Killing initial data

Abstract

We study spacetime Killing vectors in terms of their `lapse and shift' relative to some spacelike slice. We give a necessary and sufficient condition in order for these lapse - shift pairs, which we call Killing initial data (KID), to form a Lie algebra under the bracket operation induced by the Lie commutator of vector fields on spacetime. This result is applied to obtain a theorem on the periodicity of orbits for a class of Killing vector fields in asymptotically flat spacetimes.