On Cyclically Symmetrical Spacetimes

TL;DR

This paper investigates the structure of Lie groups with cyclic symmetry in spacetimes, proving that certain symmetries imply the groups are Abelian and restricting possible Bianchi types, thus deepening understanding of symmetric spacetime geometries.

Contribution

It extends previous results by weakening assumptions and applies the findings to classify Lie groups compatible with cyclic symmetry in spacetime models.

Findings

Two-dimensional Lie groups with circular orbits are Abelian.

Restrictions on Bianchi types for groups with closed orbits.

Severe limitations on four-dimensional Lie groups with cyclic symmetry.

Abstract

In a recent paper Carot et al. considered 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 paper a similar result, valid under considerably weaker assumptions, is derived: 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 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, II, III, VII_0, VIII or IX. Some results on n-dimensional Lie groups are also derived and applied to show there are severe restrictions on the structure of the allowed four-dimensional Lie transformation groups compatible with cyclic symmetry.