On Geodesic Dynamics in Deformed Black-Hole Fields
“Almost all” seems to be known about isolated stationary black holes in asymptotically flat space-times and about the behaviour of test matter and fields in their backgrounds. The black holes likely present in galactic nuclei and in some X-ray binaries ar
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Abstract “Almost all” seems to be known about isolated stationary black holes in asymptotically flat space-times and about the behaviour of test matter and fields in their backgrounds. The black holes likely present in galactic nuclei and in some X-ray binaries are commonly being represented by the Kerr metric, but actually they are not isolated (they are detected only thanks to a strong interaction with the surroundings), they are not stationary (black-hole sources are rather strongly variable) and they also probably do not live in an asymptotically flat universe. Such “perturbations” may query the classical black-hole theorems (how robust are the latter against them?) and certainly affect particles and fields around, which can have observational consequences. In the present contribution we examine how the geodesic structure of the static and axially symmetric black-hole space-time responds to the presence of an additional matter in the form of a thin disc or ring. We use several different methods to show that geodesic motion may become chaotic, to reveal the strength and type of this irregularity and its dependence on parameters. The relevance of such an analysis for galactic nuclei is briefly commented on.
1 Introduction Geodesic structure is a very comprehensive and demonstrative attribute of space (-time). As traversing regions of all possible sizes, geodesics can unveil a local behaviour of a given system as well as tiny tendencies only discernible over an extensive span of time. A default example of the latter are weak irregularities attending a lack of O. Semerák (B) Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic e-mail: [email protected] URL: utf.mff.cuni.cz/en/index.html P. Suková Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected] URL: http://www.cft.edu.pl/en/index.php © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_17
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a complete set of integrals of the motion. Actually, the long-term geodesic dynamics is a suitable tool how to detect, illustrate, evaluate, classify and compare different deviations of a chosen system from a certain simple, “regular” ideal. In mathematics and physic, such an ideal is represented by linear systems (the finite-dimensional in particular). However, within the last 150 years it has become clear that even in these highly abstract fields the linear systems represent just marginal tips within a vast non-linear tangle which is typically prone to “irregularities” and which can display “chaotic” behaviour even in rather simple settings. The modern theory of chaos was apparently inspired by Henri Poincaré’s treatment of a three-body system, and our interest will also focus on systems driven by gravitational interaction in this contribution. Specifically, we consider a si
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