Spectrum of a scalar field in a Gödel cosmological model

Abstract

A study is made of the spectrum of a scalar field for the general case of a causal metric of the Gödel type. Its substantial difference from the spectrum of a noncausal metric is demonstrated. In a causal metric, the spectrum of scalar particles is divided into three regions: 1) a region of low energy and a discrete spectrum; 2) a region of intermediate energy and a continuous spectrum, with the same dispersion relation; 3) a region of high energy and likewise a continuous spectrum, but with a different dispersion relation. In the latter case, the dispersion relation is symmetrical with respect to a sign change of the particle energy ω, whereas in the first two cases symmetry is conserved only for a simultaneous sign change of both ω and the projection of the orbital moment onto the z axis. For nonzero mass and a quite low z component of the particle momentum, the upper and lower continua of the states merge, and the energy gap disappears.