An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

TL;DR

This paper develops an integral spectral representation for the scalar wave propagator in Kerr spacetime, enabling detailed analysis of wave dynamics near rotating black holes.

Contribution

It introduces a new integral representation of the wave solution in Kerr geometry and proves the completeness of the separated ODE solutions.

Findings

Integral representation of the wave propagator derived

Completeness of separated ODE solutions established

Provides a foundation for analyzing long-term wave behavior in Kerr spacetime

Abstract

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry.