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