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Chaotic Motion around Black Holes
Suková, Petra
As a non-linear theory of space-time, general relativity deals with interesting dynamical systems which can be expected more prone to chaos than their Newtonian counter-parts. In this thesis, we study the dynamics of time- like geodesics in the static and axisymmetric field of a Schwarzschild black hole surrounded, in a concentric way, by a massive thin disc or ring. We reveal the rise (and/or decline) of geodesic chaos in dependence on parameters of the sys- tem (the disc/ring mass and position and the test-particle energy and angular momentum), (i) on Poincaré sections, (ii) on time series of position and their power spectra, (iii) by applying two simple yet powerful recurrence methods, and (iv) by computing Lyapunov exponents and two other related quantifiers of or- bital divergence. We mainly focus on "sticky" orbits whose different parts show different degrees of chaoticity and which offer the best possibility to test and compare different methods. We also add a treatment of classical but dissipative system, namely the evolution of a class of mechanical oscillators described by non-standard constitutive relations.
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Superluminal motion in general relativity
Gattermann, Rico ; Ledvinka, Tomáš (advisor) ; Krtouš, Pavel (referee)
We show how superluminal travel can be achieved by means of the Alcubierre warp drive. In this spacetime a spaceship locally at rest is surrounded by a "bubble" moving faster than the speed of light. We derive the equations of motion for photons and massive particles and illustrate properties of their solutions. We will find that warp drives cause frequency shifts and refraction of light passing the bubble wall, which affects the view of the outside universe seen by a traveller on spaceship. As for superluminal warp drives, existence of horizons will be shown. We will discuss that the stress-energy tensor, generating a warp corridor in spacetime, is not related to any classical field or matter, and attempts to interpret it via quantum mechanics resulted in extreme amounts of matter required. Powered by TCPDF (www.tcpdf.org)
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Chaotic Motion around Black Holes
Suková, Petra ; Semerák, Oldřich (advisor) ; Šubr, Ladislav (referee) ; Loukes-Gerakopoulos, Georgios (referee)
As a non-linear theory of space-time, general relativity deals with interesting dynamical systems which can be expected more prone to chaos than their Newtonian counter-parts. In this thesis, we study the dynamics of time- like geodesics in the static and axisymmetric field of a Schwarzschild black hole surrounded, in a concentric way, by a massive thin disc or ring. We reveal the rise (and/or decline) of geodesic chaos in dependence on parameters of the sys- tem (the disc/ring mass and position and the test-particle energy and angular momentum), (i) on Poincaré sections, (ii) on time series of position and their power spectra, (iii) by applying two simple yet powerful recurrence methods, and (iv) by computing Lyapunov exponents and two other related quantifiers of or- bital divergence. We mainly focus on "sticky" orbits whose different parts show different degrees of chaoticity and which offer the best possibility to test and compare different methods. We also add a treatment of classical but dissipative system, namely the evolution of a class of mechanical oscillators described by non-standard constitutive relations.
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Superluminal motion in general relativity
Gattermann, Rico ; Ledvinka, Tomáš (advisor) ; Krtouš, Pavel (referee)
We show how superluminal travel can be achieved by means of the Alcubierre warp drive. In this spacetime a spaceship locally at rest is surrounded by a "bubble" moving faster than the speed of light. We derive the equations of motion for photons and massive particles and illustrate properties of their solutions. We will find that warp drives cause frequency shifts and refraction of light passing the bubble wall, which affects the view of the outside universe seen by a traveller on spaceship. As for superluminal warp drives, existence of horizons will be shown. We will discuss that the stress-energy tensor, generating a warp corridor in spacetime, is not related to any classical field or matter, and attempts to interpret it via quantum mechanics resulted in extreme amounts of matter required. Powered by TCPDF (www.tcpdf.org)
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Dark matter
Vraštil, Michal ; Mészáros, Attila (advisor) ; Heyrovský, David (referee)
The dark matter constituting approximately 85% of the mass is an integral part of our universe. As many astronomical observations at different scales of space so the theoretical models show that there is more matter than we can see directly. This overview describes the major historical events and observational data from the time of Zwicky to the present leading to today's view on the dark matter. The paper further describes the cosmological implications of the presence of dark matter - its impact on the formation of structures in the universe and reflection of fluctuations in the cosmic background radiation. Here I describe possible candidates for dark matter - a small contribution of baryonic matter and the main candidates among non-baryonic matter. In addition to possible new particles explaining the missing mass I describe alternatives to the theory of gravity, which do not require any extra matter, in particular, I deal with a very successful theory of MOND. At the end I mention a brief overview of today's possibilities of direct or indirect observation of dark matter.
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Principles of gravitational-wave detection
Přeučil, Filip ; Ledvinka, Tomáš (advisor) ; Kofroň, David (referee)
In the present work we investigate the impact of weak gravitational wave in the linearized theory of gravity on a simple model of an interferometric gravitational wave detector, whose individual parts (mirrors, and electromagnetic field) are idealised by free test particles. After a necessary, fairly popularly conceived introduction to gravitational wave theory in the linearized gravity, the astrophysical sources of the gravitational waves, the possibilities of their detection and the principles of the detectors, we provide a mathematical survey of the indispensable parts of general relativity and of the linearized theory of gravity. After that, we finally deal with the model itself. In the linear approximation with respect to the perturbations, we solve the equations of motion of the individual components and derive the detector response to a gravitational wave. Finally, we present a few comments, including a proof of gauge invariance of the derived formula.
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Transition from regular to chaotic motion in black hole magnetospheres
Kopáček, Ondřej ; Karas, Vladimír (advisor) ; Kulhánek, Petr (referee) ; Rezzolla, Luciano (referee)
Cosmic black holes can act as agents of particle acceleration. We study properties of a system consisting of a rotating black hole immersed in a large-scale organized magnetic field. Electrically charged particles in the immediate neighborhood of the horizon are influenced by strong gravity acting together with magnetic and induced electric components. We relax several constraints which were often imposed in previous works: the magnetic field does not have to share a common symmetry axis with the spin of the black hole but they can be inclined with respect to each other, thus violating the axial symmetry. Also, the black hole does not have to remain at rest but it can instead perform fast translational motion together with rotation. We demonstrate that the generalization brings new effects. Starting from uniform electro-vacuum fields in the curved spacetime, we find separatrices and identify magnetic neutral points forming in certain circumstances. We suggest that these structures can represent signatures of magnetic reconnection triggered by frame-dragging effects in the ergosphere. We further investigate the motion of charged particles in these black hole magnetospheres. We concentrate on the transition from the regular motion to chaos, and in this context we explore the characteristics of chaos in...
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Modeling the Mach's principle in the post-Minkowskian approximation to general relativity
Schmidt, Tibor ; Ledvinka, Tomáš (advisor) ; Kofroň, David (referee)
The aim of this thesis is the simulation of relativistic phenomena in post- Minkowskian approximation. In the introduction the terms of Mach principle and gravitomagnetism are presented. Afterwards the principles of numeric solution of ordinary differential equations are summarized. Consequently, we get acquainted with the first post-Minkowskian approximation in canonical formalism and with elementary examples of its use. In the next chapter the results of performed simulations of classical General Relativity tests are described. The last chapter is devoted to the simulation of gravitomagnetism and of the system of rotating particles.
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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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Solving test-particle equations of motion near a black hole
Ryston, Matěj ; Ledvinka, Tomáš (advisor) ; Suková, Petra (referee)
Bachelor thesis Matěj Ryston 2011/2012 Abstract in English This work aims to give a well-arranged summary of the description and solving the equations of motion of particles outside a black hole (a star) with emphasis on numerical solutions. For that purpose a summary of numerical methods for solving ordinary differential equations, together with a review and comparison of chosen methods, is given. In the second chapter follows a brief recall of the foundations of General Relativity as well as the description of the geometry of Schwarzschild solution of the Einstein equations. After that equations of motion are formulated. In conclusion, selected numerical methods are used on solving said equations of motion of a test particle or those describing bending of light rays in closeness to a black hole.
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