  We study Killing vector fields in asymptotically flat space-times. We prove
the following result, implicitly assumed in the uniqueness theory of stationary
black holes. If the conditions of the rigidity part of the positive energy
theorem are met, then in such space-times there are no asymptotically null
Killing vector fields except if the initial data set can be embedded in
Minkowski space-time. We also give a proof of the non-existence of non-singular
(in an appropriate sense) asymptotically flat space-times which satisfy an
energy condition and which have a null ADM four-momentum, under conditions
weaker than previously considered.
