Complete analytic solution of the geodesic equation in Schwarzschild--(anti) de Sitter space--times

TL;DR

This paper provides a comprehensive analytic solution to the geodesic equations in Schwarzschild--(anti) de Sitter space--times, expressed through advanced mathematical functions, and classifies solutions based on physical parameters.

Contribution

It introduces a complete analytic solution framework for geodesics in Schwarzschild--(anti) de Sitter space--times using Kleinian sigma functions, extending previous partial solutions.

Findings

Solutions expressed via derivatives of Kleinian sigma functions

Classification of geodesics based on zeros of characteristic polynomial

Analytic solutions applicable to various physical parameters

Abstract

The complete set of analytic solutions of the geodesic equation in a Schwarzschild--(anti) de Sitter space--time is presented. The solutions are derived from the Jacobi inversion problem restricted to the theta--divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The solutions are completely classified by the structure of the zeros of the characteristic polynomial which depends on the energy, angular momentum, and the cosmological constant.