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Representation of dynamical black hole spacetimes in numerical simulations
Khirnov, Anton ; Ledvinka, Tomáš (advisor) ; Hilditch, David (referee) ; Liebling, Steven L. (referee)
Choptuik's unexpected discovery, almost 30 years ago, of critical behavior in grav- itational collapse opened a whole new research area within numerical relativity. While critical collapse in spherical symmetry has been thoroughly investigated and is reasonably well understood, progress for axial symmetry has been much slower. In this thesis, we study axially symmetric gravitational collapse of gravi- tational waves using numerical simulations. We construct several very different initial data families and investigate their behavior close to the threshold of collapse. We compare them against each other and also against other published results. We use invariant quantities to look for signs of self-similarity and universality.
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Replacement of singularities in static spacetimes
Došek, Jan ; Ledvinka, Tomáš (advisor) ; Žofka, Martin (referee)
This bachelor thesis deals with the possibility of using the variational principle in the se- arch for internal solutions replacing singularities in static spacetimes. In the first instance, the thesis introduces the problematics on the simple case of classical Newton's gravity, where it also shows its practical application on several simple examples of symmetric potentials, and then it tries to generalize the theory for Einstein's gravity. It turns out that the relativistic problem could be loosely related to the so-called quadratic gravity as the internal solution is being found as a minimizer of a functional composed of quadrates of the Ricci tensor, the Weyl tensor and scalar curvature. Subsequently, it is verified that the newly presented theory in the newtonian limit yields the same results as the classical one. Finally, the thesis deals with the use of this theory to find an internal solution replacing the singularity in Schwarzschild spacetime and discusses its properties. 1
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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 decision-making 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 cross-border commercial disputes. The reader is first introduced to the context and evidence-dependency 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 cross-border 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...
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Numerical solution of the Ernst equation
Pospíšil, Marek ; Ledvinka, Tomáš (advisor) ; Svítek, Otakar (referee)
This work is concerned with solving the Ernst equation using numerical techniques, namely pseudospectral methods. In theoretical chapters, we summarize the properties of some black-hole space-times. The work then cites the derivation of the Ernst equation and the Kerr solution. Afterwards we present pseudospectral techniques on the example of a numerical solution of the Laplace equation with a boundary condition at infinity. Finally we solve a non-linear differential equation, thus proving, that pseudospectral methods might be used even on the Ernst equation. 1
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Beahvior of the solutions to the wave equation in compactified hyperboloidal slicing
Ivánek, Richard ; Ledvinka, Tomáš (advisor) ; Kofroň, David (referee)
In this bachelor thesis we discuss the effects of compactification and hyperboloidal slicing of spacetime in the numerical solution of wave equation primarily for their appli- cation in numerical relativity. The aim was to find the pros and cons of these concepts, to illustrate expected problems using diagrams and to rate the results obtained in spe- cific model problems. A brief explanation and demonstration of relevant numerical me- thods, hyperbolic Cauchy hypersurfaces, compactification and causal diagrams is a part of the thesis. As a conclusion, the effect of compactification and slicing on the accuracy of differential and integrational schemes was compared as well as the effect of discrete representation on the quality of initial data. 1
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Interpretation of the sources of the known solutions of Teukolsky equations
Mikeska, Václav ; Kofroň, David (advisor) ; Ledvinka, Tomáš (referee)
Many realistic astrophysical problems can be treated as perturbations. It turns out that the NP formalism is a very successful tool in electromagnetic perturbations on Kerr background. We investigate stationary axisymmetric test electromagnetic field generated by static axisymmetric charge distribution and stationary axisymmetric tangential cur- rents around Kerr black hole. We found a simple relation between electromagnetic field NP scalars φ0 and φ2 and then we got an explicit formula for the third NP scalar φ1 by solving Maxwell equations with electromagnetic sources. Next, we investigate the problem of visualisation of electromagnetic field and develop a visualisation method on arbitrary background, which emphasize local field measured by an observer. We illustrate this method with several examples of electromagnetic fields. 1
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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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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.
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Coordinate choice in the Oppenheimer-Snyder 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
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