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The wave equation in a general spherically symmetric black hole geometry. (English) Zbl 1261.83020

Summary: 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 decays in \(L_{\text{loc}}^\infty\) 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.

MSC:

83C57 Black holes
35L05 Wave equation
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics
83C75 Space-time singularities, cosmic censorship, etc.
81T13 Yang-Mills and other gauge theories in quantum field theory
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