A Vector Field Method for Non-Trapping, Radiating Spacetimes

TL;DR

This paper establishes decay properties of solutions to the wave equation on time-dependent, weakly asymptotically flat spacetimes, linking decay rates to local energy decay assumptions.

Contribution

It introduces a vector field method to prove bounded conformal energy and decay estimates for wave solutions in non-trapping, dynamical spacetimes.

Findings

Bounded conformal energy for solutions

Global $L^{ abla}$ decay bounds

Reduction of pointwise decay proof to local energy decay

Abstract

We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy. As an application we also show a conformal energy estimate with vector fields applied to the solution as well as a global $L^{\infty}$ decay bound in terms of a weighted norm on initial data. For solutions to the wave equation in these dynamical backgrounds, our results reduce the problem of establishing the classical pointwise decay rate t^{-3/2} in the interior and t^{-1} along outgoing null cones to simply proving that local energy decay holds.