A Quantum Mechanical Model of the Reissner-Nordstrom Black Hole

TL;DR

This paper develops a quantum model for Reissner-Nordstrom black holes, revealing discrete spectra for mass and charge, and connecting these to Bekenstein's horizon area quantization proposal.

Contribution

It introduces a Hamiltonian quantum framework for charged black holes, deriving their discrete mass and charge spectra and relating them to horizon area quantization.

Findings

Mass and charge spectra are discrete and bounded below.

The spectrum of M^2 - Q^2 is strictly positive.

Large eigenvalues of sqrt(M^2 - Q^2) are proportional to sqrt(2n).

Abstract

We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the Reissner-Nordstr\"{o}m black hole, together with the corresponding canonical momenta. In this four-dimensional phase space, we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordstr\"{o}m black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator, and an eigenvalue equation for the ADM mass of the hole, from the point of view of a distant observer at rest, is obtained. Our eigenvalue equation implies that the ADM mass and the electric charge spectra of the hole are discrete, and the mass spectrum is bounded below. Moreover, the spectrum of the quantity $M^2-Q^2$ is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of the quantity $\sqrt{M^2-Q^2}$ are of the form $\sqrt{2n}$, where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes.