SPHERICALLY SYMMETRIC GRAVITATIONAL FIELDS: BLACK HOLES AND MIDISUPERSPACE QUANTIZATION NEAR THE APPARENT HORIZON

Abstract

The purpose of this paper is to investigate the quantum vacua directly implied by the wave function of a gravitational configuration characterized by the presence of an apparent horizon, namely the Vaidya space–time solution. Spherical symmetry is a main feature of this configuration, with a scalar field constituting a source [a Klein–Gordon geon or Berger–Chitre–Moncrief–Nutku (BCMN) type model]. The subsequent analysis requires solving a Wheeler–DeWitt equation near the apparent horizon (following the guidelinesintroduced by A. Tomimatsu,18; M. Pollock,19 and developed by A. Hosoya and I. Oda20,21) with the scalar field herein expanded in terms of S2 spherical harmonics: midisuperspace quantization. The main results present in this paper are as follows. It is found that the mass function characteristic of the Vaidya metric is positive definite within this quantum approach. Furthermore, the inhomogeneous matter sector determines a descrip-tion in terms of open quantum (sub)systems, namely in the form of an harmonic oscillator whose frequency depends on the mass function. For this open (sub)system, a twofold approach is employed. On the one hand, an exact invariant observable is obtained from the effective Hamiltonian for the inhomogeneous matter modes. It is shown that this invariant admits a set of discrete eigenvalues which depend on the mass function. The corresponding set of eigenstates is constructed from a particular vacuum state. On the other hand, exact solutions are found for the Schrädinger equation associated with the inhomogeneous matter modes. This paper is concluded with a discussion, where two other issues are raised: (i) the possible application to realistic black hole dynamics of the results obtained for a simplified (BCMN) model and (ii) whether such vacuum states could be related with others defined instead within scalar field theories constructed in classical backgrounds.