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-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 $Q\to M$. In the Appendix we also argue that Hawking radiation will not lead to an extremal Reissner-Nordström black hole. All our results are consistent with the third law of black hole dynamics, as currently understood.

References in corpus (3)

• One-Loop Corrected Thermodynamics of the Extremal and Non-Extremal Spinning BTZ Black Hole
• Nonthermal nature of incipient extremal black holes
• Nonthermal nature of extremal Kerr black holes

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