Price's Law on Nonstationary Spacetimes

Abstract

In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate (Price's law \cite{MR0376103}) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric. This result was previously established by the second author in \cite{Tat} on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.

32 pages, no figures, typos corrected

References in corpus (20)

• The black hole stability problem for linear scalar perturbations
• A new physical-space approach to decay for the wave equation with applications to black hole spacetimes
• Late Time Tail of Wave Propagation on Curved Spacetime
• 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
• Decay of Solutions of the Wave Equation in the Kerr Geometry
• Lectures on black holes and linear waves
• 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
• Improved decay for solutions to the linear wave equation on a Schwarzschild black hole
• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• Quasi-Normal Mode Expansion for Linearized Waves in Gravitational Systems
• A note on energy currents and decay for the wave equation on a Schwarzschild background
• 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
• Local energy estimate on Kerr black hole backgrounds
• A Time Independent Energy Estimate for Outgoing Scalar Waves in the Kerr Geometry
• Strichartz estimates on Kerr black hole backgrounds
• Weighted-$L^{\infty }$ and pointwise space-time decay estimates for wave equations with potentials and initial data of low regularity
• Global parametrices and dispersive estimates for variable coefficient wave equations

Cited by in corpus (3)

• Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations
• Linear Stability of the Non-Extreme Kerr Black Hole
• Numerical evolution of the interior geometry of charged black holes