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 → 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.

Sharp Bounds on the Critical Stability Radius for Relativistic Charged Spheres

“…One possibility to obtain an extremal black hole is to produce one from the collapse of an already extremal object. Previous mainly numerical studies ( [12], [7]) have concluded that when Q < M collapse always takes place at a critical radius R c outside the outer horizon, and as Q approaches M, this value approaches the horizon. This is similar to the non-charged case where the Buchdahl inequality implies that collapse will take place when R < 9M/4, i.e., R c = 9M/4, cf.…”

“…In the charged case the critical value is expected to be smaller due to the Coulomb repulsion, and this is also shown to be the case below and in particular as Q → M the stability radius does approach the outer horizon. For more information on the relation of this topic to extremal black holes and black hole thermodynamics we refer to [7], [16], [13] and [11] and the references therein.…”

Numeric simulation of relativistic stellar core collapse and the formation of Reissner-Nordström black holes

“…On the other hand, the formation of Reissner-Nordström (RN) black holes, is in connection with the third law of black hole thermodynamics [2], [17], [14], because an extreme RN black hole has zero temperature. So, there is an interest on the formation of charged black holes from a pure theoretical point of view [49].…”

Nonadiabatic charged spherical gravitational collapse

“…Although some of these works refer to static situations ( [3]- [5], [9,19,23,24,29,30], [32]- [34], [36,37,39,41]) there have been important efforts in describing dynamical situations too ( [6]- [8], [10]- [14], [16,17,21,22,26,40,42]). Particularly relevant for the present paper are references [14,22] and [40].…”