Killing Initial Data

TL;DR

This paper characterizes Killing initial data in terms of lapse and shift on spacelike slices, providing conditions for their Lie algebra structure and applying this to analyze orbit periodicity in asymptotically flat space-times.

Contribution

It offers a necessary and sufficient condition for Killing initial data to form a Lie algebra, linking initial data to space-time symmetries.

Findings

Killing initial data form a Lie algebra under specific conditions

Theorem on periodicity of orbits for certain Killing vectors

Application to asymptotically flat space-times

Abstract

We study space-time 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'S), to form a Lie algebra under the bracket operation induced by the Lie commutator of vector fields on space-time. This result is applied to obtain a theorem on the periodicity of orbits for a class of Killing vector fields in asymptotically flat space-times.