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The cosmological constant on the non-cosmological scales
Liška, Andrej ; Mészáros, Attila (advisor) ; Kofroň, David (referee)
The cosmological constant Λ was first added to the gravitational field equations in 1917 by Albert Einstein. Einstein preferred the static universe, whereas field equations without the cosmological constant did not allow for such a scenario. A series of later observations mainly by Slipher, Lemaitre and Hubble showed the universe to be dynamic, which led to the cosmological constant being ne- glected from Einstein's field equations. In the early 1990s, it became clear that the expansion of the universe accelerates and the cosmological constant emerged in the field equations again, as an explanatory element. Based on a study by Perlmutter and Riess who observed distant type Ia supernovae, the cosmological constant is positive with a value of 10−56 cm−2 . The 2011 Nobel Prize was awarded for this discovery. Within the limit of weak gravitational fields and low velocities, Einstein's theory of gravitation must be reduced into Newtonian theory of gravity, the so-called Newtonian limit of Einstein equations. The full Einstein equations of the gravitational field, in the Newtonian limit, are not reduced exactly to Poisson's equation of Newtonian theory of the gravitational field. The Newtonian limit contains two additional terms with the cosmological constant, which the classical theory of gravity does not account for....
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C-metric as a limit of photon rocket
Hauser, Vít ; Kofroň, David (advisor) ; Švarc, Robert (referee)
This thesis deals with the study of properties of selected exact solutions to Einstein's equations. The first part states the basic properties of the examined solutions. It briefly summarizes basic models of black holes, whose generalizations will be important in the following part of the work. Next, it discusses a simple model of an infinite cosmic string and presents one of its possible constructions. After that, it reviews the properties of a class of Robinson-Trautman solutions, which includes photon rockets. These are used to describe sources accelerated by the emission of null dust. The final section presents the C-metric describing a pair of uniformly accelerating black holes. The second part of the work presents the constructions of cosmic strings, which arise both in C-metric and the simpler model of Schwarzschild black hole, pierced by a cosmic string. The presented construction is based on a sequence of generalized photon rockets and allows to determine the energy-momentum tensor of strings, which exists as a generalized function. 1
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Magnetic field of current loops around black holes
Vrba, Šimon ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
The magnetic field of a testing current loop in the equatorial plane around a Schwarzschild and Kerr black holes is visualized. In particular, the cases of an extreme black hole and of a Kerr naked singularity are analyzed. The simplest models of massless thin and thick current disks around a Schwarzschild black hole are presented. 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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C-metric as a limit of photon rocket
Hauser, Vít ; Kofroň, David (advisor) ; Švarc, Robert (referee)
Tato práce se zabývá oblastí přesných řešení Einstenových rovnic. Toto téma zůstává po delší dobu předmětem soustředěného studia studia na MFF UK. Zaměřil jsem se na studium kosmické struny u metriky popisující akceleraci dvou černých děr, tzv. C-metriky. Cílem práce je přehledně shrnout vlastnosti několika skupin řešení - Robinson-Trautmanových řešení obecně; specificky fotonových raket a C-metriky. Následně pak ověřit možnost pře- vodu řešení založených na modelu fotonových raket na C-metriku. Zajímavou podúlohou je fokusace záření umožňující popis vakuové C-metriky. K řešení otázek v této oblasti se využívá systémů počítačové algebry k zjednodušování složitých výrazů. V druhé části je předložen problém nalezení alternativního popisu strun v rámci těchto řešení. Rozveden je i jednodušší problém Schwarzschildovy metriky proťaté kosmickou strunou. Podstatou řešení je hledání způsobu přechodu k těmto metrikám. Práce prezen- tuje a diskutuje řešení těchto úloh. Autorovi se, nicméně, nepodařilo popsat systematický způsob jak nalézat řešení s požadovanými vlastnostmi výsledné metriky. 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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Black holes under the influence of strong sources of gravitation
Kotlařík, Petr ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
In this thesis we study a deformation of a black-hole spacetime due to another strong sources of gravity. Keeping within static and axially symmetric metrics, we consider a binary of Schwarzschild black holes held apart from each other by a repulsive effect of an Appell ring. After verifying that such a system can rest in static equilibrium (without any supporting struts), we compute its several basic geometric characteristics and we plot simple invariants determined by the metric functions (especially lapse, or, equivalently, potential) and by their first and second derivatives (gravitational acceleration and Kretschmann scalar). Then we extend the analysis below the black-hole horizon and inspect the behaviour of the scalars inside. The geometry turns out to be deformed in a non-trivial way, we even find regions of negative Kretschmann scalar in some cases. In the second part, we present a summary of the perturbative solution describing a slowly rotating system of a black hole surrounded by a thin finite circular disc, and an analysis of equatorial circular geodesics in such a spacetime. 1
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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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Comparison of Brill waves with the fields of singular rings
Sychrovský, David ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
Circular matter rings are a natural zero approximation of stationary and axially symmetric structures which appear in astrophysics. If the rings are infinitesimally thin (line sources), they are singular, which in the general relativistic description typically implies weird deformation of space in their vicinity. In particular, and contrary to the Newtonian picture, such rings even tend to behave in a strongly directional manner. One solution is to consider non-singular, extended sources (toroids), which may however be difficult to treat exactly and/or be unsatisfactory in other respects. In this thesis we check another option, namely to abandon the "real matter" completely and consider a non-singular source represented by mere curvature arranged, at least at some instant, in a pattern possessing the above symmetries. One such solution of Einstein's equations is known as the Brill waves; we study its properties at the moment of time symmetry (when it is momentarily static), in order to compare it with the space-times of matter rings. 1
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