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.
The authors were supported in part by A.V.’s National Science Foundation grants DMS-1068742 and DMS-1361432, and P.H. was supported in part by a Gerhard Casper Stanford Graduate Fellowship.
"Asymptotics for the wave equation on differential forms on Kerr–de Sitter space." J. Differential Geom. 110 (2) 221 - 279, October 2018. https://doi.org/10.4310/jdg/1538791244