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

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 $\ensuremath{\wp}$, $\ensuremath{\zeta}$, and $\ensuremath{\sigma}$ functions as well as hyperelliptic Kleinian $\ensuremath{\sigma}$ functions restricted to the one-dimensional $\ensuremath{\theta}$ 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.