A comment on a paper by Carot et al

TL;DR

This paper discusses the conditions under which certain Lie groups related to symmetries in spacetime are Abelian, extending previous results and classifying possible Lie algebra types based on subgroup properties.

Contribution

It presents a weaker assumption-based proof that two-dimensional Lie groups with circle orbits are Abelian and extends the approach to three-dimensional groups, classifying Bianchi types.

Findings

Two-dimensional Lie groups with circle orbits are Abelian.

Existence of a subgroup with closed orbits restricts Bianchi types to I, II, III, VII_0, VIII, or IX.

The approach relates to and extends previous symmetry results in spacetime geometry.

Abstract

In a recent paper Carot et al. considered carefully the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian group. In this comment a similar result, valid under considerably weaker assumptions, is recalled: any two-dimensional Lie transformation group which contains a one-dimensional subgroup whose orbits are circles, must be Abelian. The method used to prove this result is extended to apply to three-dimensional Lie transformation groups. It is shown that the existence of a one-dimensional subgroup with closed orbits restricts the Bianchi type of the associated Lie algebra to be I (Abelian), II, III, VII_0, VIII or IX. The relationship between the present approach and that of the original paper is discussed.