Analytic treatment of complete geodesics in a static cylindrically symmetric conformal spacetime

TL;DR

This paper analytically solves the equations of motion for particles and light in a static cylindrically symmetric conformal spacetime, characterizing possible orbits using advanced mathematical functions and effective potential analysis.

Contribution

It provides explicit analytical solutions for geodesics in this spacetime using Weierstrass and Kleinian functions, enhancing understanding of particle trajectories.

Findings

Analytical solutions for geodesics derived

Classification of orbits based on energy and angular momentum

Examples illustrating different orbit types

Abstract

We consider the motion of test particles and light rays in a static cylindrically symmetric conformal spacetime given by Said et al [1]. We derive the equations of motion and present their analytical solutions in terms of the Weierstrass {\wp} function and the Kleinian {\sigma} function. Using parametric diagrams and effective potentials we analyze the possible orbits and characterize them in terms of the energy and the angular momentum of the test particles. Finally we show some examples of orbits.