A note on one-parameter subgroups of SO(3,2)

TL;DR

This work extends the classification of one-parameter subgroups from $SO(2,2)$ to $SO(3,2)$ in three-dimensional conformal gravity, revealing two new classes ($I_d$ and $V$) and providing explicit constructions. Through the Chern-Simons formulation, it connects these subgroups to concrete spacetime geometries, including a conformal extension of the BTZ black hole and a spacetime with constant Ricci scalar and vanishing Cotton tensor. The study highlights how the additional zero eigenvalue in $SO(3,2)$ alters the landscape of solutions and outlines directions for a complete geometric classification. The results have potential implications for CG solutions, their causal structure, and identifications leading to quotient spacetimes with novel properties.

Abstract

We analyze the structure of one-parameter subgroups of SO(3,2). We find two new types of subgroups in comparison with the structure of the one-parameter subgroups of SO(2,2), and we construct explicit examples for these subgroups. We also comment on the placement of existing conformal gravity solutions within this classification.