
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 galaxycluster 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, FrenkWhite 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 haloparameter influence on gravi tational lensing regimes of the combined model yields a complete parameterspace 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


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


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 blackhole 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 blackhole horizon and inspect the behaviour of the scalars inside. The geometry turns out to be deformed in a nontrivial 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


Spacetimes of ring sources
Pešta, Milan ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee)
Marginally outertrapped surfaces (MOTSs) are found for a family of spacelike hypersurfaces described by the BrillLindquist 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.


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 nonsingular, 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 nonsingular 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 spacetimes of matter rings. 1


Geodesic chaos in a perturbed Schwarzschild field
Polcar, Lukáš ; Semerák, Oldřich (advisor) ; Kopáček, Ondřej (referee)
We study the dynamics of timelike geodesics in the field of black holes perturbed by a circular ring or disc, restricting to static and axisymmetric class of spacetimes. Two analytical methods are tested which do not require solving the equations of motion: (i) the socalled geometric criterion of chaos based on eigenvalues of the Riemann tensor, and (ii) the method of Melnikov which detects the chaotic layer arising by breakup of a homoclinic orbit. Predictions of both methods are compared with numerical results in order to learn how accurate and reliable they are.


Stationary fields in blackhole spacetimes
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 BelinskiiZakharov 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 BachWeyl ring. 1


Stationary fields in blackhole spacetimes
Čížek, Pavel ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee) ; Gürlebeck, Norman (referee)
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 BelinskiiZakharov 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 BachWeyl ring. 1


Rotating thin disc around a Schwarzschild black hole: properties of perturbative solution
Kotlařík, Petr ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee)
In 1974, Will presented a solution for the perturbation of a Schwarzschild black hole due to a slowly rotating and light thin disc given in terms of a multipole expansion of the perturbation series. In a recently submitted paper, P. Čížek and O. Semerák generalized this procedure to the perturbation by a slowly rotating finite thin disc, using closed forms of Green functions rather than the multipole expansion. The method is illustrated there, in the first perturbation order, on the constantdensity disc. In this thesis, we summarize, check and plot some of the obtained properties, and show how the presence of the disc changes the geometry of a horizon and the position of significant circular orbits. 1


Gravitational lensing by combined continuous and discrete matter
Timko, Lukáš ; Heyrovský, David (advisor) ; Semerák, Oldřich (referee)
The aim of this work is to investigate the influence of perturbation by point mass on the caustic structure of the NavarroFrenkWhite model using the inverse ray shooting method. We specifically focus on the description of metamorphoses between different caustic topologies when changing the relative mass and position of the point. It turns out that in the combined model of discrete and continuous matter there appear some types of metamorphoses, such as elliptical umbilic, lips and probably also hyperbolic umbilic, that do not exist in purely discrete models. The main, and somewhat surprising, result of the work is the finding that even at the relative mass of the point 104 103 the perturbation is strong enough to cause changes in the caustic structure which are in size comparable to the original caustics.
