The interior of dynamical extremal black holes in spherical symmetry

TL;DR

This paper investigates the nonlinear stability of the Cauchy horizon inside extremal Reissner-Nordström black holes under spherical symmetry, showing that solutions can extend beyond the horizon with certain regularity, unlike in the subextremal case.

Contribution

It demonstrates that under specific conditions, solutions extend beyond the Cauchy horizon with regularity, revealing non-uniqueness and stability properties in extremal black hole interiors.

Findings

Solutions extend beyond the Cauchy horizon with $C^{0,1/2}$ and $W^{1,2}_{loc}$ regularity.

Existence of non-unique spherically symmetric extensions solving Einstein-Maxwell-Klein-Gordon.

In the chargeless, massless scalar field case, extensions can keep the scalar field Lipschitz.

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.