On the Closed Form Solution for the Geodesics in SdS Space

TL;DR

This paper derives a closed form solution for geodesics in Schwarzschild-de Sitter space using advanced hyperelliptic functions, solving longstanding mathematical problems with significant astrophysical implications.

Contribution

It provides the first closed form solutions for geodesics in SdS space, involving hyperelliptic and hypergeometric functions, and solves the classical Inversion Problem for genus two integrals.

Findings

Closed form solutions for geodesics in SdS space derived.

Identification of branch points on genus two Riemann surface.

Mathematical solutions applicable to measuring the cosmological constant.

Abstract

The closed form solution for the geodesics of classical particles in SdS space are obtained in terms of hyperelliptic modular functions and multiple hypergeometric functions. The closed form solution for the five roots of the fifth degree polynomial are found giving the branch places on the genus two Riemann surface. The `Inversion Problem', for the genus two hyperelliptic integral is solved in a closed form. Thus a couple of mathematical problems that have been around for a couple of centuries are solved. The solution is important in astrophysical applications of measuring the cosmological constant.