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

TL;DR

This paper derives a unified formula for all non-radial timelike and null geodesics in Schwarzschild spacetime using Weierstrass elliptic functions, enabling comprehensive analysis of particle trajectories.

Contribution

The authors provide a novel, single formula for all non-radial geodesics in Schwarzschild spacetime expressed via Weierstrass elliptic functions, including trajectories passing through turning points.

Findings

Unified formula for all non-radial geodesics

Explicit expressions for proper and coordinate time

Application of Weierstrass functions to Schwarzschild geodesics

Abstract

The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null 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.