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

Eva Hackmann and Claus Lämmerzahl
Phys. Rev. D 78, 024035 – Published 22 July 2008

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.

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  • Received 15 May 2008

DOI:https://doi.org/10.1103/PhysRevD.78.024035

©2008 American Physical Society

Authors & Affiliations

Eva Hackmann* and Claus Lämmerzahl

  • ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany

  • *hackmann@zarm.uni-bremen.de
  • laemmerzahl@zarm.uni-bremen.de

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Issue

Vol. 78, Iss. 2 — 15 July 2008

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