Charge Fluctuations of a Schwarzschild Black-Hole

Abstract

In this paper, we consider charge fluctuations of a Schwarzschild black hole of mass $M$ in thermal equilibrium with a plasma consisting of photons, electrons, and positrons confined inside a cavity . The final result is astonishingly simple: charge fluctuations have two independent contributions: one that comes from the black hole itself, with no respect to the plasma, and another one that arises from the plasma; both are Universal, do not depend on the black hole mass and none of them depends on the charge of the charge carriers. They differ by a numerical factor of order one if we assume the cut-off parameter to be Planckian. Therefore, we regard them as physically identical. Interestingly enough, the black hole's charge (squared) fluctuations are of the order $\hbar$ which is not too far from the fine structure constant. The physical origin of charge fluctuations remains raises a puzzle. Since it does not depend upon the charge of the charge carriers, it cannot be regarded as resulting from the trade of charges between the black hole and the quantum fields and is truly of quantum gravity nature (albeit in a semi-classical approximation). There are at least two possible ways of interpreting our results: either charge is trapped inside the black hole and the average must be regarded over an ensemble of black-holes or eventuality charge fluctuations arise from particles crossing the Cauchy horizon $\overline{r_-} \sim \hbar/2M$ in both directions.