Pointwise decay for the wave equation on nonstationary spacetimes

TL;DR

This paper establishes decay rates for solutions to the linear wave equation on nonstationary spacetimes with variable coefficients, using local energy decay and reduction techniques, extending understanding of wave behavior in dynamic geometries.

Contribution

It proves decay estimates for wave equations on nonstationary spacetimes assuming a weak local energy decay, incorporating variable coefficients, first-order terms, and potentials.

Findings

Decay rates depend on how rapidly metric and potential decay at infinity

Results apply to both stationary and nonstationary metrics

Uses local energy decay and reduction methods to establish decay

Abstract

The first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak local energy decay estimate holds and proves that solutions to the linear wave equation with variable coefficients in $\mathbb R^{1+3}$, first-order terms, and a potential decay at a rate depending on how rapidly the vector fields of the metric, first-order terms, and potential decay at spatial infinity. We prove results for both stationary and nonstationary metrics. The proof uses local energy decay to prove an initial decay rate, and then uses the one-dimensional reduction repeatedly to achieve the full decay rate.