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

Abstract

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.