Instability of Extremal Relativistic Charged Spheres

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-Nordström 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-Nordtröm 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 → M . In the Appendix we also argue that Hawking radiation will not lead to an extremal ReissnerNordström black hole. All our results are consistent with the third law of black hole dynamics, as currently understood. PACS: 04.70, 04.70 Bw, 97.60.Lf,