Quantum Evaporating Black Holes in Higher Dimensions

We study an N +1 dimensional generalization of the Schwarzschild black hole from the quantum mechanical viewpoint. It is shown that the mass loss rate of this higher dimensional black hole due to the black hole radiation is proportional to 1 rN−1 where r is the radial coordinate. This fact implies that except in four dimensions the quantum formalism developed in this letter gives a different result with respect to the mass loss rate from the semiclassical formalism where the Stefan-Bolzmann formula for a blackbody radiation is used. As shown previously in the study of the four dimensional Schwarzschild metric, the wave function in this case has also quite curious features that it is singular owing to strong quantum fluctuation of the gravitational field at the singularity while completely regular in the other regions of the spacetime. † E-mail address: [email protected] Despite much impressive effort, establishing a quantum field theory which unifies the gravitational interaction with the other interactions existing in nature remains to be one of unsolved problems in the modern theoretical physics. Motivated by the development of superstring theories [1], it seems that most recent works in this search have been directed at studying theories where the number of spacetime dimension is greater than four. In particular, in more recent works it is widely expected that black holes would play a very important role in understanding the non-perturbative features of quantum gravity [2]. Hence examining quantum aspects of black holes in N +1 dimensions with N ≥ 3 is certainly of importance in obtaining useful informations about quantum gravity when we attempt to construct a unified theory in future. In this letter, we would like to study the Hawking radiation [3] of the higher dimensional analog of the 3+1 dimensional Schwarzschild black hole [4, 5] in terms of the recently developed formalism [6-11] which goes beyond the semiclassical analysis [3] and is purely quantum mechanical at least in the geometry involving a black hole. Although we deal with a specific higher dimensional black hole in general relativity for the sake of simplicity, it is straightforward to apply the present formalism to the more complicated black holes with several nontrivial charges which have recently been found in the low energy effective theory of superstrings [2]. Our motivations in this letter are twofold. On the one hand, by extending the previous formalism constructed in dimensions equal to or lower than four to the higher dimensional Schwarzschild black hole we would like to understand some quantum aspects of the black hole evaporation in an arbitrary dimension. On the other hand, it was shown in the previous works [6-11] that in two and four dimensions the mass loss rate of quantum evaporating black holes coincides with the result obtained in the semiclassical analysis. Then one is naturally led to ask whether this coincidence remains true even in higher spacetime dimensions. We will see that this is not the case in general. This fact makes it clear that our quantum mechanical approach is surely different from the semiclassical one.