Int J Theor Phys (2012) 51:2427–2432 DOI 10.1007/s10773-012-1122-x Massless Spin–Zero Particle and the Classical Action via Hamilton–Jacobi Equation in Gödel Universe A.F. Bahrehbakhsh · D. Momeni · R. Myrzakulov Received: 28 December 2011 / Accepted: 27 February 2012 / Published online: 17 March 2012 © Springer Science+Business Media, LLC 2012 Abstract In this letter we investigate the separability of the Klein–Gordon and Hamilton– Jacobi equation in Gödel universe. We show that the Klein–Gordon eigen modes are quan- tized and the complete spectrum of the particle’s energy is a mixture of an azimuthal quan- tum number, m and a principal quantum number, n and a continuous wave number k. We also show that the Hamilton–Jacobi equation gives a closed function for classical action. These results may be used to calculate the Casimir vacuum energy in Gödel universe. Keywords Gödel universe · Integrability · Exact solutions 1 Introduction Gödel universe is a homogeneous and stationary exact solution of Einstein field equations of gravitation which admits some interest but pathologic physical properties [1]. In Gödel universe there is a closed time-like curve which pictures a candidate for time traveling. Gödel solution, unlike the Friedmann–Robertson–Walker (FRW) cosmological solution, is incompatible with the Weyl’s postulate [2] and does not allow the possibility of defining a universal cosmic time. It is known that this solution may be arisen from other general Lagrangian [3]. Also, there is a Gödel universe with a negative cosmological constant and a Chern–Simons term in some extended models of gravity in higher dimensions [4]. Even A.F. Bahrehbakhsh ( ) Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran e-mail: af-bahrehbakhsh@sbu.ac.ir D. Momeni · R. Myrzakulov Eurasian International Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan D. Momeni e-mail: d.momeni@yahoo.com R. Myrzakulov e-mail: rmyrzakulov@gmail.com