  It is proven that a solution to the Einstein-Maxwell equations whose
gravitational and electromagnetic radiation fields vanish is in fact stationary
in a neighbourhood of spatial infinity. That is, if the Weyl and Faraday
tensors decay suitably fast, then there exists a time-like Killing vector field
in the region outside the bifurcate horizon of a sphere of sufficiently large
radius. In particular, truly dynamical time-periodic electrovacuum spacetimes
do not exist. This is an extension of earlier work by Alexakis and Schlue and
Bi\v{c}\'{a}k, Sholtz and Tod to include electromagnetism.
