The wave equation in a general spherically symmetric black hole geometry

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

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Published 24 October 2012