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.

c 1 c 2 c 1 + L ′2 . For nonvanishingñ, E, andL the maximum of the parabola is no longer located at ξ = 0 or, equivalently, the zeros are no longer symmetric with respect to ξ = 0. Only for vanishingñ, E, orL both cones are symmetric with respect to the equatorial plane.The ϑ-motion can be classified according to the sign of c 2 − L ′2 :1. If c 2 − L ′2 < 0 then Θ ξ has 2 positive zeros for L ′ Eñ > 0 and ϑ ∈ (0, π/2), so that the particle moves above the equatorial plane without crossing it. If L ′ Eñ < 0 then ϑ ∈ (π/2, π). TO CEO EO D 0 C B O BO EO ⋆ ⋆ ⋆ EO D 0 CBO ⋆ ⋆ ⋆ ⋆ ⋆ BO EO (b) Taub-NUT space-time II. The CEO (red) starts at the horizon r−, the CBO (blue) starts at the horizon r− and terminates at the horizon r+. FIG. 10: Topology of orbits in Carter-Penrose diagrams of Taub-NUT space-time. The orbits drawn in black are standard orbits with infinite proper time, the orbits in red, blue, and green are geodesically incomplete.crossing one of the horizons r − or r + , it cannot cross this horizon a second time, because ψ(γ) diverges there. At

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.

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 ℘, ζ, and σ functions as well as hyperelliptic Kleinian σ functions restricted to the one-dimensional θ-divisor. We analyse 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.PACS numbers:

The complete analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild, Schwarzschild-(anti)de Sitter, Reissner-Nordström and Reissner-Nordström-(anti)de Sitter space-times are presented. Using the Jacobi inversion problem restricted to the theta divisor the explicit solution is given in terms of Kleinian sigma functions. The derived orbits depend on the structure of the roots of the characteristic polynomials which depend on the particle's energy and angular momentum, on the mass and the charge of the gravitational source, and the cosmological constant. We discuss the general structure of the orbits and show that due to the specific dimension-independent form of the angular momentum and the cosmological force a rich variety of orbits can emerge only in four and five dimensions. We present explicit analytical solutions for orbits up to 11 dimensions. A particular feature of Reissner-Nordström space-times is that bound and escape orbits traverse through different universes.