Towards the mass spectrum of quantum black holes and wormholes

TL;DR

This paper develops a quantum model for black holes and wormholes using a self-gravitating dust shell, deriving discrete spectra and revealing quantum hairs that influence black hole entropy.

Contribution

It introduces a finite difference Schroedinger equation approach and finds discrete spectra depending on two quantum numbers, highlighting quantum hairs and their role in black hole entropy.

Findings

Discrete spectra depend on two quantum numbers.

Quantum hairs exist at Planckian distances.

Black hole entropy relates to non-measurable quantum parameters.

Abstract

The quantum black hole model with a self-gravitating spherically symmetric thin dust shell as a source is considered. The shell Hamiltonian constraint is written and the corresponding Schroedinger equation is obtained. This equation appeared to be a finite differences equation. Its solutions are required to be analytic functions on the relevant Riemannian surface. The method of finding discrete spectra is suggested based on the analytic properties of the solutions. The large black hole approximation is considered and the discrete spectra for bound states of quantum black holes and wormholes are found. They depend on two quantum numbers and are, in fact, quasi-continuous. The quantum black hole bound state depends not only on mass but also on additional quantum number, and black holes with the same mass have different quantum hairs. These hairs exhibit themselves at the Planckian distances near the black hole horizon. For the observer who can not measure the distances smaller than the Planckian length the black hole has the only parameter, its mass. The other, non-measurable parameter leads to existence of the black hole entropy.