Geodesic equation in Schwarzschild--(anti-)de Sitter space--times: Analytical solutions and applications

TL;DR

This paper provides complete analytical solutions to the geodesic equations in Schwarzschild--(anti-)de Sitter space--times, characterizing orbits and exploring implications for phenomena like the Pioneer Anomaly.

Contribution

It introduces a method to solve geodesic equations analytically using theta and sigma functions, applicable to higher-dimensional Schwarzschild space--times.

Findings

Analytic solutions expressed via Kleinian sigma functions.

Characterization of orbit types based on conserved quantities and cosmological constant.

Analysis of the Pioneer Anomaly in the context of the cosmological constant.

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 set of zeros of the theta function, called the theta divisor. In its final form the solutions can be expressed in terms of derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant. Using the analytical solution, the question whether the cosmological constant could be a cause of the Pioneer Anomaly is addressed. The periastron shift and its post--Schwarzschild limit is derived. The developed method can also be applied to the geodesic equation in higher dimensional Schwarzschild space--times.