Killing vectors in asymptotically flat space–times: I. 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 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. Supported by Fonds zur Förderung der wissenschaftlichen Forschung, Project P9376– PHY.E–mail: [email protected] On leave of absence from the Institute of Mathematics, Polish Academy of Sciences, Warsaw. Supported in part by KBN grant # 2P30105007, by the Humboldt Foundation and by the Federal Ministry of Science and Research, Austria. E–mail: [email protected]