On the quantization of the extremal Reissner-Nordstrom black hole

Abstract

Following Rosen's quantization rules, two of the Authors (CC and FF) 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-Nordstrom 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, show the existence of oscillatory regimes or exponential damping for the evolution of a small perturbation from a stable state.

Time-covariant Schrödinger equation and invariant decay probability: The $Λ$-Kantowski-Sachs universe

“…Quantization in a larger space through an Eisenhart lift was recently employed in [50]. An analogy between the Schwarzschild-Reissner-Nordström black holes and hydrogen atoms were drawn in [51,52], where their energy spectrum was obtained based on some constructed Schrödinger type equation. A study related to a mass operator and existence of non-zero mass uncertainty can be found here [53,54].…”

Schrodinger theory of black holes

“…Bekenstein was the first physicist who observed that, in some respects, BHs play the same role in gravitation that atoms played in the nascent quantum mechanics [1]. This analogy [2] implies that BH energy could have a discrete spectrum [1]. Hence, BHs combine in some sense both the "hydrogen atom" and the "quasithermal" emission in quantum gravity [3].…”

“…Hence, BHs combine in some sense both the "hydrogen atom" and the "quasithermal" emission in quantum gravity [3]. As a consequence, BH quantization could be the key to a quantum theory of gravity [1][2][3]. Thus, starting from the pioneering works of Bekenstein [4] and Hawking [5], BH quantization became, and currently remains, one of the most important research fields in theoretical physics of the last 50 years.…”

“…In this framework, the Author developed a semiclassical approach to BH quantization [20,21] somewhat similar to the historical semi-classical approach to the structure of a hydrogen atom introduced by Bohr in 1913 [22,23]. Recently, the Author and collaborators [2] improved the analysis by adapting an approach to quantization of the famous collaborator of Einstein, N. Rosen [24] to the historical Oppenheimer and Snyder gravitational collapse [25]. The approach in [2] seems consistent with a similar one of Hajicek and Kiefer [12,13], and permits to write down, explicitly, the gravitational potential and the Schrodinger equation for the SBH as (hereafter Planck units will be used, i.e.…”

“…Thus, at the quantum level the SBH is interpreted as a two-particle system where the two components strongly interact with each other through the quantum gravitational interaction of Eq. ( 1) [2]. The two-particle system BH seems to be nonsingular from the quantum point of view and is analogous to the hydrogen atom because it consists of a "nucleus" and an "electron" [2].…”