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

TL;DR

This paper investigates Killing vector fields in asymptotically flat space-times, establishing conditions under which such vectors imply the space-time is Minkowski, and proving non-existence results for certain null ADM four-momentum configurations.

Contribution

It proves that under the rigidity conditions of the positive energy theorem, asymptotically null Killing vectors only exist in Minkowski space, and shows non-existence of certain non-singular space-times with null ADM four-momentum.

Findings

No asymptotically null Killing vectors unless the space-time is Minkowski.

Non-existence of non-singular asymptotically flat space-times with null ADM four-momentum under weaker conditions.

Supports the uniqueness of Minkowski space in the context of asymptotically flat solutions.

Abstract

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.