Riemannian Space-times of G\"odel Type in Five Dimensions

TL;DR

This paper classifies five-dimensional Riemannian G"odel-type manifolds using Cartan's equivalence method, identifying their parameters, symmetries, and causality properties, and establishing conditions for local homogeneity and equivalence.

Contribution

It provides a comprehensive classification of 5D Riemannian G"odel-type manifolds, including their symmetry groups and causality features, based on essential parameters and local homogeneity conditions.

Findings

Identified conditions for local homogeneity of 5D G"odel-type manifolds.

Classified manifolds based on parameters m^2 and ω, and determined their isometry groups.

Analyzed causality breakdown in different classes of these manifolds.

Abstract

The five-dimensional (5D) Riemannian G\"odel-type manifolds are examined in light of the equivalence problem techniques, as formulated by Cartan. The necessary and sufficient conditions for local homogeneity of these 5D manifolds are derived. The local equivalence of these homogeneous Riemannian manifolds is studied. It is found that they are characterized by two essential parameters $m^2$ and $\omega $: identical pairs $(m^2, \omega)$ correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian G\"odel-type metrics are exhibited. A classification of these manifolds based on the essential parameters is presented, and the Killing vector fields as well as the corresponding Lie algebra of each class are determined. It is shown that apart from the $(m^2= 4 \omega^2, \omega\not=0)$ and $(m^2\not=0, \omega = 0)$ classes the homogeneous Riemannian G\"odel-type manifolds admit a seven-parameter maximal group of isometry ($G_7$). The special class $(m^2= 4 \omega^2, \omega\not=0)$ and the degenerated G\"odel-type class $(m^2\not=0, \omega=0)$ are shown to have a $G_9$ as maximal group of motion. The breakdown of causality in these classes of homogeneous G\"odel-type manifolds are also examined.