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Gravitational sources in the vicinity of black holes
Kotlařík, Petr ; Semerák, Oldřich (advisor) ; Hackmann, Eva (referee) ; Mach, Patryk (referee)
Black holes are among the most intriguing objects in the Universe. When isolated and stationary, they are described by the simple Kerr(-Newman) metric. However, astro- physical black holes are seldom isolated; in fact, we can only know of them through their interaction with the environment. Consequently, the actual gravitational field differs from the Kerr(-Newman) ideal. This thesis investigates the influence of the environment on the gravitational field of black holes. In the stationary (or static) and axially symmet- ric setting, we derive several analytical models describing a black hole surrounded by a disc or ring. The static problem is solved exactly, yielding the metric in closed form in some cases. Besides the basic physical properties of the results, we analyze the scalar- field quasinormal modes in such a deformed black-hole geometry. Our findings indicate a universal behaviour of the QNM response, which may help in distinguishing the envi- ronmental effects from those of modified theories of gravity. In the stationary case, we employ perturbation theory in the tetrad formalism. By introducing the Debye potential, we find the electromagnetic field of a ring source on the rotating black-hole background. We conclude with a discussion of similar treatment of gravitational perturbations. 1
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Geodesics in the field of a perturbed black hole: where appears chaos?
Polcar, Lukáš ; Semerák, Oldřich (advisor) ; Suková, Petra (referee)
It is widely known that the motion around Schwarzshild black hole is completely integrable. However, after adding a disc or a ring one of the symmetries of the system is broken and the motion may become chaotic for some values of parameters. The aim of this thesis is to identify where appears chaos in static, axially symmetric spacetime by using the geometrical method based on the geodesic deviation equation. Is it possible to predict chaotic behaviour in general relativity solely from local geometrical properties of spacetime, without explicitly solving the geodesic equation? Powered by TCPDF (www.tcpdf.org)
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On curvature decomposition in circular space-times
Kříž, Jan ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
Calculation of scalars obtained from the Riemann curvature tensor in co- ordinate components is not always efficient, this is true even in very simple spacetimes. Firstly, the calculation is not intuitive and secondly, many terms involved in such calculations tend to strongly diverge for example on black hole horizons, even though they should precisely "cancel out". Motivated by [1], main focus of this thesis are circular space times and introduction of 2+1+1 decomposition. The latter allows for rewriting curvature scalar using sectional curvatures, exterior curvatures and geometrically significant timelike congru- ences. Crucial part of this thesis is software implementation and verification of this approach in Wolfram Mathematica using xAct package. 1
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Magnetic fields of current loops around black holes
Vrba, Šimon ; Semerák, Oldřich (advisor) ; Karas, Vladimír (referee)
We summarize and explain the mathematical procedure that allows us to find the closed form of the magnetic field generated by a test current loop in Kerr spacetime. We consider axisymmetric placement of the loop for all three cases of the Kerr background: below-extreme black hole, extreme black hole, and naked singularity. The field is obtained by differentiating the effective Green function of the Debye potential, which is expressed in terms of elliptic integrals. 1
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Geodesic dynamics in the fields of black holes surrounded by discs
Kraus, Karel ; Semerák, Oldřich (advisor) ; Čížek, Martin (referee)
One of the basic tasks of general relativity is to calculate the motion of free test particles by integrating the geodesics equation. In the field of an iso- lated stationary black hole, the problem is fully integrable. However, the pres- ence of any other source of gravitation disrupts this property and the geodesic motion may become chaotic. In the following work we study the dynamics of motion around a Schwarzschild black hole surrounded by inverted Kuzmin- Toomre disks. To integrate the geodesics equation, we have developed a new code and investigated in detail some numerical methods, in particular, we com- pared Runge-Kutta methods and modified symplectic integrators. 1
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Homoclinic orbits in perturbed black-hole fields
Feireisl, Jan ; Semerák, Oldřich (advisor) ; Witzany, Vojtěch (referee)
In order to generate observable electromagnetic signatures, astrophysical black holes have to interact with matter. Arround the black hole, matter typically forms into a symmetric disc through which it gradually inspirals towards the black hole. If the disc is dense enough, it can significantly perturb the motion of free test particles. The perturbation makes the originally completely integrable dynamical system prone to chaos. In this thesis, we focus on finding the homoclinic orbits which are the 'seeds of chaos' in the geodesic motion around black holes. Specifically, we find the homoclinic orbits in the Schwarzschild and in the extreme Reissner-Nordström space-times, and analyse how they behave under perturbation by a Kuzmin-Toomre disc and by a Majumdar-Papapetrou ring, respectively. 1
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GHP and Weyl formalism for gravitational perturbations
Mikeska, Václav ; Kofroň, David (advisor) ; Semerák, Oldřich (referee)
The exact analytical solutions of Einstein's equations describing systems of astrophy- sical interest have not been found yet, and thus they have to be studied only as perturba- tions of known spacetimes. There are various ways to investigate these perturbations. One can look directly for perturbations of metric of the exact solution of Einstein's equations. In vacuum spacetimes of type D, it has proved advantageous to investigate perturbations in the GHP formalism by introducing the Debye potential. In this paper, we discuss the connection between these two approaches. We present a general procedure for translating the results from the Debye potential formalism to stationary axisymmetric perturbations of the Kerr metric. This procedure requires solving for a calibration vector. We show that both approaches lead to the same perturbation of the radiative components of the Weyl tensor, and we find a simple relation between these components. 1
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Stationary fields in black-hole space-times
Čížek, Pavel ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee) ; Gürlebeck, Norman (referee)
Motivated by modelling of astrophysical black holes surrounded by accretion structures, as well as by theoretical interest, we study two methods how to ob- tain, within stationary and axisymmetric solutions of general relativity, a metric describing the black hole encircled by a thin ring or a disc. The first is a suitable perturbation of a Schwarzschild black hole. Starting from the seminal paper by Will (1974), we showed that it is possible to express the Green functions of the problem in a closed form, which can then be employed to obtain, e.g., a reason- able linear perturbation for a black hole surrounded by a thin finite disc. In the second part we tackle the same problem using the Belinskii-Zakharov generating algorithm, showing/confirming that in a stationary case its outcome is unphysi- cal, yet at least obtaining a modest new result for the (static) "superposition" of a Schwarzschild black hole with the Bach-Weyl ring. 1
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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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