The interior of dynamical extremal black holes in spherical symmetry

Abstract

We study the nonlinear stability of the Cauchy horizon in the interior of extremal Reissner–Nordström black holes under spherical symmetry. We consider the Einstein–Maxwell–Klein–Gordon system such that the charge of the scalar field is appropriately small in terms of the mass of the background extremal Reissner–Nordström black hole. Given spherically symmetric characteristic initial data which approach the event horizon of extremal Reissner–Nordström sufficiently fast, we prove that the solution extends beyond the Cauchy horizon in $C^{0,\frac{1}{2}}\cap W_{loc}^{1,2}$, in contrast to the subextremal case (where generically the solution is $C^0\setminus \left(C^{0,\frac{1}{2}}\cap W_{loc}^{1,2}\right)$). In particular, there exist nonunique spherically symmetric extensions which are moreover solutions to the Einstein–Maxwell–Klein–Gordon system. Finally, in the case that the scalar field is chargeless and massless, we additionally show that the extension can be chosen so that the scalar field remains Lipschitz.