Quantum oscillations in the black hole horizon

Abstract

By applying Rosen's quantization approach to the historical Oppenheimer and Snyder gravitational collapse and by setting the constraints for the formation of the Schwarzschild black hole (SBH), in a previous paper [1] two of the Authors (CC and FF) found the gravitational potential, the Schrödinger equation, the solution for the energy levels, the area quantum and the quantum representation of the ground state at the Planck scale of the SBH. Such results are consistent with previous ones in the literature. It was also shown that the traditional classical singularity in the core of the SBH is replaced by a quantum oscillator describing a non-singular two-particle system where the two components, named the "nucleus" and the "electron", strongly interact with each other through a quantum gravitational interaction. In agreement with the de Broglie hypothesis, the "electron" is interpreted in terms of the quantum oscillations of the BH horizon. In other words, the SBH should be the gravitational analogous of the hydrogen atom. In this paper, it is shown that these results allow us to compute the SBH entropy as a function of the BH principal quantum number in terms of Bekenstein-Hawking entropy and three sub-leading corrections. In addition, the coefficient of the formula of Bekenstein-Hawking entropy is reduced to a quarter of the traditional value. Then, it is shown that, by performing a correct rescaling of the energy levels, the semi-classical Bohr-like approach to BH quantum physics, previously developed by one of the Authors (CC), is consistent with the obtained results for large values of the BH principal quantum number. After this, Hawking radiation will be analysed by discussing its connection with the BH quantum structure. Finally, it is shown that the time evolution of the above mentioned system solves the BH information paradox.