Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 10

The quantum black hole as a gravitational hydrogen atom

Pages: 3437 – 3562

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n10.a4

Authors

Christian Corda (International Institute for Applicable Mathematics & Information Sciences (IIAMIS), B.M. Birla Science Centre Adarsh Nagar, Hyderabad, India; and Dipartimento di Matematica e Fisica, Istituto Livi, Prato, Italy)

Fabiano Feleppa (Institute for Theoretical Physics, Utrecht University, Utrecht, The Netherlands)

Abstract

Nathan 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 allows us 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”.

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Published 25 March 2024