Linear waves on constant radius limits of cosmological black hole spacetimes

TL;DR

This paper studies the behavior of solutions to the Klein-Gordon equation on specific constant radius black hole spacetimes, proving boundedness and decay properties depending on the mass, with implications for de Sitter backgrounds.

Contribution

It provides the first detailed analysis of Klein-Gordon solutions on Nariai and Plebański-Hacyan spacetimes, including decay results and conservation laws for massless modes.

Findings

Solutions are bounded for non-negative mass.

Decay occurs for massive solutions with angular momentum l ≥ 1.

Massless l=0 modes do not decay and obey conservation laws.

Abstract

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 Pleba\'nski-Hacyan, and can be considered constant radius limits of Reissner-Nordstr\"om-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 greater or equal to 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.