Geodesic Structure in Horizonless and Charged Spacetimes: A Comparative Study of Timelike and Null Orbits in Bertrand and Reissner- Nordström Geometries

Abstract

The investigation of geodesics is pivotal for elucidating the geometric structure of spacetime. This article delves into the properties of timelike and null geodesics within horizonless and charged spacetimes, concentrating on Bertrand and Reissner-Nordström solutions. We investigate the effective potential and the differential equations governing circular timelike and null geodesics, providing critical insights into the trajectories of free-falling particles in these contexts. By focusing on three-dimensional spacetime, we derive the solutions to the geodesic equations restricted to the equatorial plane. For the case of the Bertrand spacetime-I (BST-I), we impose the specific condition \(\varvec{D = 0}\) and adopt the upper sign for our analysis.