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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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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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Gravitational perturbations in NP/GHP formalism
Pospíšil, Marek ; Kofroň, David (advisor) ; Ledvinka, Tomáš (referee)
This work gets the reader acquainted with tetrad approaches to general theory of re- lativity (GTR), namely with Newman - Penrose (NP) and Geroch - Held - Perose (GHP) formalisms. Obtained expressions are expressed in coordinates. Subsequently, this work concerns with the perturbation techniques in GTR, including both metric perturbation and the method of superpotential, the latter being more common in GHP formalism. Following is the recapitulation of important results based on this method. The core of the work is in generalization of known results for spacetimes with ac- celeration (so called C-metric) and checking previous results using computation in the Wolfram Mathematica program. In order not to work just with rather complicated GHP and NP formalisms, the main result is checked also by transcription into coordinate par- tial derivatives. 1
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Interaction of point charge with moving conductors
Rosman, Viktor ; Kofroň, David (advisor) ; Švarc, Robert (referee)
In this thesis we deal with interaction of a point charge with conductors. The behaviour of conductors in the electrostatic field of a point charge (but also in the general electrodynamic field) is well known. The presence of a conductor creates a force that causes the free point charge to move. However, the solution of this electrodynamic problem is given very little attention and is therefore the subject of our study. Specifically, we focus on the study of the motion of a point charge placed in the presence of a conductor, which is not generally ideal (it has certain non-zero electrical resistance). In most cases, we solve the problem in a quasi-static approximation (within classical physics). In only one case, and the point charge is located above an ideally conductive plane, we will try to find a relativistic solution. We will try to extend the acquired knowledge about the behaviour of a system of moving point charge in the presence of a conductor to the case of a moving conductor. So, our last subject of study will be the interaction point charge with moving conductors.
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Gravitační vlny v expandujících vesmírech
Pavičevič, Mak ; Bičák, Jiří (advisor) ; Kofroň, David (referee)
in English In the present work we are analysing the asymptotic behaviour of vacuum spacetimes describing gravitational waves in universes with anisotropic expansion. The thesis is divided into two main parts. In the first chapter we review cylindrical gravitational waves and their cosmological generalizations; Bondi-Sachs formalism, Penrose's null infinity and the peeling property of the Riemann tensor are summarized. Symmetry reduction, a (2+1)-dimensional formalism, is presented in detail and used in calculations. Generalization of the symmetry reduction to other dimensions and relation to Brans-Dicke theory is carried out. In the second chapter we analyse solutions describing cosmological gravitational waves proposed in the literature and discuss their relation to boost-rotation symmetric spacetimes. After utilizing the symmetry reduction, Kasner and Bianchi VI models are chosen as background solutions. The behaviour of the metric is encoded in a single function, a solution to the linear wave equation. The full solution is then a superposition of the gravitational wave representing, for example, pulses, and the background one. We calculate the Riemann and the Cotton tensor components of the 3-spacetimes and discuss the conformal completion. A comparison with four-dimensional behaviour is also...
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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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Superradiance on accelerated systems
Žlábek, Martin ; Kofroň, David (advisor) ; Tahamtan, Tayebeh (referee)
In this work we will study the electromagnetic superradiance phenomenon on acceler- ated systems. We will briefly cover superradiance on a cylinder, which was thermodynam- ically proven by Zel'dovich. Then we will attempt to formulate the problem in accelerated coordinates, namely the flat-spacetime limit of the C-metric. Briefly introducing the C- metric, the Newman-Penrose formalism and the Geroch-Held-Penrose formalism along the way. Using the formalism of vector spherical harmonics, we point out the complica- tions which arise from not respecting the spherical symmetry of the spherical coordinate system. 1
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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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