The Quantum Black Hole as a Gravitational Hydrogen Atom

Abstract

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

Stretched Horizon as a Quantum Gravity Beam Splitter

“…This holds, at least to a good degree of approximation, even though spacetime shearing may complicate this Huygens' principle. For vacuum spacetimes the self-dual and anti self-dual Weyl curvatures in complex [9] variables preserve the phase space volume in general relativity [10]. This holds if the shear curvature is conformal.…”

Time-covariant Schrödinger equation and invariant decay probability: the $$\Lambda $$-Kantowski–Sachs universe

“…where we substituted the as defined from (52). This form of solution mimics the classical one (47) for c 2 = −c 1 and…”

“…Quantization in a larger space through an Eisenhart lift was recently employed in [51]. An analogy between the Schwarzschild-Reissner-Nordström black holes and hydrogen atoms were drawn in [52,53], where their energy spectrum was obtained based on some constructed Schrödinger type of equation. A study related to a mass operator and the existence of non-zero mass uncertainty can be found in [54,55].…”

“Time”-Covariant Schrödinger Equation and the Canonical Quantization of the Reissner–Nordström Black Hole

“…The authors studied the contributions to the Bekenstein black hole entropy due to quantum corrections. Yet another different approach can be found in [48,49], where a Schrödinger-type equation was found for the Schwarzschild and Reissner-Nordström black holes correspondingly. The quantum black holes were treated as a gravitational hydrogen atom, and their energy spectrum was found.…”