Charge Fluctuations of an Uncharged Black Hole

Abstract

In this paper we calculate charge fluctuations of a Schwarzschild black-hole of mass $M$ confined within a perfectly reflecting cavity of radius R in thermal equilibrium with various species of radiation and fermions . Charge conservation is constrained by a Lagrange multiplier (the chemical potential). Black hole charge fluctuations are expected owing to continuous absorption and emission of particles by the black hole. For black holes much more massive than $10^{16} g$ , these fluctuations are exponentially suppressed. For black holes lighter than this, the Schwarzschild black hole is unstable under charge fluctuations for almost every possible size of the confining vessel. The stability regime and the fluctuations are calculated through the second derivative of the entropy with respect to the charge. The expression obtained contains many puzzling terms besides the expected thermodynamical fluctuations: terms corresponding to instabilities that do not depend on the specific value of charge of the charge carriers and one of them depends on Newton's constant instead. One of the contributions to the charge fluctuations $\hbar/4\pi$ does not depend neither on number of species, nor on the the specific charge or even the size of the confining vessel. As a matter of fact, this term emerges from the second derivative of the black hole entropy alone, which means that it corresponds to a genuine quantum mechanical property of the black hole itself. Such a contribution would cause the event horizon to recede from $2M$ to $2M-T_{BH}$ or equivalently, by $(4\pi)^{-1}$ of the black hole' s Compton wave length. Similarly, a Cauchy horizon emerges at the same distance the event horizon receded.