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In [M. Dafermos and I. Rodnianski, A new physical-space approach to decay forthe wave equation with applications to black hole spacetimes, in XVIthInternational Congress on Mathematical Physics, Pavel Exner ed., Prague 2009pp. 421-433, 2009, arXiv:0910.4957], Dafermos and Rodnianski presented a novelapproach to establish uniform decay rates for solutions $\phi$ to the scalarwave equation $\square_{g}\phi=0$ on Minkowski, Schwarzschild and otherasymptotically flat backgrounds. This paper generalises the methods and resultsof 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 generaldimension $d+1$, $d\ge3$. As a soft corollary, it is shown that the Friedlanderradiation field for $\phi$ is well defined on future null infinity. Moreover,polynomial decay rates are established for $\phi$, provided that an integratedlocal energy decay statement (possibly with a finite loss of derivatives) holdsand the near region of $(\mathcal{M},g)$ satisfies some mild geometricconditions. The latter conditions allow for $(\mathcal{M},g)$ to be theexterior of a black hole spacetime with a non-degenerate event horizon (havingpossibly complicated topology) or the exterior of a compact moving obstacle inan ambient globally hyperbolic spacetime satisfying suitable geometricconditions.

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