Analytical solution of the geodesic equation in Kerr-(anti) de Sitter space-times

TL;DR

This paper derives complete analytical solutions for geodesic equations in Kerr-(anti) de Sitter space-times, expressing them with elliptic and hyperelliptic functions, and analyzes orbital dynamics and effects influenced by the cosmological constant.

Contribution

It provides the first comprehensive analytical solutions for geodesics in Kerr-(anti) de Sitter space-times using advanced special functions, extending previous Kerr space-time analyses.

Findings

Explicit formulas for geodesics in Kerr-(anti) de Sitter space-times.

Systematic identification of last stable spherical and circular orbits.

Expressions for deflection angles, orbital frequencies, and relativistic effects.

Abstract

The complete analytical solutions of the geodesic equations in Kerr-de Sitter and Kerr-anti-de Sitter space-times are presented. They are expressed in terms of Weierstrass elliptic p, zeta, and sigma functions as well as hyperelliptic Kleinian sigma functions restricted to the one-dimensional theta-divisor. We analyze the dependency of timelike geodesics on the parameters of the space-time metric and the test-particle and compare the results with the situation in Kerr space-time with vanishing cosmological constant. Furthermore, we systematically can find all last stable spherical and circular orbits and derive the expressions of the deflection angle of flyby orbits, the orbital frequencies of bound orbits, the periastron shift, and the Lense-Thirring effect.