The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

TL;DR

This paper proves the linear stability of weakly charged, slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations, showing solutions decay polynomially to a Kerr-Newman state.

Contribution

It extends stability analysis to charged, rotating black holes using advanced microlocal analysis and resolvent techniques, building on prior vacuum stability frameworks.

Findings

Solutions decay at an inverse polynomial rate.

Linearized solutions approach a Kerr-Newman solution plus gauge terms.

The analysis handles low-frequency resolvent behavior.

Abstract

In this paper, we prove the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. This work builds on the framework developed by H\"{a}fner-Hintz-Vasy for the study of the Einstein vacuum equations. We work in the generalized wave map and Lorenz gauge. The proof involves the analysis of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator on asymptotically flat spaces, which relies on recent advances in microlocal analysis and non-elliptic Fredholm theory developed by Vasy. The most delicate part of the proof is the description of the resolvent at low frequencies.