Late-time asymptotics for geometric wave equations with inverse-square potentials

Abstract

We introduce a new, physical-space-based method for deriving the precise leading-order late-time behaviour of solutions to geometric wave equations on asymptotically flat spacetime backgrounds and apply it to the setting of wave equations with asymptotically inverse-square potentials on Schwarzschild black holes. This provides a useful toy model setting for introducing methods that are applicable to more general linear and nonlinear geometric wave equations, such as wave equations for electromagnetically charged scalar fields, wave equations on extremal Kerr black holes and geometric wave equations in even space dimensions, where existing proofs for deriving precise late-time asymptotics might not apply. The method we introduce relies on exploiting the spatial decay properties of time integrals of solutions to derive the existence and precise genericity properties of asymptotic late-time tails and obtain sharp, uniform decay estimates in time.

76 pages, 1 figure, fixed typos and added publication data

References in corpus (4)

• Horizon Instability of Extremal Kerr Black Holes: Nonaxisymmetric Modes and Enhanced Growth Rate
• On pointwise decay of waves
• On the Relation Between Asymptotic Charges, the Failure of Peeling and Late-time Tails
• The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $\ell$-Modes of Linear Waves on a Schwarzschild Background

Cited by in corpus (5)

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• The Case Against Smooth Null Infinity V: Early-Time Asymptotics of Linearised Gravity Around Schwarzschild for Fixed Spherical Harmonic Modes
• Scattering for wave equations with sources close to the lightcone and prescribed radiation fields
• Conformal Scattering of Maxwell Potentials
• Price's law on Minkowski space in the presence of an inverse square potential