Riemannian Space-times of Gödel Type in Five Dimensions

Abstract

The five-dimensional (5D) Riemannian Gödel-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 $ω$: identical pairs $(m^2, ω)$ correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian Gödel-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 $(m^2= 4 ω^2, ω\not=0)$ and $(m^2\not=0, ω= 0)$ classes the homogeneous Riemannian Gödel-type manifolds admit a seven-parameter maximal group of isometry ($G_7$). The special class $(m^2= 4 ω^2, ω\not=0)$ and the degenerated Gödel-type class $(m^2\not=0, ω=0)$ are shown to have a $G_9$ as maximal group of motion. The breakdown of causality in these classes of homogeneous Gödel-type manifolds are also examined.