ON THE WHEELER-DEWITT EQUATION FOR BLACK HOLES

Abstract

Integration over the angular coordinates of the evaporating, four-dimensional Schwarzschild black hole leads to a two-dimensional action, for which the Wheeler-DeWitt equation has been found by Tomimatsu, on the apparent horizon, where the Vaidya metric is valid, using the Hamiltonian formalism of Hajicek. For the Einstein theory of gravity coupled to a massless scalar field ζ, the wave function Ψ obeys the Schrödinger equation [Formula: see text], where M is the mass of the hole. The solution is [Formula: see text], where k2 is the separation constant, and for k2>0 the hole evaporates at the rate Ṁ=−k2/4M2, in agreement with the result of Hawking. Here, this analysis is generalized to the two-dimensional theory [Formula: see text], which subsumes the spherical black holes formulated in D≥4 dimensions, when A = ½ (D - 2) (D - 3)ϕ2 (D - 4)/(D - 2), B=2(D−3)/(D−2), C=1, and also the twodimensional black hole identified by Witten and by Gautam et al., when A=4/α′, B=2, C=1/8π, c=+8/α′ being (minus) the central charge. In all cases an analogous Schrödinger equation is obtained. The evaporation rate is [Formula: see text] when D≥4 and [Formula: see text] when D=2. Since Ψ evolves without violation of unitarity, there is no loss of information during the evaporation process, in accord with the principle of black-hole complementarity introduced by Susskind et al. Finally, comparison with the four-dimensional, cosmological Schrödinger equation, obtained by reduction of the ten-dimensional heterotic superstring theory including terms [Formula: see text], shows in both cases that there is a positive semi-definite potential which evolves to zero, this corresponding to the ground state, which is Minkowski space.