Universal charge-mass relation: From black holes to atomic nuclei

Abstract

The cosmic censorship hypothesis, introduced by Penrose forty years ago, is one of the corner stones of general relativity. This conjecture asserts that spacetime singularities that arise in gravitational collapse are always hidden inside of black holes. The elimination of a black-hole horizon is ruled out by this principle because that would expose naked singularities to distant observers. We test the consistency of this prediction in a gedanken experiment in which a charged object is swallowed by a charged black hole. We find that the validity of the cosmic censorship conjecture requires the existence of a charge-mass bound of the form $q\leqμ^{2/3}E^{-1/3}_c$, where $q$ and $μ$ are the charge and mass of the physical system respectively, and $E_c$ is the critical electric field for pair-production. Applying this bound to charged atomic nuclei, one finds an upper limit on the number $Z$ of protons in a nucleus of given mass number $A$: $Z\leq Z^*=α^{-1/3}A^{2/3}$, where $α=e^2/\hbar$ is the fine structure constant. We test the validity of this novel bound against the $(Z,A)$-relation of atomic nuclei as deduced from the Weizsäcker semi-empirical mass formula.