Decay of linear waves on higher dimensional Schwarzschild black holes

Abstract

In this paper we consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation \Box_gϕ=0 on the domain of outer communications of the Schwarzschild spacetime manifold (M^n_m, g) (where n >= 3 is the spatial dimension, and m > 0 is the mass of the black hole) the associated energy flux E[ϕ](Σ_τ) through a foliation of hypersurfaces (Σ_τ) (terminating at future null infinity and to the future of the bifurcation sphere) decays, E[ϕ](Σ_τ) <= CD/τ^2, where C is a constant only depending on n and m, and D < \infty is a suitable higher order initial energy on Σ_0; moreover we improve the decay rate for the first order energy to E[\partial_tϕ](Σ_τ^R) <= CD/τ^(4-2δ) for any δ> 0 where Σ_τ^R denotes the hypersurface (Σ_τ) truncated at an arbitrarily large fixed radius R < \infty provided the higher order energy D_δon Σ_0 is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate |ϕ|_{Σ_τ^R} <= (C D'_δ) / τ^(3/2-δ). In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski.

93 pages, 7 figures

References in corpus (5)

• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I
• 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
• Improved decay for solutions to the linear wave equation on a Schwarzschild black hole
• Global solutions of the Einstein-Maxwell equations in higher dimensions

Cited by in corpus (17)

• 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
• Quasinormal modes in extremal Reissner-Nordström spacetimes
• Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies
• Long time existence for semilinear wave equations on asymptotically flat space-times
• Boundedness of massless scalar waves on Kerr interior backgrounds
• The Strauss conjecture on asymptotically flat space-times
• Sharp decay estimates for massless Dirac fields on a Schwarzschild background
• The Stable Trapping Phenomenon for Black Strings and Black Rings and its Obstructions on the Decay of Linear Waves
• News vs Information
• A non-degenerate scattering theory for the wave equation on extremal Reissner-Nordstrom
• A local energy estimate for wave equations on metrics asymptotically close to Kerr
• Late-time asymptotics for geometric wave equations with inverse-square potentials
• Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere
• Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes
• On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime
• Scattering for wave equations with sources close to the lightcone and prescribed radiation fields