
Regular sources of spacetimes with singularities
Papajčík, Matúš ; Ledvinka, Tomáš (advisor) ; Žofka, Martin (referee)
Since the formulation of Einstein's equations of general relativity, analytical methods were aplied to find their solutions. The complexity and the nonlinear character of the equations meant big difficulty of searching for solutions. Only recently the field of numerical relativity has been developed, which offered a much wider means of research of the properties o these equations. In this thesis we firstly solved the problem of regularization of newtonian sin gular potential by the method of binding potentials. Next we aplied the methods in general theory of relativity, where we found a suitable source and its pressu res of the same spherically symmetrical problem. Further we investigated this known Schwarzschild solution in Weyl coordinates for better understanding and comparison of Bonnor's results.


Dynamic elektromagnetic fields in the Kerr spacetime
Skoupý, Viktor ; Ledvinka, Tomáš (advisor) ; Kofroň, David (referee)
In this thesis we study a test electromagnetic field in the vicinity of Kerr black hole and with methods of extraction of its rotational energy. We are investigating a process in which a particle moves in an electromagnetic resonator around Kerr black hole. The energy of the particle is transferred to the electromagnetic field and the particle falls into the black hole with negative energy. We begin with the derivation of Maxwell's and Teukolsky equations and their numerical solutions. We derive a boundary condition for an electromagnetic field on a spherical mirror around the black hole, find the field that satisfies this condition, and describe the procedure for numerical calculation. Next, we calculate the trajectories of charged test particles in such a field and find particles that fall into the black hole with negative energy. We have found that it is possible for the particle to fall into the black hole with the energy of −124% of its rest mass, and the parameters of the electromagnetic field and trajectory of the particle need to be carefully selected.


Coordinate choice in the OppenheimerSnyder model of gravitational collapse
Honsa, Lukáš ; Ledvinka, Tomáš (advisor) ; Žofka, Martin (referee)
The thesis investigate a simple model of a gravitational collapse. The mo del considers a dust of constant density and zero pressure. In the first part of the thesis we cogitate over well known analytical description of the model under investigation. We elucidate the more difficult mathematical steps and the more complicated parts of general relativity. In the second part of the thesis we con struct coordinates which cover both the collapsing dust and the outer parts of space  vacuum. We discuss interesting aspects of general relativity portrayed by the chosen description. 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.


Physical interpretation of special solutions of EinsteinMaxwell equations
Ryzner, Jiří ; Žofka, Martin (advisor) ; Ledvinka, Tomáš (referee)
V klasické fyzice m·že být ustavena statická rovnováha v soustavě, která obsahuje extrémně nabité zdroje gravitačního a elektromagnetického pole. Udivujícím faktem je, že tato situace m·že nastat i pro černé díry v relativis tické fyzice. Tato práce vyšetřuje speciální případ nekonečně dlouhé, extrémně nabité struny, zkoumá geometrii prostoročasu, elektrogeodetiky, vlastnosti zdroje a srovnává řešení se situací v klasické fyzice. Dále se zabýváme analogickou situací v dynamickém prostoročase s kosmologickou konstantou, a řešení porovnáváme s jeho statickou verzí. Nakonec zkoumáme periodické řešení Laplaceovy rovnice, které odpovídá nekonečně mnoha extremálním bodovým zdroj·m rozloženým v pravidelném rozestupu podél přímky. Vyšetřujeme vlastnosti elektrostatického potenciálu a ukazujeme, že v limitě velké vzdálenosti od osy tvořené zdroji pře chází toto řešení v nabitou strunu. 1


Model of relativistic spinning system
Slezák, Daniel ; Ledvinka, Tomáš (advisor) ; Loukes Gerakopoulos, Georgios (referee)
Contrary to massive point particles, a description of extended bodies dynamics inclu des higher mass moments, the first of which is spin. In this manner, Mathisson PapapetrouDixon (MPD) equations has to be used instead of the geodesic equation to capture the more complicated evolution of the system. In this work, an extended system is represented by a set of freely moving, occasionally colliding point particles. As an aid in the construction of the model, some of these particles carry negative mass so it is possible to enclose their trajectories by elastic collisions. We then define a system's representative quantities, such as mass, momentum and spin. However, their relativistic theory requires to solve mainly the problems of parallel transport and the choice of a reference frame. Finally  from the known movement of the in dividual particles, we can show that the whole system obeys the MPD equations. For that we use the simplification of small spacetime curvature along with a more extensive use of parallel transport instead of stressenergy tensor dynamic equation, the significance of which we limit to the behaviour of the component particles.


Gravitational collapse of magnetized matter
Bucko, Jozef ; Ledvinka, Tomáš (advisor) ; Scholtz, Martin (referee)
The scope of interest of this work is to study the gravitational collapse of magnetized matter. We consider homogeneous, isotropic and ideally conduct ing star threaded by the test magnetic field. This field is chosen to be a relativistic generalization of dipole field outside the star and homogeneous field under the stellar surface. Dynamics of the collapse is described by OppenheimerSnyder solution so we consider the framework in which the motion of stellar matter re mains unaffected by the magnetic field. In Schwarzschild's coordinates we review the exact solution of Maxwell's equations for electric and magnetic field inside the star and find the evolution equations for both fields outside the star that have to be treated numerically. Then we introduce Novikov coordinates and find the respective fields evolution equations. We test the equations using known exact so lution for timedependent magnetic dipole. Finally, we adopt chosen coordinates to become singularityavoiding and integrate the respective differential equations numerically. 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


Archaic, Traditional Law and Modern Commercial Law: A Study of Their Comparisons
Ledvinka, Tomáš ; Sokol, Jan (advisor) ; Kandert, Josef (referee) ; Brezina, Peter (referee)
The old anthropological question of the comparison between an archaic or traditional commercial law on one hand and a modern commercial law on the other is revisited using a conceptualization of an empirical study of legal comparisons performed within the real decisionmaking processes at work in the current Czech justice system. Commercial law is represented by a single legal institution  the law of reciprocity (comitas gentium)  which regulates the cooperation between various legal authorities and legal systems potentially entangled in crossborder commercial disputes. The reader is first introduced to the context and evidencedependency of any legal comparison ranging from the representation of law and feud in Yemen at an asylum trial, to the legal systems regulating exchange contracts in Afghanistan involving crossborder disputes. The idea of comparing legal systems as two autonomous social units is abandoned in favor of the study of the comparative practices of a small population of Czech legal authorities, which furnishes readers with plenty of questions about the social organization of legal cognition. The dissertation refrains from drawing final conclusions using legal comparisons, instead it focuses on the limitations and barriers of marshalling evidence (symbolic representations) of...


Kinematics of particle collisions in the ergosphere of Kerr black hole
Skoupý, Viktor ; Ledvinka, Tomáš (advisor) ; Žofka, Martin (referee)
In the thesis we deal with an effect which can be used to extract energy from a rotating black hole, socalled collisional Penrose process. First we investigate the ways to find the equations of motion in the general relativity using Hamilto nian. Then we examine the equations of motion and their consequences in several coordinate systems for the spacetime in the vicinity of a rotating black hole. Fi nally we look into ways to create a particle capable to escape to infinity with as big energy as possible using Compton scattering and annihilation. The biggest energy found is approximately 14 times the energy of the incoming particles. The efficiency decreases with the distance from the horizon and with the decreasing specific angular momentum of the black hole. 1
