Timelike and Isotropic Geodesics of the Gödel Universe as a Lie Group with Left-Invariant Lorentz Metric

Studying the Gödel universe as a Lie group with left-invariant Lorentz metricand using the methodsof geometric theory of the optimal control for the search of geodesics on Lie groupswith left-invariant (sub-)Lorentz metrics,we find the expressions for timelike and isotropic geodesics in elementary functions.Also, we pay special attention to proving that the Gödel universe has no closed timelike or isotropic geodesics.