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

TL;DR

This paper investigates the conditions under which asymptotically flat space-times admit translational Killing vectors, establishing that certain borderline cases in the positive energy theorem are only realizable through embeddings in Minkowski space, especially in higher dimensions.

Contribution

It extends the positive energy theorem to higher-dimensional spin manifolds, characterizing when initial data correspond to Minkowski embeddings based on asymptotic symmetries.

Findings

Borderline cases occur only for data from Minkowski embeddings

The positive energy theorem holds under specified asymptotic conditions in higher dimensions

Characterization of asymptotic translational symmetries in higher-dimensional space-times

Abstract

We show that the borderline cases in the proof of the positive energy theorem for initial data sets, on spin manifolds, in dimensions $n\ge 3$, are only possible for initial data arising from embeddings in Minkowski space-time.