Schrödinger and Klein–Gordon theories of black holes from the quantization of the Oppenheimer and Snyder gravitational collapse

Abstract

The Schrödinger equation of the Schwarzschild black hole (BH) has been recently derived by the author and collaborators. The BH is composed of a particle, the ‘electron’, interacting with a central field, the ‘nucleus’. Via de Broglie’s hypothesis, one interprets the ‘electron’ in terms of BH horizon’s modes. Quantum gravity effects modify the BH semi-classical structure at the Schwarzschild scale rather than at the Planck scale. The analogy between this BH Schrödinger equation and the Schrödinger equation of the s states of the hydrogen atom permits us to solve the same equation. The quantum gravitational quantities analogous of the fine structure constant and of the Rydberg constant are not constants, but the dynamical quantities have well-defined discrete spectra. The spectrum of the ‘gravitational fine structure constant’ is the set of non-zero natural numbers. Therefore, BHs are well-defined quantum gravitational systems obeying Schrödinger’s theory: the ‘gravitational hydrogen atoms’. By identifying the potential energy in the BH Schrödinger equation as being the gravitational energy of a spherically symmetric shell, a different nature of the quantum BH seems to surface. BHs are self-interacting, highly excited, spherically symmetric, massive quantum shells generated by matter condensing on the apparent horizon, concretely realizing the membrane paradigm. The quantum BH described as a ‘gravitational hydrogen atom’ is a fictitious mathematical representation of the real, quantum BH, a quantum massive shell having a radius equal to the oscillating gravitational radius. Nontrivial consequences emerge from this result: (i) BHs have neither horizons nor singularities; (ii) there is neither information loss in BH evaporation, nor BH complementarity, nor firewall paradox. These results are consistent with previous ones by Hawking, Vaz, Mitra and others. Finally, the special relativistic corrections to the BH Schrödinger equation give the BH Klein–Gordon equation and the corresponding eigenvalues.