Landau Quantization for Relativistic Vector Bosons in a Gödel-Type Geometric Background

We consider a relativistic spin-1 particle under the effect of external magnetic field in a Gödel-type spacetime, which is characterized through the vorticity in the presence of a point source (topological defect) in the background, and determine the relativistic Landau levels for massive spin-1 particles in this background by performing exact solution of the corresponding relativistic vector boson equation derived as an excited state of Zitterbewegung. This equation gives $3\times 3$ matrix equation since the corresponding spinor is a symmetric spinor of rank-two. By solving this matrix equation we arrive at a wave equation and obtain solution function in terms of Whittaker function. Then, we analyse the results with and without considering the presence of external magnetic field. Furthermore, we compare the effects of both vorticity and external magnetic field on the spectra of massive vector bosons and discuss the results with details in several physical limits.