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We study Killing vector fields in asymptotically flat space-times. We provethe following result, implicitly assumed in the uniqueness theory of stationaryblack holes. If the conditions of the rigidity part of the positive energytheorem are met, then in such space-times there are no asymptotically nullKilling vector fields except if the initial data set can be embedded inMinkowski space-time. We also give a proof of the non-existence of non-singular(in an appropriate sense) asymptotically flat space-times which satisfy anenergy condition and which have a null ADM four-momentum, under conditionsweaker than previously considered.Comment: 30 page

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