A note on the existence of standard splittings for conformally stationary spacetimes

TL;DR

This paper establishes that a spacetime with a complete timelike conformal Killing vector field splits globally as a standard conformastationary spacetime if and only if it is distinguishing, linking causal properties with geometric structure.

Contribution

It proves a necessary and sufficient condition for the global splitting of conformally stationary spacetimes based on their causal properties, utilizing recent advances on smoothability of time functions.

Findings

Spacetimes with complete timelike conformal Killing vectors split iff they are distinguishing.

Causal but non-distinguishing spacetimes with complete stationary vector fields are also constructed.

The proof employs recent results on the smoothability of time functions and temporal functions.

Abstract

Let $(M,g)$ be a spacetime which admits a complete timelike conformal Killing vector field $K$. We prove that $(M,g)$ splits globally as a standard conformastationary spacetime with respect to $K$ if and only if $(M,g)$ is distinguishing (and, thus causally continuous). Causal but non-distinguishing spacetimes with complete stationary vector fields are also exhibited. For the proof, the recently solved "folk problems" on smoothability of time functions (moreover, the existence of a {\em temporal} function) are used.