Hyperelliptic functions and motion in general relativity

TL;DR

This paper reviews the use of hyperelliptic functions to solve geodesic equations in complex black hole spacetimes, extending previous elliptic function methods to more general cases involving higher dimensions and cosmological constants.

Contribution

It provides a comprehensive overview of solving geodesic equations with hyperelliptic functions, including methods and examples for genus 2 cases, advancing the mathematical tools for black hole analysis.

Findings

Method for solving hyperelliptic geodesic equations explained

Examples of genus 2 solutions provided

Framework applicable to complex black hole spacetimes

Abstract

Analysis of black hole spacetimes requires study of the motion of particles and light in these spacetimes. Here exact solutions of the geodesic equations are the means of choice. Numerous interesting black hole spacetimes have been analyzed in terms of elliptic functions. However, the presence of a cosmological constant, higher dimensions or alternative gravity theories often necessitate an analysis in terms of hyperelliptic functions. Here we review the method and current status for solving the geodesic equations for the general hyperelliptic case, illustrating it with set a of examples of genus $g=2$.