A new physical-space approach to decay for the wave equation with applications to black hole spacetimes
Mihalis Dafermos, Igor Rodnianski
TL;DR
This paper introduces a physical-space method to prove decay for the wave equation on Lorentzian backgrounds by starting from integrated local energy decay and building a foliation-based energy hierarchy that yields $\tau^{-2}$ energy decay and corresponding pointwise bounds. It demonstrates the approach in Minkowski and Schwarzschild spacetimes and outlines generalization to Kerr, including how red-shift and trapping are handled and how higher-order commutators can enhance decay. The framework avoids t-weighted multipliers/commutators, aiming for robustness suitable for nonlinear stability analyses and offering a roadmap for full decay through refinements. The results unify energy and pointwise decay proofs across black hole exteriors and obstacle settings, with clear pathways to extend to general Kerr spacetimes and sharper decay via higher-order estimates.
Abstract
We present a new general method for proving global decay of energy through a suitable spacetime foliation, as well as pointwise decay, starting from an integrated local energy decay estimate. The method is quite robust, requiring only physical space techniques, and circumvents use of multipliers or commutators with weights growing in t. In particular, the method applies to a wide class of perturbations of Minkowski space as well as to Schwarzschild and Kerr black hole exteriors.