Wave Function of Evaporating Black Holes

Abstract

We study some quantum mechanical aspects of dynamical black holes where the Vaidya metric is used as a model representing evaporating black holes. It is shown that in this model the Wheeler-DeWitt equation is solvable in whole region of spacetime, provided that one considers the ingoing (or outgoing) Vaidya metric and selects a suitable coordinate frame. This wave function has curious features in that near the curvature singularity it oscillates violently owing to large quantum effects while in the other regions the wave function exhibits a rather benign and completely regular behavior. The general formula concerning the black hole radiation, which reduces to the Hawking's semiclassical result when $r = 2M$ is chosen, is derived by means of purely quantum mechanical approach. The present formulation can be applied essentially to any system with a spherically symmetric black hole in an arbitrary spacetime dimension.