The $$r^{p}$$ r p -Weighted Energy Method of Dafermos and Rodnianski in General Asymptotically Flat Spacetimes and Applications

Abstract

In [11], Dafermos and Rodnianski presented a novel approach to establish uniform decay rates for solutions $${\upvarphi }$$ to the scalar wave equation $$\square _{g}{\upvarphi }=0$$ on Minkowski, Schwarzschild and other asymptotically flat backgrounds. This paper generalises the methods and results of [11] to a broad class of asymptotically flat spacetimes $$(\mathcal {M},g)$$ , including Kerr spacetimes in the full subextremal range $$|a|<M$$ , but also radiating spacetimes with no exact symmetries in general dimension $$d+1$$ , $$d\ge 3$$ . As a soft corollary, it is shown that the Friedlander radiation field for $${\upvarphi }$$ is well defined on future null infinity. Moreover, polynomial decay rates are established for $${\upvarphi }$$ , provided that an integrated local energy decay statement (possibly with a finite loss of derivatives) holds and the near region of $$(\mathcal {M},g)$$ satisfies some mild geometric conditions. The latter conditions allow for $$(\mathcal {M},g)$$ to be the exterior of a black hole spacetime with a non-degenerate event horizon (having possibly complicated topology) or the exterior of a compact moving obstacle in an ambient globally hyperbolic spacetime satisfying suitable geometric conditions.