The black hole stability problem for linear scalar perturbations

TL;DR

This paper reviews recent progress on the linear stability of Kerr black holes under scalar perturbations, establishing boundedness and decay results that are crucial for understanding their potential nonlinear stability.

Contribution

It provides the first comprehensive proof of decay bounds for scalar wave solutions on the full subextremal Kerr spacetime.

Findings

Boundedness of scalar wave solutions established

Decay of solutions proven for slowly rotating Kerr

Definitive decay bounds for the general subextremal range obtained

Abstract

We review our recent work on linear stability for scalar perturbations of Kerr spacetimes, that is to say, boundedness and decay properties for solutions of the scalar wave equation \Box_g{\psi} = 0 on Kerr exterior backgrounds. We begin with the very slowly rotating case |a| \ll M, where first boundedness and then decay has been shown in rapid developments over the last two years, following earlier progress in the Schwarzschild case a = 0. We then turn to the general subextremal range |a| < M, where we give here for the first time the essential elements of a proof of definitive decay bounds for solutions {\psi}. These developments give hope that the problem of the non-linear stability of the Kerr family of black holes might soon be addressed. This paper accompanies a talk by one of the authors (I.R.) at the 12th Marcel Grossmann Meeting, Paris, June 2009.