Instability of extremal relativistic charged spheres

Abstract

With the question ``Can relativistic charged spheres form extremal black holes?'' in mind, we investigate the properties of such spheres from a classical point of view. The investigation is carried out numerically by integrating the Oppenheimer-Volkov equation for relativistic charged fluid spheres and finding interior Reissner-Nordstr\"om solutions for these objects. We consider both constant density and adiabatic equations of state, as well as several possible charge distributions, and examine stability by both a normal mode and an energy analysis. In all cases, the stability limit for these spheres lies between the extremal $(Q=M)$ limit and the black hole limit ${(R=R}_{+}).$ That is, we find that charged spheres undergo gravitational collapse before they reach $Q=M,$ suggesting that extremal Reissner-Nordstr\"om black holes produced by collapse are ruled out. A general proof of this statement would support a strong form of the cosmic censorship hypothesis, excluding not only stable naked singularities, but stable extremal black holes. The numerical results also indicate that although the interior mass-energy $m(R)$ obeys the usual $m/R<4/9$ stability limit for the Schwarzschild interior solution, the gravitational mass M does not. Indeed, the stability limit approaches ${R}_{+}$ as $Q\ensuremath{\rightarrow}M.$ In the Appendix we also argue that Hawking radiation will not lead to an extremal Reissner-Nordstr\"om black hole. All our results are consistent with the third law of black hole dynamics, as currently understood.