Wave Function for the Reissner-Nordstr\"om Black-Hole

Abstract

We study the quantum behaviour of Reissner-Nordstr\"om (RN) black-holes interacting with a complex scalar field. A Maxwell field is also present. Our analysis is based on M. Pollock's method and is characterized by solving a Wheeler-DeWitt equation in the proximity of an apparent horizon of the RN space-time. Subsequently, we obtain a wave-function $\Psi_{RN}[M, Q]$ representing the RN black-hole when its charge, $\mid Q \mid$, is small in comparison with its mass, $M$. We then compare quantum-mechanically the cases of $(i)$ $Q = 0$ and $(ii)$ $M \geq \mid Q \mid \neq 0 $. A special emphasis is given to the evolution of the mass-charge rate affected by Hawking radiation.