A vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary spacetimes

TL;DR

This paper introduces a novel vector field method to improve decay estimates for wave equations on spherically symmetric, stationary spacetimes, achieving near-optimal decay rates and advancing understanding of wave behavior near black holes.

Contribution

The paper develops a new hierarchy of weighted energy estimates using commutator vector fields, enhancing decay results beyond traditional methods on certain black hole backgrounds.

Findings

Derived higher-order weighted energy estimates.

Achieved almost-sharp decay rates for wave solutions.

Provided foundational tools for precise late-time asymptotics.

Abstract

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner-Nordstrom black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds.