Analytical solution methods for geodesic motion

TL;DR

This paper reviews analytical methods for solving geodesic equations in various black hole and higher-dimensional spacetimes, highlighting mathematical tools like elliptic functions and summarizing existing solutions.

Contribution

It compiles and discusses the current state of analytical solutions to geodesic equations across different spacetime geometries, including black holes and cosmic strings.

Findings

List of known analytical solutions in literature

Application of elliptic and hyperelliptic functions

Solutions in higher-dimensional and string-inspired spacetimes

Abstract

The observation of the motion of particles and light near a gravitating object is until now the only way to explore and to measure the gravitational field. In the case of exact black hole solutions of the Einstein equations the gravitational field is characterized by a small number of parameters which can be read off from the observables related to the orbits of test particles and light rays. Here we review the state of the art of analytical solutions of geodesic equations in various space--times. In particular we consider the four dimensional black hole space--times of Pleba\'nski--Demia\'nski type as far as the geodesic equation separates, as well as solutions in higher dimensions, and also solutions with cosmic strings. The mathematical tools used are elliptic and hyperelliptic functions. We present a list of analytic solutions which can be found in the literature.