On Geodesic Congruences and the Raychaudhuri Equations in $\textrm{SAdS}_4$ Spacetime

Abstract

In this article, we look into geodesics in the Schwarzschild-Anti-de Sitter metric in (3+1) spacetime dimensions. We investigate the class of marginally bound geodesics (timelike and null), while comparing their behavior with the normal Schwarzschild metric. Using $\textit{Mathematica}$, we calculate the shear and rotation tensors, along with other components of the Raychaudhuri equation in this metric and we argue that marginally bound timelike geodesics, in the equatorial plane, always have a turning point, while their null analogues have at least one family of geodesics that are unbound. We also present associated plots for the geodesics and geodesic congruences, in the equatorial plane.