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Space-times with toroidal horizons
Pešta, Milan ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee)
Basic results of the uniqueness theorems and the topological-censorship theorem are presented. Properties of the well-known solution of Einstein's equations with a toroidal event horizon are examined and one of possible visualizations of the coordinate system that helps to better understand the behaviour of various invariants in the vicinity of the singularity is suggested. Apart from this solution, two solutions with a ring singularity are introduced as potential candidates for space-times with toroidal horizons whose properties are interpreted intuitively using the toroidal or Weyl coordinate systems. Last part is devoted to apparent horizons of the considered solutions and the differential equation for the apparent horizon of an arbitrary solution of the Weyl class is derived. The numerical solution of this equation is not presented.
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Chaos in deformed black-hole fields
Witzany, Vojtěch ; Semerák, Oldřich (advisor) ; Kopáček, Ondřej (referee)
The consequences of two key approximations of accretion-disc physics near black holes are studied in this thesis. First, the question of effective ``pseudo-Newtonian" potentials mimicking a black hole is investigated both through numerical simulations and analytical means, and second, the neglect of additional gravitating matter near accreted-upon black holes and its consequences are put to test. After some broader discussion of integrability, resonance and chaos, a general "pseudo-Newtonian" limit for geodesic motion is derived, and applied for the case of null geodesics near a glowing toroid and for time-like geodesics in the Kerr metric. Afterwards, a new Newtonian gravitational potential for non- singular toroids is proposed and its usefulness for the so-called Weyl space-times is discussed. Finally, a new pseudo-Newtonian potential is introduced and applied alongside already known potentials in models of free test particle motion in the field of a black hole with a disc or ring, in complete analogy with previous exact-relativistic studies, and the previous conclusion of chaos in disc/ring-hole models is confirmed. Overall, the pseudo-Newtonian framework is able to reproduce a number of key features of the original systems with notable differences arising only as a consequence of extremely strong or...
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Magnetic fields of current loops around black holes
Pejcha, Jakub ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee)
Magnetic field of equatorial current loop around Schwarzschild (or Kerr) black hole has been studied in many papers and solutions expressed in different forms. In this work we summarize derivations of some of these solutions, illustrate them on specific examples and compare these examples. We also indicate analytic com- parison of some of the formulas. Published formulas lead, as expected, to same results, but some of them are more convenient for numerical evaluation. 1
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Symmetries of systems in spaces related to high-dimensional black hole spacetime
Kolář, Ivan ; Krtouš, Pavel (advisor) ; Kubizňák, David (referee)
In this work we study properties of the higher-dimensional generally rotating black hole space-time so-called Kerr-NUT-(A)dS and the related spaces with the same explicit and hidden symetries as the Kerr-NUT-(A)dS spacetime. First, we search commuta- tivity conditions for classical (charged) observables and their operator analogues, then we investigate a fulfilment of these conditions in the metioned spaces. We calculate the curvature of these spaces and solve the charged Hamilton-Jacobi and Klein-Gordon equations by the separation of the variables for an electromagnetic field, which pre- serves integrability of motion of a charged particle and mutual commutativity of the corresponding operators.
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Chaos v porušených polích černých děr
Witzany, Vojtěch ; Semerák, Oldřich (advisor) ; Heyrovský, David (referee)
The loss of complete geodesic integrability is one of the important consequences (and thus indicators) of deviation from the Kerr-type space-time. Indeed, it has been confirmed many times in the literature that even a highly symmetric perturbation of the Kerr or Schwarzschild metric can make the free test-particle motion chaotic. In this thesis, we study the test-particle dynamics in the field of a Schwarzschild black hole surrounded by a thin disc or ring, using, however, Newton's gravity with a simple "pseudo- Newtonian" potential to mimic the black hole. The Poincaré sections show that the (pseudo-)Newtonian system is slightly more chaotic than the general relativistic one. The difference seems to be correlated with the phase-space allowed region being more open towards the center in the pseudo-Newtonian case. Powered by TCPDF (www.tcpdf.org)
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Částice a pole v křivých prostoročasech (vybrané problémy)
Hejda, Filip ; Bičák, Jiří (advisor) ; Krtouš, Pavel (referee)
In 2009 Bañados, Silk and West described the possibility of principally unbounded collision energies in the centre-of-mass frame for the particle collisions in the vicinity of black holes. Their work attracted a big response. This thesis aims to summarise the results of a number of the articles about the topic and puts these results into a new, broader context. It also presents some generalisations of the existing results, especially for models of magnetised black holes. The main subject of interest is the question, whether the unbounded collision energies can be achieved in a single-scattering or multiple-scattering process which was first formulated by Grib, Pavlov and Zaslavskii. Variety of methods is summarised. A considerable attention is paid to the limiting near-horizon description, which is further developed in order to derive new links and relations among known results, particularly between the purely theoretical work dealing with the geometry of degenerate horizons and their vicinity and more astrophysical articles about magnetic fluxes. Powered by TCPDF (www.tcpdf.org)
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Numerical evolution of black-hole spacetimes
Khirnov, Anton ; Ledvinka, Tomáš (advisor) ; Palenzuela, Carlos (referee)
吀e so-called "trumpet" initial data has recently received mu挀 a琀ention as a potential candidate for the natural black hole initial data to be used in 3+1 numerical relativity simulations with 1+log foliation. In this work we first derive a variant of the maximal trumpet initial data that is made to move on the numerical grid by the means of a Lorentz boost and write a numerical code that constructs this boosted trumpet initial data. We also write a numerical code for calculating the Krets挀mann scalar from the 3+1 variables, to be used in analysing the data from our simulations. With the help of those two codes, we study the behaviour of the boosted trumpet initial data when evolved with the BSSN formulation of the Einstein equations, using 1+log slicing and the Γ-driver shi昀 condition.
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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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Geometry inside deformed black holes
Basovník, Marek ; Semerák, Oldřich (advisor) ; Svítek, Otakar (referee)
In this thesis we study exact general relativistic space-times generated by a black hole and an additional source of gravity, while restricting to two classes of static and axially symmetric solutions: the Majumdar-Papapetrou solution for a couple (in general, a multiple system) of extremally charged black holes and the "superposition" of a Schwarzschild black hole with the Bach-Weyl thin ring. We follow the effect of the additional source on the geometry of black-hole space-time on the behaviour of important invariants, in particular of the simplest scalars obtained from the Riemann and possibly also Ricci tensor. We have plotted the invariants both outside and inside the black hole; in the case of a Schwarzschild black hole with ring, we found, to this end, an extension of the metric below the horizon. It turns out that the external source may affect the geometry inside the black hole considerably, even in the vicinity of singularity, although the singularity itself remains point-like in both solutions studied here.
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The Black Holes
JURČA, Miroslav
The main aim of this bachelor thesis is to summarize the knowledge about the final phases of stellar evolution with a diferent masses especially to a very massive stars. We describe the Universe since the Big Bang, the formation of protons and neutrons, forming the stars and their final phases. In particular, we emphasise to a very massive stars which collapses itself, under their own gravity to the event horizon and may become the black holes. This work also describes the possible detection of black holes and gravitational waves, generated by the relative motion of very massive stars in a gravitational field.
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