Quantum Fluctuations of Black Hole Geometry

Abstract

By using the minisuperspace model for the interior metric of static black holes, we solve the Wheeler·DeWitt equation to study quantum mechanics of the horizon geometry. Our basic idea is to introduce the gravitational mass and th,e expansions of null rays as quantum operators. Then, the exact wave function is found as a mass eigenstate, and the radius of the apparent horizon is quantum·mechanically defined. In the evolution of the metric variables, the wave function changes from a WKB solution giving the classical trajectories to a tunneling solution. By virtue of the quantum fluctuations of the metric evolution beyond the WKB approximation, we can observe a static black hole state with the apparent horizon separating from the event horizon. Since the discovery of the Hawking radiation, much work has been devoted to the analysis of the quantum evaporation of black holes. In particular, the problem of final fates of evaporating black holes has been much debated. In Hawking'S semi­ classical calculation, the emitted radiation was found to be exactly thermal. Then, if a black hole evaporates completely, an initially pure quantum state must evolve to a mixed state. This is known as the information loss paradox. As was emphasized by Preskill, I) it is very difficult to resolve the serious puzzle in quantum mechanics and general relativity. Before reaching the final resolution, we must develop quantum theories of the black hole geometry. A possible way of treating the horizon as a quantum system is to apply the Wheeler-DeWitt equation to spherically symmetric spacetimes. In the superspace canonical formulation, the Hamiltonian and momentum constraints work as quantum equations for the physical state IJf which is a functional of the metric variables. To make the calculation tractable, Rodrigues et al. 2 ) proposed a black hole minisuper­ space model and derived the simplified Hamiltonian constraint. Unfortunately their model was found to be incompatible with the momentum constraint. 3 ) The compati­ bility between the two constraints can be recovered if we consider a local minisuper­ space model valid near the apparent horizon. 4 ) From the wave function dependent on the local metric near the apparent horizon, we can derive the mass-loss rate due to the back-reaction of Hawking radiation and show the breakdown of the semi-classical result at the final stage of complete evaporation. The Wheeler-DeWitt approach will be viable as a quantum theory of the horizon. To advance this prospect, in this paper, we want to clarify another quantum feature of the horizon from the Wheeler­ DeWitt equation. We will consider static states of a spherically symmetric black hole instead of its evolutionary states. In classical relativity the apparent horizon is always located just on the event horizon. The degeneracy of the two horizons can be removed owing to quantum fluctuations of the metric. This was first pointed out by Y ork 5 ) under the