Gödel-type Spacetimes in Induced Matter Gravity Theory

Abstract

A five-dimensional (5D) generalized Gödel-type manifolds are examined in the 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 three essential parameters $k$, $m^2$ and $ω$: identical triads $(k, m^2, ω)$ correspond to locally equivalent 5D manifolds. An irreducible set of isometrically nonequivalent 5D locally homogeneous Riemannian generalized 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 the generalized Gödel-type 5D manifolds admit maximal group of isometry $G_r$ with $r=7$, $r=9$ or $r=15$ depending on the essential parameters $k$, $m^2$ and $ω$. The breakdown of causality in all these classes of homogeneous Gödel-type manifolds are also examined. It is found that in three out of the six irreducible classes the causality can be violated. The unique generalized Gödel-type solution of the induced matter (IM) field equations is found. The question as to whether the induced matter version of general relativity is an effective therapy for these type of causal anomalies of general relativity is also discussed in connection with a recent article by Romero, Tavakol and Zalaletdinov.