Phase Space Analysis on some Black Hole Manifolds

Abstract

The Schwarzschild and Reissner-Nordstrom solutions to Einstein's equations describe space- times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space- time of this type. We show that for solutions with initial data which decay at infinite, a weighted $L^6$ norm in space decays like $t^{-1/3}$. This weight vanishes at the event horizon, but not at infinite.

63 pages (correction to Morawetz multiplier- 14 August 2006)

References in corpus (8)

• Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space
• Decay of Solutions of the Wave Equation in the Kerr Geometry
• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• A note on energy currents and decay for the wave equation on a Schwarzschild background
• 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''
• Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole
• A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold
• The red-shift effect and radiation decay on black hole spacetimes

Cited by in corpus (39)

• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I
• The global non-linear stability of the Kerr-de Sitter family of black holes
• Slowly Decaying Waves on Spherically Symmetric Spacetimes and an Instability of Ultracompact Neutron Stars
• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II
• A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds
• Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space
• Strichartz estimates on Schwarzschild black hole backgrounds
• Gravitational instability of an extreme Kerr black hole
• 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
• Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces
• A note on instabilities of extremal black holes under scalar perturbations from afar
• Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds
• Characterisation of the Energy of Gaussian Beams on Lorentzian Manifolds - with Applications to Black Hole Spacetimes
• Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity
• 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 of solutions to the Maxwell equation on the Schwarzschild background
• On pointwise decay of waves
• Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces
• Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole I: Maxwell field
• Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole
• The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Stability and Instability Results
• Radiation fields on Schwarzschild spacetime
• Horizon instability of massless scalar perturbations of an extreme Reissner-Nordström-AdS black hole
• Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior
• A local energy estimate for wave equations on metrics asymptotically close to Kerr
• Abstract theory of decay estimates: perturbed Hamiltonians
• Late-time asymptotics for geometric wave equations with inverse-square potentials
• Monotonic Local Decay Estimates
• Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes
• Decay of Maxwell Fields on Reissner-Nordstrøm-de Sitter Black Holes
• Localized energy estimates for wave equations on high dimensional Schwarzschild space-times
• Decay of the Maxwell field on the Schwarzschild manifold
• Multi-center Vector Field Methods for Wave Equations
• Nonlinear Wave Equations With Null Condition On Extremal Reissner-Nordström Spacetimes I: Spherical Symmetry
• Localized energy estimates on Myers-Perry space-times
• Linear Instability of the Reissner-Nordström Cauchy Horizon
• Differential Equations of Quantum Mechanics
• Remark On The Notion Of Adapted Conformal And Other Estimates