A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

TL;DR

This paper proves that solutions to the wave equation remain uniformly bounded on a broad class of slowly rotating Kerr and related black hole spacetimes, including near the event horizon, without relying on separability.

Contribution

It establishes uniform boundedness of wave solutions on general stationary axisymmetric black hole backgrounds close to Schwarzschild, without using separability properties.

Findings

Solutions are uniformly bounded on Kerr and similar spacetimes.

Bound holds up to the event horizon without unphysical restrictions.

Bound derived from positive definite energy flux.

Abstract

We consider Kerr spacetimes with parameters a and M such that |a|<< M, Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M>0, with appropriate geometric assumptions on the plane spanned by the Killing fields. We show uniform boundedness on the exterior for sufficiently regular solutions to the scalar homogeneous wave equation. In particular, the bound holds up to and including the event horizon. No unphysical restrictions are imposed on the behaviour of the solution near the bifurcation surface of the event horizon. The pointwise estimate derives in fact from the uniform boundedness of a positive definite energy flux. Note that in view of the very general assumptions, the separability properties of the wave equation on the Kerr background are not used.