Geodesic equations in black hole space-times with cosmological constant

Abstract

The physics of a gravitational field can be explored by studying the geodesic motion of test particles and light. Although the majority of gravitational effects can be discussed using approximations and numerics, a systematic study of all effects can only be achieved by using analytical methods. In particular, exact analytic treatments can finally answer the question whether the cosmological expansion, modeled here by a cosmological constant, has an observable influence on effects as the Pioneer anomaly or the creation of gravitational waves.This thesis is devoted to the study of the geodesic motion in space-times with a nonvanishing cosmological constant using analytical methods. In each space-time considered here, the discussion of geodesics takes place on two different levels: The first is the classification of orbits in terms of the black hole and test particle or light parameters, which is used to compare the geodesic motion in different space-times and, in particular, to study the influence of the cosmological constant on a geodesic. On the second level, the analytical solutions of geodesic equations in terms of elliptic or hyperelliptic functions are derived and used to determine analytical expressions for observables.All space-times considered in this thesis are special cases of the general class of Plebanski-Demianski space-times, which includes the static and spherically symmetric Schwarzschild solution as the most simple case. The complete discussion of geodesics in this space-time provides the basic reference for comparisons with more complex space-times, i.e. the Reissner-Nordstr\"om space-time, the generalization of both these space-times to the case of a nonvanishing cosmological constant, as well as the stationary and axially symmetric Kerr and Kerr-(anti-)-de Sitter space-times. In a short excursus it is shown that the methods presented so far can also be applied to higher-dimensional space-times. Finally, the analytical solution methods developed in this thesis are applied to the general class of Plebanski-Demianski space-times itself for the case of a vanishing acceleration of the gravitating object, hereby showing that the analytical solutions of all integrable geodesic equations in black hole electrovac space-times without acceleration can be given explicitly.