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GR: Fachverband Gravitation und Relativitätstheorie

GR 4: Klassische Allgemeine Relativitätstheorie I

GR 4.5: Talk

Tuesday, March 10, 2009, 15:20–15:40, A214

Analytic solutions of the geodesic equation in higher dimensional static spherically symmetric space–times — Eva Hackmann¹, •Valeria Kagramanova², Jutta Kunz², and Claus Lämmerzahl¹ — ¹ZARM, Universität Bremen, Am Fallturm, D–28359 Bremen — ²Institut für Physik, Universität Oldenburg, D–26111 Oldenburg

We present 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. 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.