Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in General Relativity

TL;DR

This paper develops a method to invert hyperelliptic integrals of arbitrary genus using algebro-geometric techniques, with applications to modeling particle motion in complex space-times within General Relativity.

Contribution

It introduces a new approach employing Klein–Weierstraß sigma functions for hyperelliptic integral inversion, applicable to arbitrary genus and exemplified with genus three cases.

Findings

All parameters for calculations are expressed via period matrices and theta-constants.

The method is detailed for genus two and three cases.

Application demonstrated with particle motion in a genus three hyperelliptic curve.

Abstract

The description of many dynamical problems like the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of a holomorphic hyperelliptic integral. The result of the inversion is defined only locally, and is done using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the $\theta$--divisor. For a representation of the hyperelliptic functions the Klein--Weierstra{\ss} multivariable sigma function is introduced. It is shown that all parameters needed for the calculations like period matrices and Abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and theta-constants. The cases of genus two and three are considered in detail. The method is exemplified by particle motion associated with a genus three hyperelliptic curve.