. The situation is sketched in Fig. 1. As all participating particles are massive we can use locally flat space-time. Of course we cannot describe the gravitational interactions of X 0 by means of quantum field theory but we can try to estimate their size from classical physics. The definition of the Schwarzschild radius is equivalent to the condition that at this radius the potential energy of a particle in the gravitational field of a

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