Area spectrum of the Schwarzschild black hole

TL;DR

This paper develops a Hamiltonian quantum gravity model for the Schwarzschild black hole, revealing a discrete, evenly spaced area spectrum consistent with Bekenstein's proposal, and extends results to charged and cosmological constant cases.

Contribution

It introduces a Hamiltonian quantization of Schwarzschild black holes with a discrete area spectrum matching Bekenstein's conjecture, including extensions to charged and cosmological constant scenarios.

Findings

The Hamiltonian spectrum is discrete and bounded below.

Large eigenvalues asymptotically behave as √(2k).

The area spectrum aligns with Bekenstein's proposal.

Abstract

We consider a Hamiltonian theory of spherically symmetric vacuum Einstein gravity under Kruskal-like boundary conditions in variables associated with the Einstein-Rosen wormhole throat. The configuration variable in the reduced classical theory is the radius of the throat, in a foliation that is frozen at the left hand side infinity but asymptotically Minkowski at the right hand side infinity, and such that the proper time at the throat agrees with the right hand side Minkowski time. The classical Hamiltonian is numerically equal to the Schwarzschild mass. Within a class of Hamiltonian quantizations, we show that the spectrum of the Hamiltonian operator is discrete and bounded below, and can be made positive definite. The large eigenvalues behave asymptotically as~$\sqrt{2k}$, where $k$ is an integer. The resulting area spectrum agrees with that proposed by Bekenstein and others. Analogous results hold in the presence of a negative cosmological constant and electric charge. The classical input that led to the quantum results is discussed.