de Broglie-Bohm Interpretation for the Wave Function of Quantum Black Holes

Abstract

We study the quantum theory of the spherically symmetric black holes. The theory yields the wave function inside the apparent horizon, where the role of time and space coordinates is interchanged. The de Broglie-Bohm interpretation is applied to the wave function and then the trajectory picture on the minisuperspace is introduced in the quantum as well as the semi-classical region. Around the horizon large quantum fluctuations on the trajectories of metrics $U$ and $V$ appear in our model, where the metrics are functions of time variable $T$ and are expressed as $ds^2=-{α^2}/U dT^2 + U dR^2 + V dΩ^2$. On the trajectories, the classical relation $U=-V^{1/2}+2Gm$ holds, and the event horizon U=0 corresponds to the classical apparent horizon on $V=2Gm$. In order to investigate the quantum fluctuation near the horizon, we study a null ray on the dBB trajectory and compare it with the one in the classical black hole geometry.