Linear waves on constant radius limits of cosmological black hole spacetimes

In this paper we consider the Klein–Gordon equation on spherically symmetric background spacetimes with a constant area radius. The spacetimes under consideration are Nariai and Plebański–Hacyan, and can be considered constant radius limits of Reissner–Nordström–de Sitter spacetimes. We prove boundedness in the case of a non-negative Klein–Gordon mass and decay unless the mass is zero. In the latter case we prove decay of solutions that are supported on all harmonic modes with angular momentum $l \geq 1$. We show that the $l = 0$ modes of solutions to the massless Klein–Gordon equation do not decay. They are subject to conservation laws along degenerate Killing horizons. We apply the estimates in Nariai to give decay of solutions to the massive Klein–Gordon equation on an $n$-dimensional de Sitter background, using only the vector field method and with no restrictions on the positive Klein–Gordon mass.

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Published 5 December 2018