On homogeneous 3-dimensional spacetimes: focus on plane waves

TL;DR

This paper classifies 3D homogeneous Lorentzian spacetimes, especially plane waves, revealing their geometric properties, completeness conditions, and the uniqueness of their compact models, thus advancing understanding of Lorentzian geometry.

Contribution

It provides a comprehensive classification of 3D homogeneous plane waves, including non-unimodular cases, and characterizes their geometric and completeness properties.

Findings

Non-unimodular elliptic plane waves are neither locally symmetric nor locally isometric to a Lie group metric.

Homogeneous plane waves are non-extendable and geodesically complete only if symmetric.

Only one non-flat plane wave admits a compact model.

Abstract

We revisit the classification of Lorentz homogeneous spaces of dimension $3$, and relax usual completeness assumptions. In particular, non-unimodular elliptic plane waves, and only them, are neither locally symmetric nor locally isometric to a left-invariant Lorentz metric on a $3$-dimensional Lie group. We characterize homogeneous plane waves in dimension $3$, and prove they are non-extendable, and geodesically complete only if they are symmetric. Finally, only one non-flat plane wave has a compact model.