Schwarzschild geodesics in terms of elliptic functions and the related red shift

TL;DR

This paper derives exact Schwarzschild geodesics using elliptic functions, enabling precise calculations of orbital precession, light deflection, and gravitational redshift, with applications in both analytical and numerical contexts.

Contribution

It introduces a simple elliptic function formulation of Schwarzschild geodesics, facilitating accurate analytical and numerical computations of relativistic effects.

Findings

Exact geodesics expressed via Weierstrass elliptic functions

Calculation of perihelion precession and light deflection in post-Einsteinian approximation

Determination of gravitational redshift for a star in Schwarzschild field

Abstract

Using Weierstrassian elliptic functions the exact geodesics in the Schwarzschild metric are expressed in a simple and most transparent form. The results are useful for analytical and numerical applications. For example we calculate the perihelion precession and the light deflection in the post-Einsteinian approximation. The bounded orbits are computed in the post-Newtonian order. As a topical application we calculate the gravitational red shift for a star moving in the Schwarzschild field.