Quantum motion of scalar particle under rainbow gravity effects in spherical symmetrical Gödel-type metric with a topological defect

Abstract

In this research contribution, we explore the relativistic quantum dynamics of spin-0 scalar particles within a curved space-time background containing topological defects, while incorporating the effects of rainbow gravity. The Klein–Gordon equation is solved in a spherically symmetrical Gödel-type space-time, considering the presence of topological defects and the influence of rainbow gravity, described by the line-element:$d{s}^2=-{\{\frac{dt}{F}+\alpha \Omega r^2{(1+\frac{r^2}{4R^2})}^{-1}\frac{d\phi }{G}\}}^2+G^{-2}\{{(1+\frac{r^2}{4R^2})}^{-2}(d{r}^2+{\alpha }^2{r}^{2}d{\phi }^2)+d{z}^2\}.$ The functions $F=F\left(\chi \right)$ and $G=G\left(\chi \right)$, where $0<\chi (=\frac{\left|E\right|}{E_p})\le 1$, E is the particle's energy, and $E_p$ the Planck's energy, are referred to as rainbow functions. Analytical solutions for the energy levels and wave-functions of scalar particles are obtained through general hypergeometric equations for the pair of rainbow functions: (i) $F\left(\chi \right)=1$ and $G\left(\chi \right)=(1+\kappa \chi )$, and (ii) $F\left(\chi \right)=\frac{(e^{{\beta }_0\chi }-1)}{{\beta }_0\chi }$ and $G\left(\chi \right)=1$. Notably, we demonstrate that the energy levels and wave-functions of scalar particles are intricately linked to the topological defect of cosmic string, characterized by the parameter α, and the constant angular speed of the rotating frame, characterized by the parameter Ω. Furthermore, the influence of the rainbow parameters $(\kappa ,{\beta }_0)$ is evident in the eigenvalue solutions, introducing modifications compared to known expressions. It is worth mentioning that the presence of topological defects breaks the degeneracy of the energy levels, resulting in substantial modifications. These findings underscore the interplay between rainbow gravity, topological defects, and the intrinsic properties of scalar particles within a curved space-time framework.