ON THE QUANTIZATION OF CHARGED BLACK HOLES

Abstract

The Wheeler–DeWitt equation for the wave function Ψ of the Schwarzschild black hole has been derived by Tomimatsu in the form of a Schrödinger equation, valid on the apparent horizon, using the two-dimensional Hamiltonian formalism of Hajicek and the radiating Vaidya metric. Here, the analysis is generalized to the Reissner–Nordström black hole. At constant charge Q, the evaporation rate is calculated from the solution for Ψ to be [Formula: see text], where k is a constant and [Formula: see text] are the radii of the outer event horizon and inner Cauchy horizon. In the extremal limit M → Q, however, the Hawking temperature [Formula: see text] tends to zero, suggesting, when the back reaction is taken into account, that the evaporation cannot occur this way and in agreement with the known discharging process of the hole via the Schwinger electron–positron pair-production mechanism. The more general charged dilaton black holes obtained from the theory L4 = [R4 - 2 (∇ Φ)2 - e-2aΦF2 ]/16π are also discussed, and it is explained why this quantization procedure cannot be applied when a is non-zero.