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On homogeneous 3-dimensional spacetimes: focus on plane waves

Souheib Allout, Abderrahmane Belkacem, Abdelghani Zeghib

TL;DR

The article advances the classification of three-dimensional Lorentzian homogeneous spaces by relaxing completeness assumptions and showing that, besides left-invariant Lie group metrics and globally symmetric spaces, there exist non-unimodular elliptic plane waves that are neither locally symmetric nor locally isometric to a 3D Lie group. It constructs and analyzes these plane waves via Heisenberg extensions $ ext{P}_ ho=G_ ho/I$, distinguishing unimodular (symmetric or Minkowski/Cahen-Wallach) from non-unimodular cases, and derives a global coordinate description $g=2dudv+ rac{b( ho)x^2}{u^2}du^2+dx^2$ with a single invariant parameter $b( ho)$. The work proves completeness and extendibility properties, shows non-existence of compact quotients except in a unique flat or solvable model, and provides a comprehensive framework tying plane waves to global homogeneous structures through a Lie-theoretic and geometric lens. This deepens understanding of Lorentzian 3-manifolds, clarifies global vs local phenomena, and sharpens the boundary between symmetric, left-invariant, and plane-wave geometries with respect to compactness and completeness.

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.

On homogeneous 3-dimensional spacetimes: focus on plane waves

TL;DR

The article advances the classification of three-dimensional Lorentzian homogeneous spaces by relaxing completeness assumptions and showing that, besides left-invariant Lie group metrics and globally symmetric spaces, there exist non-unimodular elliptic plane waves that are neither locally symmetric nor locally isometric to a 3D Lie group. It constructs and analyzes these plane waves via Heisenberg extensions , distinguishing unimodular (symmetric or Minkowski/Cahen-Wallach) from non-unimodular cases, and derives a global coordinate description with a single invariant parameter . The work proves completeness and extendibility properties, shows non-existence of compact quotients except in a unique flat or solvable model, and provides a comprehensive framework tying plane waves to global homogeneous structures through a Lie-theoretic and geometric lens. This deepens understanding of Lorentzian 3-manifolds, clarifies global vs local phenomena, and sharpens the boundary between symmetric, left-invariant, and plane-wave geometries with respect to compactness and completeness.

Abstract

We revisit the classification of Lorentz homogeneous spaces of dimension , 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 -dimensional Lie group. We characterize homogeneous plane waves in dimension , 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.
Paper Structure (52 sections, 22 theorems, 71 equations)

This paper contains 52 sections, 22 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Results on plane waves
  3. Heisenberg extensions
  4. Plane waves
  5. Literature
  6. Some terminology
  7. Global coordinates system
  8. Compact models of homogeneous plane waves
  9. Completeness and extendibility
  10. Classification of homogeneous Lorentz 3-spaces
  11. Homogeneous locally symmetric case
  12. Locally homogeneous compact Lorentz manifolds
  13. Organization of the article
  14. What's new in the non-Riemannian case.
  15. The Riemannian case
  16. ...and 37 more sections

Key Result

Theorem 1.1

Let $I$ be any non-central one parameter subgroup of ${\operatorname{Heis}}$ generated by a vector $W \in {\mathfrak{heis}}$ and put $\operatorname{P}_{\rho}=G_{\rho}/I$. Then we have

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10: DZ
  • ...and 34 more