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This paper establishes that on the domain of outer communications of ageneral class of stationary and asymptotically flat Lorentzian manifolds ofdimension $d+1$, $d\ge3$, the local energy of solutions to the scalar waveequation $\square_{g}\psi=0$ decays at least with an inverse logarithmic rate.This class of Lorentzian manifolds includes (non-extremal) black holespacetimes with no restriction on the nature of the trapped set. Spacetimes inthis class are moreover allowed to have a small ergoregion but are required tosatisfy an energy boundedness statement. Without making further assumptions,this logarithmic decay rate is shown to be sharp. Our results can be viewed asa generalisation of a result of Burq, dealing with the case of the waveequation on flat space outside compact obstacles, and results ofRodnianski--Tao for asymptotically conic product Lorentzian manifolds. Theproof will bridge ideas of Rodnianski--Tao with techniques developed in theblack hole setting by Dafermos--Rodnianski. As a soft corollary of our results,we will infer an asymptotic completeness statement for the wave equation on thespacetimes considered, in the case where no ergoregion is present.

Logarithmic local energy decay for scalar waves on a general class of asymptotically flat spacetimes