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Homoclinic Chaos in Black-hole Fields
Hájková, Tereza-Marie ; Semerák, Oldřich (advisor) ; Suková, Petra (referee)
The existence of homoclinic orbits is an intrinsic feature of static black hole's space- time. The regularity of the geodesic motion around these black holes can therefore be quickly disrupted due to the small changes in the original space-time. The nature of the dynamics around a perturbed orbit depends on the manner of intersection of the surrounding stable and unstable manifolds. If they intersect transversally, the homoclinic orbit splits into chaotic layers. In this thesis, the mathematical formulation of chaotic dynamical systems and main properties of the geodesic motion in circular space-times are discussed. Thereupon, the space-time around a static black hole is reproduced by classical approximations by using Paczyński-Wiita and logarithmic pseudo-Newtonian potentials. By means of the effective potential method, the homoclinic orbits are found for these potentials. In addition, the analysis of the general circular space-time is done and the equations of geodesic motion in axially symmetric space-times are examined. Finally, the motion in a Schwarzschild space-time with a static axially symmetric external source is inspected. 1
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Shape of the Kerr gravitational field
Tynianskaia, Valeriia ; Semerák, Oldřich (advisor) ; Švarc, Robert (referee)
Kerr metric is one of the most well-known and useful exact solutions of Einstein equations. We study various geometric properties of the Kerr spacetime in order to gain intuition for its spatial shape. In the review part we summarize basic features of the Kerr geometry, we write down Carter equations for geodesic motion in the Kerr spacetime, and we introduce kinematic characteristics of time-like and light-like congruences, such as expansion, shear and twist. In the second part of the thesis we calculate scalars for acceleration, expansion, shear and twist - and plot the corresponding "equipotential" surfaces - for several privi- leged congruences, namely the Carter observers, the static observers, the zero-angular- momentum observers, the principal null congruence and the recently found non-twisting null congruence(s). We also draw surfaces radially equidistant from the horizon and sur- faces spatially orthogonal to the PNC and to the twist-free congruences, as well as the surfaces of constant energy and redshift for the important time-like congruences. 1
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Study of geodesic chaos by fractal methods
Sychrovský, David ; Semerák, Oldřich (advisor) ; Čížek, Martin (referee)
We study the dynamics of free test particles in a field of Schwarzschild black hole surrounded by an external exact thin axisymmetric solutions of Einstein's equations. Specifically, we use the Bach-Weyl ring and two member of the inverted Morgan-Morgan family of solutions as the additional sources. The fractal basin boundary and other meth- ods are used to detect and quantify chaos in time-like geodesic motion of the particles, primarily by computing box-counting dimension of said basin boundary. Our results mainly consist of the dependence of the chaoticity of these systems on mass and radius of the additional source as well as conserved energy and angular momentum of the test particles. We compare our results to literature and expand on them. 1
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Magnetic field of current loops around black holes
Vrba, Šimon ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
The magnetic field of a testing current loop in the equatorial plane around a Schwarzschild and Kerr black holes is visualized. In particular, the cases of an extreme black hole and of a Kerr naked singularity are analyzed. The simplest models of massless thin and thick current disks around a Schwarzschild black hole are presented. 1
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Stationary fields in black-hole space-times
Čížek, Pavel ; Semerák, Oldřich (advisor)
Motivated by modelling of astrophysical black holes surrounded by accretion structures, as well as by theoretical interest, we study two methods how to ob- tain, within stationary and axisymmetric solutions of general relativity, a metric describing the black hole encircled by a thin ring or a disc. The first is a suitable perturbation of a Schwarzschild black hole. Starting from the seminal paper by Will (1974), we showed that it is possible to express the Green functions of the problem in a closed form, which can then be employed to obtain, e.g., a reason- able linear perturbation for a black hole surrounded by a thin finite disc. In the second part we tackle the same problem using the Belinskii-Zakharov generating algorithm, showing/confirming that in a stationary case its outcome is unphysi- cal, yet at least obtaining a modest new result for the (static) "superposition" of a Schwarzschild black hole with the Bach-Weyl ring. 1
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Imprint of individual galaxies on gravitational lensing by galaxy clusters
Střeleček, Jan ; Heyrovský, David (advisor) ; Semerák, Oldřich (referee)
The aim of this work is to study gravitational lensing of galaxy-cluster halos in- fluenced by individual galaxies in the halo. The halo mass is generally dominated by dark matter, which can be described using a Navarro, Frenk-White density profile. In our model, we use a spherical halo defined by this model and point masses as simplest approximation for individual galaxies. The analysis of halo-parameter influence on gravi- tational lensing regimes of the combined model yields a complete parameter-space map of critical curves and caustics. In addition we present an adaptation that makes this model numerically more advantageous and we illustrate special cases of the combined influence of two galaxies. 1
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The fields of current loops around black holes
Vlasáková, Zuzana ; Semerák, Oldřich (advisor) ; Karas, Vladimír (referee)
The field of a test current loop placed symmetrically in the equatorial plane around a Kerr black hole has been studied several times and solutions have been published in different forms. We compare these results and determine their limits in important places - in radial infinity, on the outer event horizon, on the static limit, in the equatorial plane and on the axis of symmetry. Furthermore, we show the behaviour of the field corresponding to the extreme black hole and verify Meissner effect. In the end we determine the field of a simple model of a current disc by a superposition of test current loops. This problem has an astrophysical motivation - the description of accretion discs in the vicinity of black holes. 1
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Black holes under the influence of strong sources of gravitation
Kotlařík, Petr ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
In this thesis we study a deformation of a black-hole spacetime due to another strong sources of gravity. Keeping within static and axially symmetric metrics, we consider a binary of Schwarzschild black holes held apart from each other by a repulsive effect of an Appell ring. After verifying that such a system can rest in static equilibrium (without any supporting struts), we compute its several basic geometric characteristics and we plot simple invariants determined by the metric functions (especially lapse, or, equivalently, potential) and by their first and second derivatives (gravitational acceleration and Kretschmann scalar). Then we extend the analysis below the black-hole horizon and inspect the behaviour of the scalars inside. The geometry turns out to be deformed in a non-trivial way, we even find regions of negative Kretschmann scalar in some cases. In the second part, we present a summary of the perturbative solution describing a slowly rotating system of a black hole surrounded by a thin finite circular disc, and an analysis of equatorial circular geodesics in such a spacetime. 1
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Space-times of ring sources
Pešta, Milan ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee)
Marginally outer-trapped surfaces (MOTSs) are found for a family of space-like hypersurfaces described by the Brill-Lindquist initial data. These hypersurfaces contain a singular ring characterized by its radius, mass and charge. Due to the ring character of the singularity, these surfaces are natural candidates for MOTSs with toroidal topology. By adjusting and employing the numerical method of geodesics, we indeed localize MOTSs of both spherical and toroidal topology, and compare the results with those obtained previously by Jaramillo & Lousto.
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Comparison of Brill waves with the fields of singular rings
Sychrovský, David ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
Circular matter rings are a natural zero approximation of stationary and axially symmetric structures which appear in astrophysics. If the rings are infinitesimally thin (line sources), they are singular, which in the general relativistic description typically implies weird deformation of space in their vicinity. In particular, and contrary to the Newtonian picture, such rings even tend to behave in a strongly directional manner. One solution is to consider non-singular, extended sources (toroids), which may however be difficult to treat exactly and/or be unsatisfactory in other respects. In this thesis we check another option, namely to abandon the "real matter" completely and consider a non-singular source represented by mere curvature arranged, at least at some instant, in a pattern possessing the above symmetries. One such solution of Einstein's equations is known as the Brill waves; we study its properties at the moment of time symmetry (when it is momentarily static), in order to compare it with the space-times of matter rings. 1
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