Revisiting timelike geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions

Abstract

The theory of Schwarzschild timelike geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, time-like trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.

Optical properties of null geodesics emerging from dynamical systems

“…The analytical calculations involving null geodesics in spherically symmetric space-times using Weierstrass elliptic a e-mail: davide.batic@ku.ac.ae (corresponding author) b e-mail: sumanto.chanda@icts.res.in c e-mail: partha.guha@ku.ac.ae functions are given in [4][5][6][7]. Null geodesics can be written in the form of Binet's equations [8], casting them as central force systems for some potentials.…”

Optical properties of null geodesics emerging from dynamical systems

“…The analytical calculations involving null geodesics in spherically symmetric space-times using Weierstrass elliptic functions are given in [4][5][6][7]. Null geodesics can be written in the form of Binet's equations [8], casting them as central force systems for some potentials.…”