Mapping Schwarzschild, de Sitter and anti-de Sitter spacetimes into flat spacetime systems with Kratzer potentials

Abstract

This work investigates a method to map systems with gravitational fields into non-relativistic flat spacetime ones. Considering a line element of the form $d{s}^2=e^{2f\left(r\right)}d{t}^2-e^{2g\left(r\right)}d{r}^2-r^{2}d{\theta }^2-r^2{sin}^2\theta d{\varphi }^2$, we constructed an external potential $V\left(r\right)$ in terms of the functions $f\left(r\right)$ and $g\left(r\right)$. We found that the non-relativistic wave function is written in terms of the relativistic wave function. As applications, we constructed non-relativistic analogous models, introducing external potentials for two types of static and non-rotating black holes: Schwarzschild, anti-de Sitter, and de Sitter. Finally, we analyzed the physics at the event horizon region for the black holes and cosmological horizon for the de Sitter spacetime, and the values coincide with those obtained for the case with a gravitational field in the relativistic regime.