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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.
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Electromagnetic Waves in Dispersiveand Refractive Relativistic Systems
Bezděková, Barbora ; Bičák, Jiří (advisor) ; Heyrovský, David (referee)
Study of light rays (light world lines) plays a significant role in many of astro- physical applications. Light rays are mainly studied in terms of so-called grav- itational lensing. However, the majority of studies are mainly focused on light propagation in vacuum. If the refractive and dispersive medium characterised by refractive index n is considered, effects occurring due to the medium presence need to be taken into account, which significantly complicates the problem. In the present thesis, rays propagating through simple refractive and dispersive systems, such as plane differentially sheared medium, are studied. In order to simplify the problem, the Hamiltonian equations of motion are used. The ray trajectories in the vicinity of Kerr black hole as well as accessible regions for the rays are also studied. Radial variation of the medium velocity is considered. Due to the recent increase of publications focused on the gravitational lensing in plasma, a detailed review summarizing the results obtained recently is included. 1
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Parameterization of the Kerr solution
Miškovský, David ; Švarc, Robert (advisor) ; Scholtz, Martin (referee)
In this thesis we are exploring basic properties of the Kerr solution using se- veral coordinate systems. Later on, we are deriving general metric form of the spacetime foliated by null hypersurfaces. Employing the formalism of optical sca- lars we shall see, that geometry of a such a spacetime is non-twisting, that is it admits existence of a non-twisting affinely parametrized null geodesic congru- ence. Subsequently, we are trying to express the Kerr solution in the form of non-twisting coordinates. This form would have many applications e.g. in forma- lism of weakly isolated horizons (WHIs) for use in more realistic astrophysical models of black holes.
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Physical interpretation of special solutions of Einstein-Maxwell equations
Ryzner, Jiří ; Žofka, Martin (advisor)
In Newtonian physics, it is possible to establish static equilibrium in a system, which consists of extremal sources of gravitational and electromagnetic field. Surprisingly, this situation can occur in general relativity for black holes, too. This work examines a special case involving an infinitely long, straight, extremally charged string, studies its geometry, electrogeodesics, properties of the source and compares the solution to Newtonian physics. We also investigate an analogous situation in a dynamic spacetime with cosmological constant, and we compare it to the static version. Finally, we investigate a periodical solution of Laplace's equation corresponding to infinitely many extremal point sources distributed at regular intervals along a straight line. We study the properties of the electrostatic potential and show that in the limit of large distances from the axis formed by the sources, the solution approaches the charged string. 1
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Physical interpretation of special solutions of Einstein-Maxwell 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
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Standard and alternative cosmological models
Pulnova, Yelyzaveta ; Acquaviva, Giovanni (advisor) ; Žofka, Martin (referee)
The main aim of this thesis is the study of the dependence of the scale factor on the cosmic time for different models of Universe's evolution in the framework of the general theory of relativity. In this thesis we consider the FLRW metric and admit nonzero curvature. The models we consider differ from each other by the equation of state of the source, hence by the composition of the cosmic fluid under study. In this thesis the following models are discussed: ΛCDM (we consider a perfect cosmic fluid consisting of the incoherent dust, radiation and a cosmological constant in a curved space-time), generalized Chaplygin gas, and, also, two kinds of the scalar field (describing separately power-law inflation and the period after recombination). The numerical and analytical results obtained are processed graphically. 1
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Termodynamika černých děr. Entropie a informace.
Liška, Marek ; Acquaviva, Giovanni (advisor) ; Scholtz, Martin (referee)
The aim of the thesis is to provide a review of black hole thermodynamics and its relation with concepts of entropy and physical information. We start by deriving the four laws of black hole thermodynamics in the context of classical general relativity. To supplement this, we use semiclassical limit of quantum mechanics to show that black holes radiate and have non-zero thermodynamic temperature. In the second part of the thesis we describe the concepts of the Shannon and von Neumann entropy and of physical information. Lastly, we discuss the use of these concepts in the context of black hole mechanics. 1
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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, so-called 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 space-time 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
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Probabilistic Spacetimes
Káninský, Jakub ; Svítek, Otakar (advisor) ; Žofka, Martin (referee)
Probabilistic Spacetime is a simple generalization of the classical model of spa- cetime in General Relativity, such that it allows to consider multiple metric field realizations endowed with probabilities. The motivation for such a generalization is a possible application in the context of some quantum gravity approaches, na- mely those using the path integral. It is argued that this model might be used to restrict the precision of the geometry on small scales without postulating discrete structure; or it may be used as an effective description of a probabilistic geometry resulting from a full-fledged quantum gravity computation.
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Gravitational collapse of scalar field
Šípka, Martin ; Ledvinka, Tomáš (advisor) ; Scholtz, Martin (referee)
In this thesis we study a critical collapse of a scalar field. We follow the work of [1] and create detailed and easy to understand figures and images describing the results and the phenomena observed during our calculations. We also try to give a detailed look at numerical methods used along with two ways of convergence testing. 1
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