Complete Analytic Solution of the Geodesic Equation in Schwarzschild–(Anti-)de Sitter Spacetimes

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 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.