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Thin discs and rings as sources of Weyl space-times
Kubíček, Jan ; Semerák, Oldřich (advisor) ; Žofka, Martin (referee)
Static and axially symmetric vacuum solutions of Einstein's equations can be descri- bed by the Weyl metric which only depends on two unknown functions, given by the Laplace equation and a line integral. In this thesis we study some properties of two Weyl space-times whose sources are one-dimensional rings - the Appell ring and the Bach-Weyl ring. On the behaviour of proper distances and geodesics in the central region we demonstrate that in Weyl coordinates these sources represent directional singularities. 1
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Geodetic structure of multi-black-hole spacetimes
Ryzner, Jiří ; Žofka, Martin (advisor) ; Svítek, Otakar (referee)
V klasické fyzice m·že být ustavena statická rovnováha v soustavě nabitých hmotných bod·, jsou-li poměry náboje a hmotnosti každého hmotného bodu stejné. Udivujícím faktem je, že tato situace m·že nastat i pro černé díry v relativistické fyzice. Obecný případ takovéhoto systému poprvé popsali Majumdar a Papapetrou nezávisle na sobě v roce 1947. Tato práce se zabývá jeho speciálním případem obsahujícím dvě nabité černé díry, zkoumá elektrogeodetiky v tomto prostoročasu a srovnává je se situací v klasické fyzice. Dále též shrnujeme situaci v případě nestatického vesmíru, kterou popsali Kastor a Traschenová v roce 1992, a tuto geometrii srovnáváme se statickou verzí. 1
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Properties of near-horizon geometry of spacetimes
Daněk, Jiří ; Žofka, Martin (advisor) ; Svítek, Otakar (referee)
Nowadays, the near-horizon regions of black holes have enjoyed great attention thanks to their role in the popular AdS/CFT correspondence and their specific geometry suitable for formulations of uniqueness theorems in higher dimensions. A strictly general-relativistic point of view reveals also many interesting phenomena taking place near black-hole horizons. Our aim was to investigate how horizon multiplicity affects near-horizon geometry, geodesical distance, radial motion of photons and massive, charged particles, and also the possibility of collision processes leading to unbound collision energies near the horizon. We chose the Reissner-Nordström-de Sitter metric, which, on the one hand, is simple thanks to being static and spherically symmetric but which, on the other hand, is rich enough to enable the existence of up to a doubly degenerate ultra-extreme horizon. After discussing the physical feasibility of the near-horizon limit, we applied it to single, double, and triple horizons, their near-horizon geometries, and local collision processes. We found continuous coordinate systems covering all types of horizons and analytic solutions for motion of radial photons and special or critical, massive, charged particles in their vicinity. We addressed particle collisions in the immediate vicinity of horizons...
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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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Centre of the Kerr and Appell space-times
Jurčík, Róbert ; Semerák, Oldřich (advisor) ; Žofka, Martin (referee)
One of the most important solutions of Einstein equations is the Kerr metric. At the very centre of this space-time, there lies a ring curvature singularity. The singularity encircles a surface which joins together two asymptotically flat sheets of the manifold. The surface is intrinsically flat and is standardly interpreted as a planar disc. However, an article has been recently published which claims that the central surface is actually a dicone, with vertex (vertices) on the symmetry axis. In this thesis we analyse various geometric characteristics of the surface, in order to check which of the pictures is more adequate. We also examine the same surface of the Appell space-time which has the same spatial structure as the Kerr one. 1
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