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

TL;DR

This paper analyzes geodesic behavior in Schwarzschild-Anti-de Sitter spacetime, focusing on marginally bound geodesics and their properties using Raychaudhuri equations and computational tools.

Contribution

It provides a detailed analysis of geodesic congruences in SAdS4, including shear and rotation calculations, and compares null and timelike geodesics with the Schwarzschild case.

Findings

Marginally bound timelike geodesics have a turning point in the equatorial plane.

Null geodesics can be unbound, with at least one family lacking a turning point.

Plots illustrate geodesic behavior and congruences in the SAdS4 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.