A note on energy currents and decay for the wave equation on a Schwarzschild background

TL;DR

This paper provides an alternative proof for decay estimates of wave solutions on Schwarzschild backgrounds, avoiding the need for spherical harmonic decomposition, thus simplifying the analysis of wave decay in black hole spacetimes.

Contribution

It introduces a new proof technique for decay bounds of wave equations on Schwarzschild backgrounds that does not rely on spherical harmonic decomposition.

Findings

Established uniform decay bounds for wave solutions on Schwarzschild spacetime.

Provided an alternative proof method that simplifies previous approaches.

Confirmed the decay estimate ||  C v_+^{-1} without harmonic decomposition.

Abstract

In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_g\phi=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|\phi|\le Cv_+^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_+=\max\{v,1\}$, and $C$ is a constant depending on a Sobolev norm of initial data. A crucial estimate in the proof required a decomposition into spherical harmonics. We here give an alternative proof of this estimate not requiring such a decomposition.