Hyperelliptic integrals and Ho\v{r}ava-Lifshitz black hole space-times

TL;DR

This paper develops a comprehensive method for inverting hyperelliptic integrals using algebro-geometric techniques, with applications to particle motion in complex space-times like Hořava-Lifshitz black holes.

Contribution

It introduces a new approach to hyperelliptic integral inversion using the Klein–Weierstraß sigma-function, applicable to various genera and relevant for black hole geodesic solutions.

Findings

Explicit formulas for period matrices and theta constants.

Detailed analysis of genus two, three, and four cases.

Application to geodesic equations in Hořava-Lifshitz black hole space-times.

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 hyperelliptic integrals of all three kinds. The result of the inversion is defined locally, 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, three and four are considered in detail. The method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves. Applications are for instance solutions to the geodesic equations in the space-times of static, spherically symmetric Ho\v{r}ava-Lifshitz black holes.