Improved decay for solutions to the linear wave equation on a Schwarzschild black hole

TL;DR

This paper establishes improved decay rates for solutions to the wave equation on Schwarzschild black holes, using a novel vector field method, enhancing understanding of wave behavior in curved spacetime.

Contribution

Introduces a new vector field commutator technique to prove sharper decay estimates for wave solutions on Schwarzschild backgrounds.

Findings

Solutions decay at rates $v_+^{-3/2+ ext{small}}$ and $v_+^{-2+ ext{small}}$

Decay estimates hold in a compact region and along the event horizon

Method improves upon previous decay rate results

Abstract

We prove that sufficiently regular solutions to the wave equation $\Box_g\phi=0$ on the exterior of the Schwarzschild black hole obey the estimates $|\phi|\leq C_\delta v_+^{-{3/2}+\delta}$ and $|\partial_t\phi|\leq C_{\delta} v_+^{-2+\delta}$ on a compact region of $r$ and along the event horizon. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite $r$ and along the event horizon.