The $r^{p}$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications

Abstract

In [M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, in XVIth International Congress on Mathematical Physics, Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arXiv:0910.4957], Dafermos and Rodnianski presented a novel approach to establish uniform decay rates for solutions $\phi$ to the scalar wave equation $\square_{g}\phi=0$ on Minkowski, Schwarzschild and other asymptotically flat backgrounds. This paper generalises the methods and results of the above paper 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 $d+1$, $d\ge3$. As a soft corollary, it is shown that the Friedlander radiation field for $\phi$ is well defined on future null infinity. Moreover, polynomial decay rates are established for $\phi$, 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.