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Special Surfaces
Ochodnický, Erik ; Vašík, Petr (referee) ; Doupovec, Miroslav (advisor)
The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.
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Geodesics
Čambalová, Kateřina ; Tomáš, Jiří (referee) ; Doupovec, Miroslav (advisor)
The goal of the thesis is to create an overivew of geodesics. At the beginning of their study, they were considered shortest paths connecting two points on surfaces. In the thesis we will show more of the complexity of the term and introduced the properties, some uses of the geodesics and methods of their computation. Later, the Clairaut patches and their geodesics will be analysed. Clairaut patches are characterized by a specific property which makes computation of geodesics simpler. 3D plots of some Clairaut patches and their geodesics are also included.
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Weyl metrics and their generalizations: classical and quantum viewpoint
Polcar, Lukáš ; Svítek, Otakar (advisor) ; Ledvinka, Tomáš (referee) ; Pound, Adam (referee)
In this thesis, we study two distinct topics both connected to stationary axially sym- metric spacetimes. The first is a study of an exact solution sourced by phantom scalar field. This solution can be derived from the well-known Curzon-Chazy metric and has several unusual features. It is a spherically symmetric wormhole which is however not symmetric with respect to its throat, it possesses a non-scalar curvature singularity and functions as a one-directional time machine. The energy content of the spacetime is ex- amined and various other properties are discussed. The remaining parts are dedicated to extreme mass ratio inspirals in two stationary axially symmetric spacetimes, perturbed Schwarzschild and Kerr. The canonical perturbation theory was used to transform the respective geodesic Hamiltonian to action-angle coordinates allowing us to evolve flux- driven inspirals in both spacetimes. 1
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Study of Exact Spacetimes
Švarc, Robert ; Podolský, Jiří (advisor) ; Pravda, Vojtěch (referee) ; Steinbauer, Roland (referee)
In this work we study various aspects of the behaviour of free test particles in Einstein's general relativity and analyze specific physical properties of the background spacetimes. In the first part we investigate geodesic motions in the four-dimensional constant curvature spacetimes, i.e., Minkowski and (anti-)de Sitter universe, with an expanding impulsive gravitational wave. We derive the simple refraction formulae for particles crossing the impulse and describe the effect of nonvanishig cosmological constant. In the second part of this work we present a general method useful for geometrical and physical interpretation of arbitrary spacetimes in any dimension. It is based on the systematic analysis of the relative motion of free test particles. The equation of geodesic deviation is rewritten with respect to the natural orthonormal frame. We discuss the contributions given by a specific algebraic structure of the curvature tensor and the matter content of the universe. This formalism is subsequently used for investigation of the large class of nontwisting spacetimes. In particular, we analyse the motions in the nonexpanding Kundt and expanding Robinson--Trautman family of solutions.
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Study of Exact Spacetimes
Švarc, Robert
In this work we study various aspects of the behaviour of free test particles in Einstein's general relativity and analyze specific physical properties of the background spacetimes. In the first part we investigate geodesic motions in the four-dimensional constant curvature spacetimes, i.e., Minkowski and (anti-)de Sitter universe, with an expanding impulsive gravitational wave. We derive the simple refraction formulae for particles crossing the impulse and describe the effect of nonvanishig cosmological constant. In the second part of this work we present a general method useful for geometrical and physical interpretation of arbitrary spacetimes in any dimension. It is based on the systematic analysis of the relative motion of free test particles. The equation of geodesic deviation is rewritten with respect to the natural orthonormal frame. We discuss the contributions given by a specific algebraic structure of the curvature tensor and the matter content of the universe. This formalism is subsequently used for investigation of the large class of nontwisting spacetimes. In particular, we analyse the motions in the nonexpanding Kundt and expanding Robinson--Trautman family of solutions.
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Impulsive gravitational waves
Karamazov, Michal ; Švarc, Robert (advisor) ; Podolský, Jiří (referee)
In the review part of this bachelor thesis, we summarize various results about solutions to Einstein's gravitational field equations which describe both non-expanding and expanding impulsive gravitation waves in spacetimes of constant curvature. Special attention will be paid to geodesic motion in these spacetimes and to geometrical methods of their construction. In the original part of the thesis, we check compatibility of a direct solution to geodesic equation in (anti-)de Sitter spacetime with non-expanding impulsive wave and refraction formulae derived under the assumption of continuity of geodesics in a specific coordinate system. We also investigate an interaction of test particles with expanding spherical impulsive wave propagating on the Minkowski background which is generated by a pair of perpendicular snapping cosmic strings. Powered by TCPDF (www.tcpdf.org)
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Study of Exact Spacetimes
Švarc, Robert
In this work we study various aspects of the behaviour of free test particles in Einstein's general relativity and analyze specific physical properties of the background spacetimes. In the first part we investigate geodesic motions in the four-dimensional constant curvature spacetimes, i.e., Minkowski and (anti-)de Sitter universe, with an expanding impulsive gravitational wave. We derive the simple refraction formulae for particles crossing the impulse and describe the effect of nonvanishig cosmological constant. In the second part of this work we present a general method useful for geometrical and physical interpretation of arbitrary spacetimes in any dimension. It is based on the systematic analysis of the relative motion of free test particles. The equation of geodesic deviation is rewritten with respect to the natural orthonormal frame. We discuss the contributions given by a specific algebraic structure of the curvature tensor and the matter content of the universe. This formalism is subsequently used for investigation of the large class of nontwisting spacetimes. In particular, we analyse the motions in the nonexpanding Kundt and expanding Robinson--Trautman family of solutions.
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Special Surfaces
Ochodnický, Erik ; Vašík, Petr (referee) ; Doupovec, Miroslav (advisor)
The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.
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Study of Exact Spacetimes
Švarc, Robert ; Podolský, Jiří (advisor) ; Pravda, Vojtěch (referee) ; Steinbauer, Roland (referee)
In this work we study various aspects of the behaviour of free test particles in Einstein's general relativity and analyze specific physical properties of the background spacetimes. In the first part we investigate geodesic motions in the four-dimensional constant curvature spacetimes, i.e., Minkowski and (anti-)de Sitter universe, with an expanding impulsive gravitational wave. We derive the simple refraction formulae for particles crossing the impulse and describe the effect of nonvanishig cosmological constant. In the second part of this work we present a general method useful for geometrical and physical interpretation of arbitrary spacetimes in any dimension. It is based on the systematic analysis of the relative motion of free test particles. The equation of geodesic deviation is rewritten with respect to the natural orthonormal frame. We discuss the contributions given by a specific algebraic structure of the curvature tensor and the matter content of the universe. This formalism is subsequently used for investigation of the large class of nontwisting spacetimes. In particular, we analyse the motions in the nonexpanding Kundt and expanding Robinson--Trautman family of solutions.
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Geodesics
Čambalová, Kateřina ; Tomáš, Jiří (referee) ; Doupovec, Miroslav (advisor)
The goal of the thesis is to create an overivew of geodesics. At the beginning of their study, they were considered shortest paths connecting two points on surfaces. In the thesis we will show more of the complexity of the term and introduced the properties, some uses of the geodesics and methods of their computation. Later, the Clairaut patches and their geodesics will be analysed. Clairaut patches are characterized by a specific property which makes computation of geodesics simpler. 3D plots of some Clairaut patches and their geodesics are also included.
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