Area spectrum of the Schwarzschild black hole

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.Comment: 30 pages, REVTeX v3.0. (Minor additions, several added references.