Energy-Morawetz estimates for the wave equation in perturbations of Kerr

TL;DR

This paper establishes energy and Morawetz estimates for scalar wave solutions in perturbed Kerr spacetimes, advancing the understanding of black hole stability by addressing low frequency control in a full subextremal range.

Contribution

It introduces a global energy-Morawetz estimate for wave equations in Kerr perturbations, crucial for proving nonlinear stability of Kerr black holes.

Findings

Proves energy and Morawetz estimates in Kerr perturbations.

Provides a framework for extending Kerr stability results.

Addresses low frequency control using microlocal multipliers.

Abstract

In this paper, we prove energy and Morawetz estimates for solutions to the scalar wave equation in spacetimes with metrics that are perturbations, compatible with nonlinear applications, of Kerr metrics in the full subextremal range. Central to our approach is the proof of a global in time energy-Morawetz estimate conditional on a low frequency control of the solution using microlocal multipliers adapted to the $r$-foliation of the spacetime. This result constitutes a first step towards extending the current proof of Kerr stability in \cite{GCM1} \cite{GCM2} \cite{KS:Kerr} \cite{GKS} \cite{Shen}, valid in the slowly rotating case, to a complete resolution of the black hole stability conjecture, i.e., the statement that the Kerr family of spacetimes is nonlinearly stable for all subextremal angular momenta.