Decay of Solutions of the Wave Equation in the Kerr Geometry

TL;DR

This paper proves that solutions to the scalar wave equation in Kerr black hole spacetime decay over time, using a mode decomposition and integral representations involving radial and angular ODEs.

Contribution

It introduces a method to demonstrate decay of wave solutions in Kerr geometry through mode decomposition and integral representations involving radial and angular equations.

Findings

Solutions decay in time in local L^ abla_ ext{loc} norm.

Representation as sum over angular momentum modes.

Uses solutions of radial and angular ODEs for analysis.

Abstract

We consider the Cauchy problem for the scalar wave equation in the Kerr geometry for smooth initial data supported outside the event horizon. We prove that the solutions decay in time in L^\infty_loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables.