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

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Johns Hopkins University
The Einstein-Maxwell system describes the interaction of gravity and electromagnetism. In Einstein-Maxwell system, Einstein's field equation describes the relation between the geometry of the spacetime and the energy momentum of an electromagnetic field, and Maxwell's equation describes how electromagnetic waves propagate in the spacetime. Among the most interesting solutions to Einstein-Maxwell equations are the families of black hole solutions. The Kerr-Newman family of solutions describes stationary, charged, and rotating black holes. In this thesis, 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 in \cite{HHV21} 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 in \cite{Vas13}. The most delicate part of the proof is the description of the resolvent at low frequencies.
Einstein-Maxwell equations, Kerr-Newman black holes