Non-linear stability of the Kerr-Newman-de Sitter family of charged black holes

TL;DR

This paper proves the nonlinear stability of slowly rotating charged black holes in de Sitter space, showing that small perturbations decay exponentially and the solution approaches a member of the Kerr-Newman-de Sitter family.

Contribution

It establishes the first proof of nonlinear stability for charged black holes in de Sitter space without symmetry assumptions, extending previous vacuum results.

Findings

Exponential decay of metric and electromagnetic field perturbations.

Automatic determination of final black hole parameters and gauge.

First proof of linear mode stability for slowly rotating Kerr-Newman-de Sitter black holes.

Abstract

We prove the global non-linear stability, without symmetry assumptions, of slowly rotating charged black holes in de Sitter spacetimes in the context of the initial value problem for the Einstein-Maxwell equations: If one perturbs the initial data of a slowly rotating Kerr-Newman-de Sitter (KNdS) black hole, then in a neighborhood of the exterior region of the black hole, the metric and the electromagnetic field decay exponentially fast to their values for a possibly different member of the KNdS family. This is a continuation of recent work of the author with Vasy on the stability of the Kerr-de Sitter family for the Einstein vacuum equations. Our non-linear iteration scheme automatically finds the final black hole parameters as well as the gauge in which the global solution exists; we work in a generalized wave coordinate/Lorenz gauge, with gauge source functions lying in a suitable finite-dimensional space. We include a self-contained proof of the linear mode stability of Reissner-Nordstr\"om-de Sitter black holes, building on work by Kodama-Ishibashi. In the course of our non-linear stability argument, we also obtain the first proof of the linear (mode) stability of slowly rotating KNdS black holes using robust perturbative techniques.