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Multi-black-hole gravitational field
Klimešová, Eliška ; Žofka, Martin (advisor) ; Kubizňák, David (referee)
We study dynamics of an extremally charged black hole on the background of two stationary, mutually orbiting extremally charged black holes, forming thus a seed binary system. We extend the two-body perturbative solution of vacuum Einstein-Maxwell equations, known from literature, to the three body system. Restricting to slow-motion approximation and motion at a large distance, we de- rive the corresponding three-body Lagrangian and investigate limits of evolution of little- and large-mass black hole. Motivated by physical intuition and in order to check our results, we compare the former with solution of geodesic equation of an extremally charged test-body on identical background. Our results mainly consist of interpretation and comparison of characteristic motions. 1
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Elastic strings in general relativity
Frühauf, Josef ; Žofka, Martin (advisor) ; Tahamtan, Tayebeh (referee)
We study a simple model of a one dimensional extended body in a gravitational field, consisting of two point particles connected by an elastic string, and derive equations of motion for this system in both classical mechanics and general relativity. We study the motion of the system with a focus on the relationship between these models, and the difference of its motion from geodesic. 1
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Prostoročasy vytvářené elektromagnetickým polem a ideální tekutinou
Doleček, Vojtěch ; Žofka, Martin (advisor) ; Veselý, Jiří (referee)
This bachelor thesis deals with the search for sources of curvature of spacetime in general theory of relativity. In particular, it seeks to find a source of curvature that can be confidently said to be composed of a perfect fluid and an electromagnetic field. The problem of finding such physical solutions is first summarized in the first two chapters. Then, for specific spacetimes, it is already trying to find exact solutions that would correspond to such a source. 1
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Geodesic deviation
Vařeka, Viktor ; Švarc, Robert (advisor) ; Žofka, Martin (referee)
In the thesis, we derive the well-known equation of geodesic deviation, and then by relaxing one of the initial assumptions we obtain its generalized form. Next, we rewrite the generalized equation in an invariant form projecting the Riemann tensor onto an orthonormal frame associated with the fiducial observer moving along the geodesic in D- dimensional spacetime. We decompose the Riemann tensor into the traceless Weyl tensor, Ricci tensor, and scalar curvature and express these quantities with respect to the null frame. In general, the projections of the Weyl tensor enable us to study the spacetime properties based on its algebraic type. Finally, we employ the Einstein field equations to relate the Ricci tensor and scalar curvature, respectively, with the matter content of the spacetime. As an explicit example, we discuss the Kundt spacetime of algebraic type II representing gravitational waves propagating on the type D background in D-dimensional Einstein's gravity.
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Nonlinear Electrodynamics
Hale, Tomáš George ; Tahamtan, Tayebeh (advisor) ; Žofka, Martin (referee)
Nonlinear electrodynamics, introduced in the 1930s to remedy divergences associated with Maxwell's theory, has become a recurring theme in theoret- ical physics. Recent developments in the area of nonlinear electrodynamics coupled to gravity have prompted the creation of an accessible ground up reformulation of the basic structure. We develop the formalism by building upon classical electromagnetism in Minkowski spacetime, deriving the funda- mental equations by the action principle before re-deriving the Lagrangians of two important models from the founding era and describing the corre- sponding regular static spherically symmetric solutions. The focus is then shifted to the examination of a recently discovered model through which we develop a basic background for the coupling of nonlinear electrodynamics to gravity and AdS black hole thermodynamics.
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Exact spacetimes and their physical properties
Veselý, Jiří ; Žofka, Martin (advisor) ; Hennigar, Robie (referee) ; Tahamtan, Tayebeh (referee)
Motivated by our desire to find generalizations of the Bonnor-Melvin spacetime, the thesis investigates seven static, cylindrically-symmetric and electrovacuum exact solutions to the Einstein-Maxwell equations. They contain a magnetic field and six of them also include the cosmological constant. After discussing some of the methods we use during our investigation, we present the basic properties of the spacetimes, and for each of them we also study charged test particle motion and their admissible shell sources composed of particle streams. We also perform numerical computations to determine whether the equations admit more general solutions than the exact ones we derived. 1
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Uniformly accelerated coordinates
Voldřich, Jakub ; Kofroň, David (advisor) ; Žofka, Martin (referee)
A coordinate system can severely impact the difficulty of computations of a given problem. The uniformly accelerated coordinates are well-suited for a description of uni- formly accelerated motions. It is usually the primary choice for expressing the C-metric, which is an exact solution to Einstein's equations. In this thesis, the coordinates are considered in a limit of a flat spacetime, where problems have analytical solutions, and a good adaptation of coordinates is blatant. A natural definition of those coordinates is presented through Rindler coordinates and Milne coordinates. First from those specific problems that display good adaptation of uniformly accelerated coordinates are null ge- odesics. Then the Born's solution is computed, followed by pictures of electric intensity, magnetic induction, and Poynting vector field in constant global time. There is also com- putation of integral curves of electric intensity. And finally, it is shown what happens if a dipole is accelerated. 1
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Macroscopic gravity
Kašpar, Petr ; Svítek, Otakar (advisor) ; Žofka, Martin (referee)
Due to the nonlinearity of the Einstein equations it is possible to obtain modified equations of the general relativity (with application in cosmology) just by averaging. One of the first covariant approaches to this problem is the theory of Macroscopic Gravity. Next proposed possibility is to first characterize spacetime by the Cartan scalars and then to proceed averaging procedure.
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Rovnice geodetiky v prostoročasech s helikální symetrií
Tomášik, Miroslav ; Scholtz, Martin (advisor) ; Žofka, Martin (referee)
In this bachelor thesis we investigate geodesics in helically symmetric spacetimes in the framework of linearized Einstein's gravity. Work is an extension of paper by Bičák, Scholtz and Bohata [2], which is under preparation. First we introduce standard numerical methods for solving systems of ordinary differential equations. Next we present helically symmetric solution of linearized Einstein's equations and numerical code solving the geodesic equation on given background. We discuss conditions of existence of helically symmetric solution and finally we present selected results obtained by numerical simulations. We give present few particular examples of geodesics, selected phase portraits obtained by the method of the Lyapunovov exponents and visualize the causal structure of helically symmetric spacetime.
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