  We prove that smooth asymptotically flat solutions to the Einstein vacuum
equations which are assumed to be periodic in time, are in fact stationary in a
neighborhood of infinity. Our result applies under physically relevant
regularity assumptions purely at the level of the initial data. In particular,
our work removes the assumption of analyticity up to null infinity in [Bicak,
Scholtz, and Tod; 2010]. The proof relies on extending a suitably constructed
"candidate" Killing vector field from null infinity, via Carleman-type
estimates obtained in [Alexakis, Schlue, Shao; 2013].
