Asymptotics for the wave equation on differential forms on Kerr-de Sitter space

Abstract

We study asymptotics for solutions of Maxwell's equations, in fact of the Hodge-de Rham equation $(d+\delta )u=0$ without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which in particular include Schwarzschild-de Sitter spaces of all spacetime dimensions $n\ge 4$. We prove that solutions decay exponentially to $0$ or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and in particular prove analogous results on Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell's equations, the Hodge-de Rham equation and the wave equation on differential forms on Kerr-de Sitter spacetimes with small angular momentum.

47 pages. v2 is the published version, with improved exposition

References in corpus (3)

• The wave equation on Schwarzschild-de Sitter spacetimes
• Resonance expansions for tensor-valued waves on asymptotically Kerr-de Sitter spaces
• Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: The cases |a| << M or axisymmetry

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• The global non-linear stability of the Kerr-de Sitter family of black holes
• Quantum Instability of the Cauchy Horizon in Reissner-Nordström-deSitter Spacetime
• Analysis of linear waves near the Cauchy horizon of cosmological black holes
• Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime
• Resonance expansions for tensor-valued waves on asymptotically Kerr-de Sitter spaces
• Quasinormal modes of small Schwarzschild-de Sitter black holes
• Quasinormal modes and dual resonant states on de Sitter space
• The decay of the $SU(2)$ Yang-Mills fields on the Schwarzschild black hole for spherically symmetric small energy initial data