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Geometry of isolated horizons
Flandera, Aleš ; Scholtz, Martin (advisor) ; Acquaviva, Giovanni (referee)
While the formalism of isolated horizons is known for some time, only quite recently the near horizon solution of Einstein's equations has been found in the Bondi-like coordinates by Krishnan in 2012. In this framework, the space-time is regarded as the characteristic initial value problem with the initial data given on the horizon and another null hypersurface. It is not clear, however, what ini- tial data reproduce the simplest physically relevant black hole solution, namely that of Kerr-Newman which describes stationary, axisymmetric black hole with charge. Moreover, Krishnan's construction employs the non-twisting null geodesic congruence and the tetrad which is parallelly propagated along this congruence. While the existence of such tetrad can be easily established in general, its explicit form can be very difficult to find and, in fact it has not been provided for the Kerr-Newman metric. The goal of this thesis was to fill this gap and provide a full description of the Kerr-Newman metric in the framework of isolated horizons. In the theoretical part of the thesis we review the spinor and Newman-Penrose formalism, basic geometry of isolated horizons and then present our results. Thesis is complemented by several appendices.
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Quasilocal horizons
Klozová, Eliška ; Svítek, Otakar (advisor) ; Scholtz, Martin (referee)
In this thesis we discuss drawbacks of the event horizon which is defined glo- bally in spacetime and we introduce a quasilocal definition of black hole boundary foliated by marginally trapped surfaces on which the expansion of the outer null normal congruence becomes zero. List of different types of quasilocal horizons follows, i.e. apparent horizon, trapping horizon and isolated and dynamical hori- zon. Subsequently we calculate and analyse quasilocal horizons in two dynamical spacetimes which are used as inhomogeneous cosmological models. We discover future and past horizon in spherically symmetric Lemaître spacetime and we come to conclusion that both are null and have locally the same geometry as the ho- rizons in the LTB spacetime. Then we study Szekeres-Szafron spacetime with no symmetries, particularly its subfamily with β,z ̸= 0, and we derive the equation of the horizon. However, because of the lack of symmetries the spacetime is not adapted to double-null foliation, therefore we were unsuccessful in our attempts to estimate the equation's solution. Only in a special case when the function Φ does not depend on the coordinate z we found a condition on the existence of the horizon, that is Φ,t Φ > 0. 1
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The mathematical theory of perturbations in cosmology
Novák, Jan ; Pravda, Vojtěch (advisor) ; Chopovsky, A. (referee) ; Scholtz, Martin (referee)
We deal with cosmological perturbation theory in my work. We investigate General Theory of Relativity in Higher Dimensions in the Chapter 1. I mention GHP-formalism and algebraical classification of spacetimes. I use spinors to show that spacetimes of dimension 4 are special. I discuss also Kundt spacetimes, which are interesting for perturbation theory of black holes. I work with perturbations of FLRW ST's in GHP formalism in Chapter 2, which we want to use in Cosmological Inflation. The final part of my thesis is connected with scalar perturbations in f(R)-cosmologies, that can be used for explaining accelerated expansion in the last 5 billion years. I investigate the Universe at the scales of 150 Mpc, where I could not use the hydrodynamical approach. Thus I work with the generalization of the Landau's mechanical approach. I need quasi-static approximation for getting the potentials Φ and Ψ, since the equations are too complicated for direct integration. I plan to use the result also for numerical simulation of motions of dwarf galaxies in these potentials. Powered by TCPDF (www.tcpdf.org)
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Calculus of variation in Physics and Geometry
Kuchařík, Jan ; Krýsl, Svatopluk (advisor) ; Scholtz, Martin (referee)
Název práce: Variační počet ve fyzice Autor: Jan Kuchařík Katedra / Ústav: Matematický ústav UK Vedoucí bakalářské práce: RNDr. Svatopluk Krýsl, Ph.D. Abstrakt: Ve své práci shrnuji některá základní použití variačního počtu v praktických aplikacích. Odvozuju zde nezbytný matematický aparát. Zavádím pojem matematického funkcionálu a jeho extremalizaci, odvozuji Euler-Lagrangeovu rovnici a její důsledek - Beltramiho identitu; dále se věnuji odvození metody řešení izoperimetrických úloh, která zobecňuje metodu Lagrangeových multiplikátorů. Ačkoliv se v práci vyskytují řešené úlohy nejrůznějšího typu, zaměřuju se na čtyři hlavní oblasti: Fermatův princip, Hamiltonův princip nejmenší akce, isoperimetrické úlohy a hledání geodetik. Title: Variational calculus in physics Author: Jan Kuchařík Department: Supervisor: RNDr. Svatopluk Krýsl, Ph.D. Abstract: In my research work, I try to collect some basic usage of variational calculus in practical applications. I derive all the necessary mathematical tools. I explain what is a fuctional and what it means to extremalize it, I derive Euler- Lagrange equation and its corollary - Beltrami identity. I also try to derive a method for solving isoperimetric problems which generalizes the one of the Lagrange multipliers. Although there is a variety of several different...
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Differential geometry and dynamics
Nárožný, Jiří ; Krýsl, Svatopluk (advisor) ; Scholtz, Martin (referee)
The aim of this thesis is to show some mathematical concepts and methods of differential geometry and Lie groups. Subsequently, we try to use this tools in physics. Selection of these two mathematical topics is not random, because these topics are close related essentials of theoretical physics. The thesis is split into two chapters. Each chapter fulfils one of this aim. In the first chapter we introduce the notion of group, which is further enriched with other notions, like group action or group product. This detailed and smooth process leads us to introduction of homogeneous space which is one of the most important notion of Klein geometry. The end of this chapter is devoted to brief introduction to this attitude to geometry. The second chapter consists formulation of physical tasks in the language of differential geometry and afterwards its solution. As the final topic in this thesis we introduce Jacobi connection, as more natural option of connection which is implemented to physical system. Powered by TCPDF (www.tcpdf.org)
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The mathematical theory of perturbations in cosmology
Novák, Jan ; Pravda, Vojtěch (advisor) ; Scholtz, Martin (referee) ; Chopovsky, A. (referee)
We have been studying Cosmological Perturbation Theory in this thesis. There was presented the Standard General Relativity in higher dimensions. Then we used the apparatus of so called GHP formalism and this is a generalization of the well-known NP-formalism. Scalar perturbations in f(R)-cosmology in the late Universe is the final topic, which was a logical step how to proceed further and to continue in work where was shown that four-dimensional spacetimes are special. We get the potentials φ and ψ for the case of a box 150 Mpc. We used the so called mechanical approach for the case of a cosmological background. Our approach of getting these potentials is in observable Universe new. It is interesting also in the context of simulations in these, so called nonlinear theories. Powered by TCPDF (www.tcpdf.org)
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Rovnice geodetiky v prostoročasech s helikální symetrií
Tomášik, Miroslav ; Scholtz, Martin (advisor) ; Žofka, Martin (referee)
In this bachelor thesis we investigate geodesics in helically symmetric spacetimes in the framework of linearized Einstein's gravity. Work is an extension of paper by Bičák, Scholtz and Bohata [2], which is under preparation. First we introduce standard numerical methods for solving systems of ordinary differential equations. Next we present helically symmetric solution of linearized Einstein's equations and numerical code solving the geodesic equation on given background. We discuss conditions of existence of helically symmetric solution and finally we present selected results obtained by numerical simulations. We give present few particular examples of geodesics, selected phase portraits obtained by the method of the Lyapunovov exponents and visualize the causal structure of helically symmetric spacetime.
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Helical symmetry and the non-existence of asymptotically flat periodic solutions in general relativity
Scholtz, Martin ; Bičák, Jiří (advisor) ; Krtouš, Pavel (referee) ; Fraundiener, Jörg (referee)
1 Title Helical symmetry and the non-existence of asymptotically flat periodic solutions in general relativity Author Martin Scholtz Department Institute of theoretical physics Faculty of Mathematics and Physics Charles University in Prague Supervisor Prof. RNDr. Jiří Bičák, DrSc., dr. h.c. Abstract. No exact helically symmetric solution in general relativity is known today. There are reasons, however, to expect that such solutions, if they exist, cannot be asymptotically flat. In the thesis presented we investigate a more general question whether there exist periodic asymptotically flat solutions of Einstein's equations. We follow the work of Gibbons and Stewart [3] who have shown that there are no periodic vacuum asymptotically flat solutions an- alytic near null infinity I. We discuss necessary corrections of Gibbons and Stewart proof and generalize their results for the system of Einstein-Maxwell, Einstein-Klein-Gordon and Einstein-conformal-scalar field equations. Thus, we show that there are no asymptotically flat periodic space-times analytic near I if as the source of gravity we take electromagnetic, Klein-Gordon or conformally invariant scalar field. The auxilliary results consist of corresponding confor- mal field equations, the Bondi mass and the Bondi massloss formula for scalar fields. We also...
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Simulace dvojrozměrného toku kolem překážek za použití "lattice-gas" celulárních automatů
Tomášik, Miroslav ; Scholtz, Martin (advisor) ; Pavelka, Michal (referee)
Cellular automata constitues original computational methods, that found its application in many disciplines. The special class of cellular automata, so called lattice gas automata were succesfull in dealing with many challenges in hydrodynamic simulations, and they bootstrap one of the most perspective CFD methods, the Lattice Boltzmann models. In the theoretical part, we follow the evolution of the lattice gas automata, explore the theory behind them, and from their microdynamics, we derive the macroscopic equations. In the practical part, we implemented two distincet types of LGCA, the pair-interaction automata and FCHC. We applied them on the flow around obstacles of various shapes. The scientifically most relevant part concerns statistical properties of the turbulent flow simmulated by LGCA, but requires further research to conclude it. Powered by TCPDF (www.tcpdf.org)
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