An exact solution of the orbit equation for a massive particle in Schwarzschild metric

TL;DR

This paper derives an exact analytical solution for the orbit of a massive particle in Schwarzschild spacetime using Jacobi's elliptic functions, revealing detailed relativistic orbital behaviors and bound regions.

Contribution

It provides the first exact solution of the orbit equation in Schwarzschild spacetime using elliptic functions, surpassing approximate methods.

Findings

Oscillatory orbital movements are confirmed analytically.

Bound regions for particle motion are characterized.

Relativistic effects are explicitly demonstrated in the exact solution.

Abstract

In this paper, we consider a spherically curved symmetric spacetime to exact solving the orbit equation of a massive particle by using Jacobi's elliptic functions. Generally, the solution of the orbit equation provides the relativistic effects on the massive particle, absents in Newtonian mechanics. Besides, we investigate the additional physical information introduced by the exact solution to the orbit equation that is not visible in the approximate solutions traditionally presented in literature. Here, we exactly solve the problem by the use an analytical methodology step by step in order to provide detailed solutions as well as demonstrate with mathematical rigour the geodesic solution in terms of Jacobi's elliptic functions. We find oscillatory movements of the orbit of the massive particle at the expected regimes without to consider any heuristic argument. Bound regions to the solution of the equation of motion is presented, finding the aspect of the geodesic when the massive particle is trapped in the gravitational field of the source.