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

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.

41 pages, 4 figures, minor corrections

References in corpus (4)

• Decay of Solutions of the Wave Equation in the Kerr Geometry
• The Long-Time Dynamics of Dirac Particles in the Kerr-Newman Black Hole Geometry
• On the stability of the Kerr metric
• Spectral Estimates and Non-Selfadjoint Perturbations of Spheroidal Wave Operators

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• Decay of Solutions of the Wave Equation in the Kerr Geometry
• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• Spectral decomposition of black-hole perturbations on hyperboloidal slices
• Decay Rates for Spherical Scalar Waves in the Schwarzschild Geometry
• Linear Stability of the Non-Extreme Kerr Black Hole
• Linear Waves in the Kerr Geometry: A Mathematical Voyage to Black Hole Physics
• Creation of fermions by rotating charged black-holes
• Spectral Estimates and Non-Selfadjoint Perturbations of Spheroidal Wave Operators
• A Time Independent Energy Estimate for Outgoing Scalar Waves in the Kerr Geometry
• A Rigorous Treatment of Energy Extraction from a Rotating Black Hole
• The Cauchy Problem for the Wave Equation in the Schwarzschild Geometry
• Recovering the mass and the charge of a Reissner-Nordström black hole by an inverse scattering experiment
• A Positive-Definite Energy Functional for Axially Symmetric Maxwell's Equations on Kerr-de Sitter Black Hole Spacetimes
• A Conserved Energy for Axially Symmetric Newman-Penrose-Maxwell Scalars on Kerr Black Holes
• Refined Error Estimates for the Riccati Equation with Applications to the Angular Teukolsky Equation
• Stationary Spacetimes and Self-Adjointness in Klein-Gordon Theory
• Decay of Maxwell Fields on Reissner-Nordstrøm-de Sitter Black Holes
• A Positive-Definite Energy Functional for the Axisymmetric Perturbations of Kerr-Newman Black Holes
• Lectures on Linear Stability of Rotating Black Holes
• Axially Symmetric Perturbations of Kerr Black Holes II: Boundary Behaviour of the Dynamics in the Orbit Space