Killing vectors in asymptotically flat space-times .1. Asymptotically translational Killing vectors and the rigid positive energy theorem

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 nonexistence of nonsingular (in an appropriate sense) asymptotically flat space-times that satisfy an energy condition and that have a null ADM four-momentum, under conditions weaker than previously considered. (C) 1996 American Institute of Physics.