Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

The Quantum Black Hole as a Gravitational Hydrogen Atom

Version 1 : Received: 12 October 2018 / Approved: 18 October 2018 / Online: 18 October 2018 (10:39:33 CEST)
Version 2 : Received: 22 November 2018 / Approved: 23 November 2018 / Online: 23 November 2018 (14:29:57 CET)
Version 3 : Received: 5 December 2019 / Approved: 6 December 2019 / Online: 6 December 2019 (04:14:35 CET)

How to cite: Corda, C.; Feleppa, F. The Quantum Black Hole as a Gravitational Hydrogen Atom. Preprints 2018, 2018100413 (doi: 10.20944/preprints201810.0413.v3). Corda, C.; Feleppa, F. The Quantum Black Hole as a Gravitational Hydrogen Atom. Preprints 2018, 2018100413 (doi: 10.20944/preprints201810.0413.v3).


In this paper only one basic assumption has been made: if we try to describe black holes, their behavior should be understood in the same language as the one we use for particles; black holes should be treated just like atoms. They must be quantum forms of matter, moving in accordance with Schrödinger equations just like everything else. In particular, Rosen’s quantization approach to the gravitational collapse is applied in the simple case of a pressureless “star of dust” by finding the gravitational potential, the Schrödinger equation and the solution for the collapse’s energy levels. By applying the constraints for a Schwarzschild black hole (BH) and by using the concept of BH effective state, previously introduced by one of the authors (CC), the BH quantum gravitational potential, Schrödinger equation and the BH energy spectrum are found. Remarkably, such an energy spectrum is in agreement (in its absolute value) with the one which was conjectured by Bekenstein in 1974 and consistent with other ones in the literature. This approach also permits to find an interesting quantum representation of the Schwarzschild BH ground state at the Planck scale. Moreover, two fundamental issues about black hole quantum physics are addressed by this model: the area quantization and the singularity resolution. As regards the former, a result similar to the one obtained by Bekenstein, but with a different coefficient, has been found. About the latter, it is shown that the traditional classical singularity in the core of the Schwarzschild BH is replaced, in a full quantum treatment, by a two-particle system where the two components strongly interact with each other via a quantum gravitational potential. The two-particle system seems to be non-singular from the quantum point of view and is analogous to the hydrogen atom because it consists of a “nucleus” and an “electron”.


gravitational atom; quantum black hole; quantum levels; Schrodinger equation; mass spectrum


PHYSICAL SCIENCES, Astronomy & Astrophysics

Comments (1)

Comment 1
Received: 6 December 2019
Commenter: Christian Corda
Commenter's Conflict of Interests: Author
Comment: The title is changed and computations and discussion are improved
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