Closed timelike curves and geodesics of Godel-type metrics
  It is shown explicitly that when the characteristic vector field that defines a Godel-type metric is also a Killing vector, there always exist closed timelike or null curves in spacetimes described by such a metric. For these geometries, the geodesic curves are also shown to be characterized by a lower dimensional Lorentz force equation for a charged point particle in the relevant Riemannian background. Moreover, two explicit examples are given for which timelike and null geodesics can never be closed. 
Reinaldo J. Gleiser, Metin Gurses, Atalay Karasu, Ozgur Sarioglu