Vol. 1, No. 2, 2019

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The interior of dynamical extremal black holes in spherical symmetry

Dejan Gajic and Jonathan Luk

Vol. 1 (2019), No. 2, 263–326

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 C0,1 2 Wloc1,2, in contrast to the subextremal case (where generically the solution is C0 (C0,1 2 Wloc1,2)). 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.

extremal black hole, black hole interior, Cauchy horizon, strong cosmic censorship conjecture
Mathematical Subject Classification 2010
Primary: 35L51, 83C05, 83C57, 83C75
Received: 12 January 2019
Accepted: 12 February 2019
Published: 20 April 2019
Dejan Gajic
Department of Mathematics
South Kensington Campus
Imperial College London
United Kingdom
Jonathan Luk
Department of Mathematics
Stanford University
Stanford, CA
United States