Analytical time-like geodesics

TL;DR

This paper provides a comprehensive classification and analytical solutions for time-like geodesics in Schwarzschild space-time, facilitating efficient modeling of phenomena near black holes using elliptic functions.

Contribution

It introduces a clear classification of time-like orbits and derives explicit analytical solutions using elliptic functions for all orbit types in Schwarzschild space-time.

Findings

Classified time-like orbits into four types.

Derived explicit solutions using elliptic functions.

Facilitated modeling of black hole phenomena.

Abstract

Time-like orbits in Schwarzschild space-time are presented and classified in a very transparent and straightforward way into four types. The analytical solutions to orbit, time, and proper time equations are given for all orbit types in the form r=r(\lambda), t=t(\chi), and \tau=\tau(\chi), where \lambda\ is the true anomaly and \chi\ is a parameter along the orbit. A very simple relation between \lambda\ and \chi\ is also shown. These solutions are very useful for modeling temporal evolution of transient phenomena near black holes since they are expressed with Jacobi elliptic functions and elliptic integrals, which can be calculated very efficiently and accurately.