The wave equation on axisymmetric stationary black hole backgrounds

TL;DR

This paper reviews recent mathematical advances in understanding linear wave behavior on black hole backgrounds, including boundedness and decay properties, with implications for stability in general relativity.

Contribution

It extends classical boundedness results to a broader class of axisymmetric stationary black hole spacetimes close to Schwarzschild.

Findings

Boundedness and decay theorems for linear waves on black hole backgrounds

Extension of results to slowly rotating Kerr and more general axisymmetric spacetimes

Progress towards understanding nonlinear stability of black holes

Abstract

Understanding the behaviour of linear waves on black hole backgrounds is a central problem in general relativity, intimately connected with the nonlinear stability of the black hole spacetimes themselves as solutions to the Einstein equations--a major open question in the subject. Nonetheless, it is only very recently that even the most basic boundedness and quantitative decay properties of linear waves have been proven in a suitably general class of black hole exterior spacetimes. This talk will review our current mathematical understanding of waves on black hole backgrounds, beginning with the classical boundedness theorem of Kay and Wald on exactly Schwarzschild exteriors and ending with very recent boundedness and decay theorems (proven in collaboration with Igor Rodnianski) on a wider class of spacetimes. This class of spacetimes includes in particular slowly rotating Kerr spacetimes, but in the case of the boundedness theorem is in fact much larger, encompassing general axisymmetric stationary spacetimes whose geometry is sufficiently close to Schwarzschild and whose Killing fields span the null generator of the horizon.