The extreme Reissner–Nordström Black Hole: New exact solutions to the Klein–Gordon equation with minimal coupling

Abstract

The omission of numerical methods to solve the Klein–Gordon equation with extremal black holes background is a well known problem. The difficulty forces the numerical investigation stop short at the near-extremal limit. In this work, we present a novel exact scalar quasibound states solutions in the extremal Reissner–Nordström black hole background. We start with the construction of the Klein–Gordon equation with minimal coupling in the extremal Reissner–Nordström black hole background and applying the separation of variables ansatz. We successfully find and present the exact solutions to the radial Klein–Gordon equation in the terms of the double confluent Heun functions. The exact eigenfrequencies expression of the scalar field bound to the extremal Reissner–Nordström black hole background, i.e., the quasibound states frequency, is subsequently determined by applying the polynomial condition to the double confluent Heun function. By utilizing the obtained exact quasibound states formula, we present an analytic proof to the nonexistence of charged scalar cloud in the extreme Reissner–Nordström black hole. And lastly, for the sake of completeness of the investigation, by utilizing the obtained exact radial wave function, we re-investigate the Hawking radiation and present a derivation to the extremal black hole horizon’s zero Hawking temperature via the Damour–Ruffini formulation.