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General Relativity and Gravitation

, Volume 44, Issue 4, pp 1057–1072 | Cite as

Analytical time-like geodesics in Schwarzschild space-time

  • Uroš Kostić
Research Article

Abstract

Time-like orbits in Schwarzschild space-time are presented and classified in a very transparent and straightforward way into four types. The analytical solutions to orbit, time, and proper time equations are given for all orbit types in the form r = r(λ), t = t(χ), and ττ (χ), where λ is the true anomaly and χ is a parameter along the orbit. A very simple relation between λ and χ is also shown. These solutions are very useful for modelling temporal evolution of transient phenomena near black holes since they are expressed with Jacobi elliptic functions and elliptic integrals, which can be calculated very efficiently and accurately.

Keywords

Schwarzschild space-time Analytical solutions Time-like geodesics 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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