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arXiv · 1709.09137

The interior of dynamical extremal black holes in spherical symmetry

Dejan Gajic·Jonathan Luk
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Abstract

We study the nonlinear stability of the Cauchy horizon in the interior of extremal Reissner-Nordstr\"om 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-Nordstr\"om black hole. Given spherically symmetric characteristic initial data which approach the event horizon of extremal Reissner-Nordstr\"om sufficiently fast, we prove that the solution extends beyond the Cauchy horizon in $C^{0,\frac{1}{2}}\cap W^{1,2}_{loc}$, in contrast to the subextremal case (where generically the solution is $C^0\setminus (C^{0,\frac{1}{2}}\cap W^{1,2}_{loc}))$. 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.

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Dejan Gajic, Jonathan Luk. 2017-09-26. The interior of dynamical extremal black holes in spherical symmetry. https://doi.org/10.2140/paa.2019.1.263

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