Geodesic equations and algebro-geometric methods

TL;DR

This paper explores solving geodesic equations in various space-times using elliptic functions, enabling detailed analysis of gravitational fields and observable effects like perihelion shift.

Contribution

It introduces algebro-geometric methods, specifically elliptic functions, for solving geodesic equations in the Plebański-Demiański family of solutions, enhancing analytical approaches.

Findings

Analytical solutions for geodesics in multiple space-times.

Calculation of observable effects such as perihelion shift.

Application of elliptic functions to gravitational field analysis.

Abstract

For an investigation of the physical properties of gravitational fields the observation of massive test particles and light is very useful. The characteristic features of a given space-time may be decoded by studying the complete set of all possible geodesic motions. Such a thorough analysis can be accomplished most effectively by using analytical methods to solve the geodesic equation. In this contribution, the use of elliptic functions and their generalizations for solving the geodesic equation in a wide range of well known space-times, which are part of the general Pleba\'nski-Demia\'nski family of solutions, will be presented. In addition, the definition and calculation of observable effects like the perihelion shift will be presented and further applications of the presented methods will be outlined.