Massless Spin-Zero Particle and the Classical Action via Hamilton-Jacobi Equation in Godel Universe

In this letter we investigate the separability of the Klein-Gordon and Hamilton-Jacobi equation in Godel universe. We show that the Klein-Gordon eigen modes are quantized and the complete spectrum of the particle's energy is a mixture of an azimuthal quantum 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 Godel universe.