The Wave Equation in a General Spherically Symmetric Black Hole Geometry

TL;DR

This paper proves the existence, uniqueness, and decay of solutions to the wave equation in general spherically symmetric black hole geometries, including those from Einstein/Yang-Mills theories, under mild conditions.

Contribution

It establishes global smoothness, decay, and support properties of solutions to the wave equation in a broad class of black hole spacetimes, extending previous results to more general geometries.

Findings

Solutions decay in $L^{ abla}_{ ext{loc}}$ as time tends to infinity.

Unique globally smooth solutions exist for compactly supported initial data.

Results apply to black hole solutions of Einstein/Yang-Mills equations.

Abstract

We consider the Cauchy problem for the wave equation in a general class of spherically symmetric black hole geometries. Under certain mild conditions on the far-field decay and the singularity, we show that there is a unique globally smooth solution to the Cauchy problem for the wave equation with data compactly supported away from the horizon that is compactly supported for all times and \emph{decays in $L^{\infty}_{\text{loc}}$ as $t$ tends to infinity}. We obtain as a corollary that in the geometry of black hole solutions of the SU(2) Einstein/Yang-Mills equations, solutions to the wave equation with compactly supported initial data decay as $t$ goes to infinity.