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

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.

23 pages, 12 figures

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• A massive binary black-hole system in OJ287 and a test of general relativity
• The Contribution of the Cosmological Constant to the Relativistic Bending of Light Revisited
• Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes
• Complete analytic solution of the geodesic equation in Schwarzschild--(anti) de Sitter space--times
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• Two-body problem with the cosmological constant and observational constraints
• Integrability and the Kerr-(A)dS black hole in five dimensions

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• Light paths of normal and phantom Einstein-Maxwell-dilaton black holes
• A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems
• Geodesic Motion in the Singly Spinning Black Ring Spacetime
• Approximating light rays in the Schwarzschild field
• Geodesic Motion in the (Charged) Doubly Spinning Black Ring Spacetime
• Coupling a Point-Like Mass to Quantum Gravity with Causal Dynamical Triangulations
• Velocity and velocity bounds in static spherically symmetric metrics