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

TL;DR

This paper analyzes the asymptotic behavior of solutions to Maxwell's equations, the Hodge-de Rham equation, and the wave equation on differential forms in Kerr-de Sitter spacetimes, showing exponential decay and stability.

Contribution

It extends decay and stability results to differential forms on Kerr-de Sitter spacetimes with small angular momentum, including new interpretations of stationary states.

Findings

Solutions decay exponentially to zero or stationary states

Stationary states relate to spacetime cohomology

Stability results hold for Kerr-de Sitter with small angular momentum

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 $n\geq 4$. 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.