
Selected exact spacetimes in Einstein's gravity
Ryzner, Jiří ; Žofka, Martin (advisor) ; McNutt, David D. (referee) ; Pravdová, Alena (referee)
The aim of this thesis is to construct exact, axially symmetric solutions of Einstein Maxwell(dilaton) equations, which possess a discrete translational symmetry along an axis. We present two possible approaches to their construction. The first one is to solve EinsteinMaxwell equations, the second one relies on a dimensional reduction from a higher dimension. We examine the geometry of the solutions, their horizons and singu larities, motions of charged test particles and compare them. 1


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.


Physics of extended objects in strong gravitational fields
Veselý, Vítek ; Žofka, Martin (advisor) ; Loukes Gerakopoulos, Georgios (referee)
We study several different models of extended bodies in gravitational fields. Firstly, we revisit the glider model of a dumbbelllike oscillating body. We develop an independent scheme to integrate the equations of motion. We study the radial fall of a Newtonian spring, calculate the position shifts of the spring and find the critical value of the spring constant which cannot overcome the tidal forces. We argue that the relativistic glider model is unphysical due to its behaviour in the critical regions. Secondly, we show that Dixon's theory of extended bodies predicts a geodesic motion of the centre of mass in maximally symmetric spacetimes. We prove that a system of test particles can be described by a conserved stressenergy tensor and we evaluate the position shifts of the glider model in the maximally symmetric spacetimes, showing its disagreement with Dixon's theory. We thus conclude again that the glider model must be rejected. And thirdly, we study a model of an extended body consisting of interacting particles, which is in accord with Dixon's theory. We calculate the position shifts for this model and show that the model does not predict any measurable swimming effect. Finally, we estimate the numerical error of the calculation by finding the position shifts of the model in maximally symmetric...


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


Physical interpretation of special solutions of EinsteinMaxwell 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


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


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 spacetime), generalized Chaplygin gas, and, also, two kinds of the scalar field (describing separately powerlaw inflation and the period after recombination). The numerical and analytical results obtained are processed graphically. 1


Canonical quantization of midisuperspace models
Černý, Jiří ; Svítek, Otakar (advisor) ; Žofka, Martin (referee)
In this work we will try to quantize midisuperspace model of spherically sym metric spacetime with massless scalar field. On this type of spacetimes we apply Dirac method of canonical quantization, leading to WheelerDeWitt equations. We will attempt to solve those equation generally for aforementioned type of spa cetimes. Our initial midisuperspace model is Roberts dynamical spacetime. As we will see later, Roberts metric behaves badly in the asymptotic region. Due to this problematic behaviour of Roberts spacetime at the boundary, we will choose to quantize its static version, the special JanisNewmanWinicour spacetime. This midisuperspace model is static, asymptotically flat spacetime with scalar field and it contains a naked timelike singularity. For special JanisNewmanWinicour spacetime we will then solve WheelerDeWitt equations.


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


Motion of extended bodies in gravitational fields
Veselý, Vítek ; Žofka, Martin (advisor) ; Tahamtan, Tayebeh (referee)
In the first chapter of this thesis we analyse the problem of a dumbbell body moving in a homogeneous field and a central gravitational field. In the homogeneous field the centre of mass of the body behaves like a point particle regardless of the force acting between the two parts of the body if we introduce an additional external force to the equations of motion. A similar method is applied to the problem of a dumbbell body in a central gravitational field. We verify the results found by Burov and Kosenko [2015] and show that the orbiting body follows Kepler's second law of motion as well as a modification of the third law. We also show that the body can keep any orientation if its length is adjusted properly and we find two numerical solutions of such cases. In the second chapter we study the problem of an oscillating dumbbell body falling into a Schwarzschild black hole as proposed by Guéron and Mosna [2007]. We verify their results and study the velocity of the body after the maneuver and the case of high and low frequencies. Furthermore, we show that the body can continue to slow its fall by further oscillations.
