Geodesic completeness of some Lorentzian simple Lie groups

TL;DR

This paper studies the conditions under which left-invariant Lorentzian metrics on simple Lie groups, especially $SL_2(\mathbb{C})$, are geodesically complete, focusing on the nature of invariant Killing vector fields.

Contribution

It establishes new criteria linking the causal character of Killing vector fields to geodesic completeness on simple Lie groups and analyzes the complex case of $SL_2(\mathbb{C})$ in detail.

Findings

Completeness when $Z$ is timelike or $G$ is strongly causal with $Z$ lightlike.

Existence of lightlike $Z$ on $SL_2(\mathbb{C})$ implies completeness.

Spacelike $Z$ on $SL_2(\mathbb{C})$ has an equivalent completeness condition.

Abstract

In this paper we investigate geodesic completeness of left-invariant Lorentzian metrics on a simple Lie group $G$ when there exists a left-invariant Killing vector field $Z$ on $G$. Among other results, it is proved that if $Z$ is timelike, or $G$ is strongly causal and $Z$ is lightlike, then the metric is complete. We then consider the special complex Lie group $SL_2(\mathbb{C})$ in more details and show that the existence of a lightlike vector field $Z$ on it, implies geodesic completeness. We also consider the existence of a spacelike vector field $Z$ on $SL_2(\mathbb{C})$ and provide an equivalent condition for the metric to be complete. This illustrates the complexity of the situation when $Z$ is spacelike.