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Properties of the extreme charged black hole near horizon
Hejda, Filip ; Krtouš, Pavel (advisor) ; Svítek, Otakar (referee)
It is known, that there exists a limiting correspondence between certain part (including the horizon) of extremal case of Reissner-Nordström space-time and Robinson-Bertotti space-time and that different generalisations of this near-horizon limit are possible. The aim of the presented work is to examine some of the properties of such limiting transitions. Firstly it is stressed how the global structure is reflected in the limit and secondly which properties of the space-time do provide that physical distances are preserved in the limit. Besides the extremal case the subextremal and hyperextremal generalisations are studied. As a complementary topic, the global extremal limit is stated. That means a transition from a generalised (non-symmetrical) conformal diagram of the subextremal case to the conformal diagram of the extremal case of Reissner-Nordström solution.
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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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De Sitter special relativity
Vrecion, Jiří ; Svítek, Otakar (advisor) ; Žofka, Martin (referee)
This thesis deals with de Sitter space as an alternative to Minkowski space, which is generally used in theories describing matter and fields (for example quantum field theory). The problem of mass in de Sitter space is analyzed in more detail. The mathematical apparatus needed in this thesis, from factor groups, through Lie groups and algebras to Casimir operators, is also mentioned. In the final part it is shown, that de Sitter space is a factorgroup dS(1, 3) = SO(1,4) SO(1,3) , which is at least from mathematical standpoint much more natural factorization than Poincaré SO(1,3) , which leads to Minkowski. Conformal cyclic cosmology developed by Roger Penrose is briefly described as a motivation for this thesis. This theory could benefit from some properties of de Sitter relativity. 1
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Quasilocal horizons
Polášková, Eliška ; Svítek, Otakar (advisor)
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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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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Quasilocal horizons
Polášková, Eliška ; Svítek, Otakar (advisor)
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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Emergence of space geometries from quantum entanglement
Lukeš, Petr ; Scholtz, Martin (advisor) ; Svítek, Otakar (referee) ; Švarc, Robert (referee)
MASTER THESIS Petr Lukeš Emergence of space geometries from quantum entanglement Institute of Theoretical Physics Supervisor of the master thesis: Mgr. Martin Scholtz, Ph.D. Study programme: Physics Study branch: Theoretical physics Prague 2019 Abstract: Connecting the field of Quantum Physics and General Relativity is one of the main interests of contemporary Theoretical Physics. This work attempts to find solution to simplified version of this problem. Firstly entropy is shown to be a good meeting point between the two different theories. Then some of entropy's less intuitive properties are shown, namely its dependence on area, not volume. This relation is studied from both Relativistic and Quantum viewpoint. After- wards there is a short description of a quantum model interpretable as geometry based on the information between its subsystems. Lastly, results of computations within this model are presented.
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Visualization of black hole spacetimes
Maixner, Michal ; Krtouš, Pavel (advisor) ; Svítek, Otakar (referee)
This work is focused on visualisation of Schwarzschild, Reissner- Nordström and Kerr black hole. The two-dimensional conformal diagram was constructed. In the case of Kerr black hole, the causal structure was visualized by intersection of chronological future of given point in spacetime with hyper- surfaces of constant value of Boyer-Lindquist coordinate t. Conformal diagram for Kerr black hole was constructed only in the neighbourhood of outer event horizon. Then the causal diagram, which is analogous to conformal diagram for Reissner-Nordström black hole was constructed. In all cases two-dimensional spa- celike hypersurfaces were chosen that were embedded into Euclidean space. The interpretation of time evolution of black hole universe was given to a sequence of such embedded hypersurfaces. In the case of Kerr black hole the embedding of outer ergosphere and outer event horizon were also constructed. 1
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