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, 1 2 ∩ W 1,2 loc , in contrast to the subextremal case (where generically the solution is C 0 \ (C 0, 1 2 ∩ W 1,2 loc )). In particular, there exist non-unique 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. * and the solution cannot be extended as a spherically symmetric solution to the Einstein-Maxwell-scalar field system 2 .Remark 1.3 (Regularity of the metric and extensions as solutions to (1.1)). The fact that we can extend the solutions beyond the Cauchy horizon is intimately connected to the regularity of the solutions up to the Cauchy horizon. In particular this relies on the fact the metric, the scalar field and the electromagnetic potential remain in (spacetime) C 0 ∩ W 1,2 loc . In fact, the solutions are at a level of regularity for which the Einstein equations are still locally well-posed 3 . One can therefore construct extensions beyond the Cauchy horizon by solving appropriate characteristic initial value problems; see Section 10.In this connection, note that we emphasised in the statement of the theorem that the solution can be extended beyond the Cauchy horizon as a spherically symmetric solution to (1.1). The emphasis on the spherical symmetry of the extension is made mostly to contrast with the situation in the subextremal case (cf. Remark 1.2). This should not be understood as implying that the extensions necessarily are spherically symmetric: In fact, with the bounds that we establish in this paper, one can in principle construct using the techniques in [23] extensions (still as solutions to (1.1)) without any symmetry assumptions (cf. Footnote 1).Remark 1.4 (Assumptions on the event horizon). The assumptions we impose on the event horizon are consistent with the expected late-time behavior of the solutions in the exterior region of the black hole, at least in the e = m = 0 case if one extrapolates from numerical results [25]. In particular, the transversal derivative of the scalar field is not required to decay along the event horizon, and is therefore consistent with the Aretakis instability [2]. Of course, in order to completely understand the structure of the interior, one needs to prove that the decay estimates along the event horizon indeed hold for general dynamical solutions approaching these extremal black holes. This remains a...

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