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.
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Notes
- 1.
We also checked that the results are similar for discs of the family with power-law density profile [5] which are however more demanding computationally.
- 2.
The series of just one variable (e.g. position) suffices actually, since the phase space can be reconstructed from a sequence of its time-delayed copies as in the WADV method summarized above.
- 3.
The exponents should reveal the nature of the flow in the vicinity of the reference world-line, hence, while time is running, the separation vector has to be renormalized whenever it reaches a certain “too large” value; the velocity deviation vector, given by difference between four-velocities of the neighbouring world-lines, has to be renormalized by the same factor.
- 4.
These points does not represent true recurrences and are usually discarded from the statistics.
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Acknowledgments
We thank for support the Czech grants GACR 14-10625S (O.S.) and SVV-267301 (P.S.). O.S. is grateful to Dirk Puetzfeld for invitation to the Bad Honnef conference and to the Wilhelm und Else Heraeus Stiftung for support there.
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Semerák, O., Suková, P. (2015). On Geodesic Dynamics in Deformed Black-Hole Fields. In: Puetzfeld, D., Lämmerzahl, C., Schutz, B. (eds) Equations of Motion in Relativistic Gravity. Fundamental Theories of Physics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-18335-0_17
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