On the quantization of the extremal Reissner-Nordström black hole

¹ International Institute for Applicable Mathematics and Information Sciences (IIAMIS), B.M. Birla Science Centre - Adarsh Nagar, Hyderabad, 500 463, India
² Institute for Theoretical Physics, Utrecht University - Princetonplein 5, 3584 CC Utrecht, The Netherlands
³ ZKM - Zentrum für Kunst und Medientechnologie - Lorentzstr. 19, D-76135, Karlsruhe, Germany
⁴ Istituto Livi - Via Antonio Marini, 9, I-59100 Prato, Italy

^(a) cordac.galilei@gmail.com
^(b) feleppa.fabiano@gmail.com
^(c) fabrizio.tamburini@gmail.com

Received: 15 July 2020
Accepted: 5 October 2020

Abstract

Following Rosen's quantization rules, two of the authors (C. Corda and F. Feleppa) recently described the Schwarzschild black hole (BH) formed after the gravitational collapse of a pressureless “star of dust” in terms of a “gravitational hydrogen atom”. Here we generalize this approach to the gravitational collapse of a charged object, namely, to the geometry of a Reissner-Nordström BH (RNBH) and calculate the gravitational potential, the Schrödinger equation and the exact solutions of the energy levels of the gravitational collapse. By using the concept of BH effective state, previously introduced by one of us (CC), we describe the quantum gravitational potential, the mass spectrum and the energy spectrum for the extremal RNBH. The area spectrum derived from the mass spectrum finds agreement with a previous result by Bekenstein. The stability of these solutions, described with the Majorana approach to the Archaic Universe scenario, shows the existence of oscillatory regimes or exponential damping for the evolution of a small perturbation from a stable state.