Geodesic equations in black hole space-times with cosmological constant

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. This includes an overview of the complete set of possible orbits as well as a discussion of special orbits. The results of this analysis are 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. In addition, series expansions for a small cosmological constant are deduced. All space-times considered in this thesis are special cases of the general class of Plebański-Demiański 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 provide the basic reference for comparisons with more complex space-times. After discussing the slightly more complicated ReissnerNordström space-time, the generalization of both these space-times to the case of a nonvanishing cosmological constant is carried out. In a short excursus it is shown that the methods presented so far can also be applied to higher-dimensional spacetimes. Subsequently, the discussion will be further generalized to the stationary and axially symmetric Kerr and Kerr-(anti-)-de Sitter space-times. For both space-times, general statements concerning the geometrical interpretation of the additional constant of motion, the Carter constant, are formulated. As in general the geodesic motion in axially symmetric space-times can not be restricted to a plane, the classification of orbits is extended to include the latitudinal direction. Finally, the analytical solution methods developed in this thesis are applied to the general class of Plebański-Demiański space-times itself for the case of a vanishing acceleration of the gravitating object. While so far the geodesic equation has only be analytically solved in Kerr-Newman space-times together with all its special cases, it is hereby shown that this can be extended to all integrable geodesic equations in black hole electrovac space-times without acceleration. This thesis also enables future work on the mathematical methods presented here as well as on analytical treatments of a range of physical problems. Possible further developments and applications are suggested at the end of the thesis. List of publications E. Hackmann and C. Lämmerzahl. Complete analytic solution of the geodesic equation in Schwarzschild-(anti) de Sitter space-times. Phys. Rev. Lett. , 100: 171101-1, 2008. E. Hackmann and C. Lämmerzahl. Geodesic equation in Schwarzschild-(anti-)de Sitter space–times: Analytical solutions and applications. Phys. Rev., D 78: 024035, 2008. E. Hackmann and C. Lämmerzahl. Geodesic equation and theta-divisor. In Recent Developments in Gravitation and Cosmology, volume 977 of AIP Conference Proceedings, pages 116-133, USA 2008. E. Hackmann and C. Lämmerzahl. Hyperelliptic functions and geodesic equations. In Proceedings in Applied Mathematics and Mechanics, volume 8, issue 1, pages 10723-10724, Wiley-VCH, Weinheim, 2008. E. Hackmann, V. Kagramanova, J. Kunz and C. Lämmerzahl. Analytic solutions of the geodesic equations in higher dimensional static spherically symmetric space– times. Phys. Rev., D 78: 124018, 2008. E. Hackmann, V. Kagramanova, J. Kunz, and C. Lämmerzahl. Analytic solutions of the geodesic equation in axially symmetric space–times. Europhys. Lett. , 88: 30008, 2009. E. Hackmann, V. Kagramanova, J. Kunz, and C. Lämmerzahl. Analytical solution of the geodesic equation in Kerr-(anti) de Sitter space-time. Phys. Rev., D 81, 044020, 2010. E. Hackmann, C. Lämmerzahl and A. Macias. Complete classification of geodesic motion in fast Kerr and Kerr-(anti-)de Sitter space-times. In New trends in statistical physics: Festschrift in honour of Leopoldo Garcia-Colin’s 80 birthday. World Scientific, Singapore, 2010, to appear. E. Hackmann, B. Hartmann, C. Lämmerzahl and P.Sirimachan. The complete set of solutions of the geodesic equations in the space-time of a Schwarzschild black hole pierced by a cosmic string. Phys. Rev., D 81, 064016, 2010. Acknowledgment First of all I would like to thank my supervisor Prof. Dr. Claus Lämmerzahl for providing me with this interesting topic, his good advice, and for always having time to help despite his considerable workload. Without our numerous interesting, detailed, and helpful discussions and his physical as well as formal explanatory information this thesis would not exist. Also I am grateful to Prof. Dr. Peter H. Richter for his interest in my research, his comprehensive introduction to hyperelliptic functions, and for directing my attention to some most useful papers. A thanks goes also to Dennis Lorek, Dr. Holger Dullin, Prof. Dr. Wolfgang Fischer, Prof. Dr. Hansjörg Dittus, Prof. Dr. Jutta Kunz, and in particular to Dr. Valeria Kagramanova for fruitful discussions and the good cooperation. Further I am much obliged to Prof. Dr. Hans Rath for the opportunity to work at the Center of Applied Space Technology and Microgravity and to my colleagues of the fundamental physics group for their support and the good time. Likewise I want to express my gratitude to my office colleagues Andreas Resch for answering my computer questions and Thorben Könemann for the hardware support. The financial support of the German Research Foundation DFG is gratefully acknowledged. Finally I am deeply grateful to my family for their constant support over all these years and to Daniel for everything.