A note on the uniqueness of global static decompositions

TL;DR

This paper investigates conditions under which static Killing vector fields lead to unique, global orthogonal spacetime decompositions, emphasizing the role of compactness and spacelike submanifolds.

Contribution

It proves the uniqueness of global static decompositions in compact cases and analyzes obstructions to such splittings.

Findings

Orthogonal splitting is unique for compact natural spaces.

Many spacelike submanifolds must lie within an orthogonal slice.

Obstructions to global splitting are identified.

Abstract

We discuss when static Killing vector fields are standard, that is, leading to a global orthogonal splitting of the spacetime. We prove that such an orthogonal splitting is unique whenever the natural space is compact. This is carried out by proving that many notable spacelike submanifolds must be contained in an orthogonal slice. Possible obstructions to the global splitting are also considered.