Spectrum of Charged Black Holes - The Big Fix Mechanism Revisited

TL;DR

This paper derives a discrete area spectrum for charged black holes using thermodynamics and quantum mechanics, revealing a relation between the fine structure constant and black hole parameters, and confirming the area as an adiabatic invariant.

Contribution

It introduces a new derivation of the black hole area spectrum incorporating charge and quantum effects, and links fundamental constants to black hole properties.

Findings

Discrete area spectrum for charged black holes derived

Relation between fine structure constant and black hole parameters established

Horizon area confirmed as an adiabatic invariant

Abstract

Following an earlier suggestion of the authors(gr-qc/9607030), we use some basic properties of Euclidean black hole thermodynamics and the quantum mechanics of systems with periodic phase space coordinate to derive the discrete two-parameter area spectrum of generic charged spherically symmetric black holes in any dimension. For the Reissner-Nordstrom black hole we get $A/4G\hbar=\pi(2n+p+1)$, where the integer p=0,1,2,.. gives the charge spectrum, with $Q=\pm\sqrt{\hbar p}$. The quantity $\pi(2n+1)$, n=0,1,... gives a measure of the excess of the mass/energy over the critical minimum (i.e. extremal) value allowed for a given fixed charge Q. The classical critical bound cannot be saturated due to vacuum fluctuations of the horizon, so that generically extremal black holes do not appear in the physical spectrum. Consistency also requires the black hole charge to be an integer multiple of any fundamental elementary particle charge: $Q= \pm me$, m=0,1,2,.... As a by-product this yields a relation between the fine structure constant and integer parameters of the black hole -- a kind of the Coleman big fix mechanism induced by black holes. In four dimensions, this relationship is $e^2/\hbar=p/m^2$ and requires the fine structure constant to be a rational number. Finally, we prove that the horizon area is an adiabatic invariant, as has been conjectured previously.