Decay of Solutions of the Wave Equation in the Kerr Geometry

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.

44 pages, 5 figures, minor corrections

References in corpus (1)

• Superradiant instabilities of rotating black branes and strings

Cited by in corpus (67)

• A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds
• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta
• Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
• Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole
• Exponentially growing finite energy solutions for the Klein-Gordon equation on sub-extremal Kerr spacetimes
• Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes
• Spectral decomposition of black-hole perturbations on hyperboloidal slices
• Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime
• Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime
• Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces
• Phase Space Analysis on some Black Hole Manifolds
• Resolvent estimates for normally hyperbolic trapped sets
• Asymptotics of linear waves and resonances with applications to black holes
• Stability for linearized gravity on the Kerr spacetime
• A scattering theory construction of dynamical vacuum black holes
• Non-linear stability of the Kerr-Newman-de Sitter family of charged black holes
• Superradiance on the Reissner-Nordstrom metric
• On the stability of the massive scalar field in Kerr space-time
• Errata for ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'', ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'', and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation''
• Decay Rates for Spherical Scalar Waves in the Schwarzschild Geometry
• Linear Stability of the Non-Extreme Kerr Black Hole
• Black-hole stability in non-local gravity
• Linear Waves in the Kerr Geometry: A Mathematical Voyage to Black Hole Physics
• On pointwise decay of waves
• Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces
• A sharp version of Price's law for wave decay on asymptotically flat spacetimes
• Linear Stability of Schwarzschild Spacetime: Decay of Metric Coefficients
• Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole
• Black Hole Instabilities and Exponential Growth
• A Time Independent Energy Estimate for Outgoing Scalar Waves in the Kerr Geometry
• Price's law for spin fields on a Schwarzschild background
• A spectral method for half-integer spin fields based on spin-weighted spherical harmonics
• Peeling on Kerr spacetime~:linear and non linear scalar fields
• Normally hyperbolic trapping on asymptotically stationary spacetimes
• On the relation between mathematical and numerical relativity
• Asymptotic completeness for superradiant Klein-Gordon equations and applications to the De Sitter Kerr metric
• Recovering the mass and the charge of a Reissner-Nordström black hole by an inverse scattering experiment
• Charges for linearized gravity
• Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric
• A Positive-Definite Energy Functional for Axially Symmetric Maxwell's Equations on Kerr-de Sitter Black Hole Spacetimes
• On pointwise decay of linear waves on a Schwarzschild black hole background
• Local decay of waves on asymptotically flat stationary space-times
• Price's Law on Nonstationary Spacetimes
• A new result on the Klein-Gordon equation in the background of a rotating black hole
• A Conserved Energy for Axially Symmetric Newman-Penrose-Maxwell Scalars on Kerr Black Holes
• Note on black hole no hair theorems for massive forms and spin-1/2 fields
• Local energy decay of massive Dirac fields in the 5D Myers-Perry metric
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation
• Decay of Maxwell Fields on Reissner-Nordstrøm-de Sitter Black Holes
• Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities
• Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations
• A vector field method on the distorted Fourier side and decay for wave equations with potentials
• A Positive-Definite Energy Functional for the Axisymmetric Perturbations of Kerr-Newman Black Holes
• Linear stability of Schwarzschild spacetime subject to axial perturbations
• Stability study of a model for the Klein-Gordon equation in Kerr spacetime
• Resolvent estimates with mild trapping
• Decay of the Maxwell field on the Schwarzschild manifold
• The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting
• Lectures on Linear Stability of Rotating Black Holes
• Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials
• Stability study of a model for the Klein-Gordon equation in Kerr space-time II
• Axially Symmetric Perturbations of Kerr Black Holes I: A gauge-invariant construction of ADM Energy
• The decay of the $SU(2)$ Yang-Mills fields on the Schwarzschild black hole for spherically symmetric small energy initial data
• Axially Symmetric Perturbations of Kerr Black Holes II: Boundary Behaviour of the Dynamics in the Orbit Space
• The wave equation on axisymmetric stationary black hole backgrounds
• The Wave Equation in a General Spherically Symmetric Particlelike Geometry
• An Integral Representation for the Dirac Propagator in the Reissner-Nordström Geometry in Eddington-Finkelstein Coordinates