A new physical-space approach to decay for the wave equation with applications to black hole spacetimes

TL;DR

This paper introduces a robust physical-space method for proving decay of wave energy and pointwise decay in black hole spacetimes, avoiding traditional multiplier techniques.

Contribution

It presents a new general approach for energy decay in wave equations that applies to various black hole backgrounds using only physical space methods.

Findings

Applicable to Minkowski, Schwarzschild, and Kerr spacetimes

Requires only local energy decay estimates

Circumvents traditional multiplier and commutator techniques

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.