Optics in the Gödel universe: Effects of global rotation and Planck-scale corrections

We investigate the propagation of a massless scalar field in the Gödel universe and demonstrate that the rotating spacetime functions as an intrinsically optically active medium, characterized by a refractive index that depends on both position and frequency. Starting from the covariant Klein-Gordon equation, we derive a generalized Helmholtz-type radial equation in which the effective refractive index n ∘ ( r, ω ) explicitly depends on the radial coordinate r and frequency ω . Consequently, the Gödel geometry manifests as a rotation-induced optical medium: scalar modes are confined within a finite radial domain bounded by turning points where . Depending on the rotation rate Ω, the frequency ω , and the azimuthal quantum number m , the medium supports either propagating or evanescent behavior. Near the rotational axis, a centrifugal singularity emerges, producing strongly evanescent regions for | m | > 1/2, whereas for , propagation persists near the axis but confinement still occurs due to the evanescent asymptotics at large r . Asymptotically, , rendering all modes evanescent. Planck-scale corrections are incorporated within the framework of rainbow gravity, employing energy-dependent functions f (ϵ) and g (ϵ), where . We consider (i) loop-quantum-gravity-inspired functions and (ii) doubly special relativity-inspired functions , which induce the frequency deformation and lead to a modified refractive index . First-order corrections ( η ϵ ≪ 1) yield analytic shifts of the turning points, while exact closed-form expressions allow precise numerical treatment. The resulting rainbow-induced deformations generate frequency-dependent modifications to the propagation thresholds, which become particularly pronounced near the axis ( r → 0) and in near-Planckian regimes. Furthermore, we demonstrate that these results generalize to z -polarized electromagnetic wave modes and show the smooth recovery of scalar-like propagation in the Gödel universe. Finally, when , we establish that in the high-frequency limit ( ω → ∞), the effective refractive index accurately reproduces the null geodesics of the spacetime.