Three Dimensional Lorentz homogeneous spaces and the Petrov classification

TL;DR

This paper verifies Petrov's classification of 3D Lorentz homogeneous spaces by establishing criteria for identification and analyzing subalgebras of Lie algebras to determine invariant Lorentz metrics.

Contribution

It provides a systematic method to identify and classify 3D Lorentz homogeneous spaces within Petrov's framework using Lie algebra substructures.

Findings

Criteria for identifying spaces in Petrov's classification

Complete list of inequivalent 1D subalgebras of real 4D Lie algebras

Identification of spaces admitting invariant Lorentz metrics

Abstract

In this note, we verify the classification of local geometries given by A.Z. Petrov. First, we determine criteria for identifying a given 3D Lorentz homogeneous space in Petrov's classification. Then, we identify all inequivalent 1D subalgebras of all real 4D Lie algebras and determine which of these give rise to a homogeneous space admitting an invariant Lorentz metric.