
Numerical solution of the Ernst equation
Pospíšil, Marek ; Ledvinka, Tomáš (advisor) ; Svítek, Otakar (referee)
This work is concerned with solving the Ernst equation using numerical techniques, namely pseudospectral methods. In theoretical chapters, we summarize the properties of some blackhole spacetimes. The work then cites the derivation of the Ernst equation and the Kerr solution. Afterwards we present pseudospectral techniques on the example of a numerical solution of the Laplace equation with a boundary condition at infinity. Finally we solve a nonlinear differential equation, thus proving, that pseudospectral methods might be used even on the Ernst equation. 1


Quasilocal horizons
Polášková, Eliška ; Svítek, Otakar (advisor)
In this thesis we discuss drawbacks of the event horizon which is defined glo bally in spacetime and we introduce a quasilocal definition of black hole boundary foliated by marginally trapped surfaces on which the expansion of the outer null normal congruence becomes zero. List of different types of quasilocal horizons follows, i.e. apparent horizon, trapping horizon and isolated and dynamical hori zon. Subsequently we calculate and analyse quasilocal horizons in two dynamical spacetimes which are used as inhomogeneous cosmological models. We discover future and past horizon in spherically symmetric Lemaître spacetime and we come to conclusion that both are null and have locally the same geometry as the ho rizons in the LTB spacetime. Then we study SzekeresSzafron spacetime with no symmetries, particularly its subfamily with β,z ̸= 0, and we derive the equation of the horizon. However, because of the lack of symmetries the spacetime is not adapted to doublenull foliation, therefore we were unsuccessful in our attempts to estimate the equation's solution. Only in a special case when the function Φ does not depend on the coordinate z we found a condition on the existence of the horizon, that is Φ,t Φ > 0. 1


Emergence of space geometries from quantum entanglement
Lukeš, Petr ; Scholtz, Martin (advisor) ; Svítek, Otakar (referee) ; Švarc, Robert (referee)
MASTER THESIS Petr Lukeš Emergence of space geometries from quantum entanglement Institute of Theoretical Physics Supervisor of the master thesis: Mgr. Martin Scholtz, Ph.D. Study programme: Physics Study branch: Theoretical physics Prague 2019 Abstract: Connecting the field of Quantum Physics and General Relativity is one of the main interests of contemporary Theoretical Physics. This work attempts to find solution to simplified version of this problem. Firstly entropy is shown to be a good meeting point between the two different theories. Then some of entropy's less intuitive properties are shown, namely its dependence on area, not volume. This relation is studied from both Relativistic and Quantum viewpoint. After wards there is a short description of a quantum model interpretable as geometry based on the information between its subsystems. Lastly, results of computations within this model are presented.


Visualization of black hole spacetimes
Maixner, Michal ; Krtouš, Pavel (advisor) ; Svítek, Otakar (referee)
This work is focused on visualisation of Schwarzschild, Reissner Nordström and Kerr black hole. The twodimensional conformal diagram was constructed. In the case of Kerr black hole, the causal structure was visualized by intersection of chronological future of given point in spacetime with hyper surfaces of constant value of BoyerLindquist coordinate t. Conformal diagram for Kerr black hole was constructed only in the neighbourhood of outer event horizon. Then the causal diagram, which is analogous to conformal diagram for ReissnerNordström black hole was constructed. In all cases twodimensional spa celike hypersurfaces were chosen that were embedded into Euclidean space. The interpretation of time evolution of black hole universe was given to a sequence of such embedded hypersurfaces. In the case of Kerr black hole the embedding of outer ergosphere and outer event horizon were also constructed. 1


Canonical quantization of midisuperspace models
Černý, Jiří ; Svítek, Otakar (advisor) ; Žofka, Martin (referee)
In this work we will try to quantize midisuperspace model of spherically sym metric spacetime with massless scalar field. On this type of spacetimes we apply Dirac method of canonical quantization, leading to WheelerDeWitt equations. We will attempt to solve those equation generally for aforementioned type of spa cetimes. Our initial midisuperspace model is Roberts dynamical spacetime. As we will see later, Roberts metric behaves badly in the asymptotic region. Due to this problematic behaviour of Roberts spacetime at the boundary, we will choose to quantize its static version, the special JanisNewmanWinicour spacetime. This midisuperspace model is static, asymptotically flat spacetime with scalar field and it contains a naked timelike singularity. For special JanisNewmanWinicour spacetime we will then solve WheelerDeWitt equations.


Probabilistic Spacetimes
Káninský, Jakub ; Svítek, Otakar (advisor) ; Žofka, Martin (referee)
Probabilistic Spacetime is a simple generalization of the classical model of spa cetime in General Relativity, such that it allows to consider multiple metric field realizations endowed with probabilities. The motivation for such a generalization is a possible application in the context of some quantum gravity approaches, na mely those using the path integral. It is argued that this model might be used to restrict the precision of the geometry on small scales without postulating discrete structure; or it may be used as an effective description of a probabilistic geometry resulting from a fullfledged quantum gravity computation.


Charged particles in spacetimes with an electromagnetic field
Veselý, Jiří ; Žofka, Martin (advisor) ; Svítek, Otakar (referee)
The subject of study of this thesis is the KerrNewman(anti)de Sitter space time, a rotating and charged exact blackhole solution of the EinsteinMaxwell equations with a nonzero cosmological constant. In the first part of the thesis we examine admissible extremal configurations, present the corresponding Penrose diagrams, and investigate the effects of framedragging. In the second part, we follow the motion of charged particles via the Lagrangian formalism, focusing on the equatorial plane and the axis where we arrived at some analytic results con cerning the trajectories. Static particles, effective potentials and  in the case of the equatorial plane  stationary circular orbits are examined. We also perform numerical simulations of particle motion to be able to check our analytic results and also to foster our intuition regarding the behaviour of the test particles. The last part concerns quantum tunnelling of particles through the spacetime's hori zons, specifically the null geodesic method. The main goal of these computations is to obtain horizon temperatures, in which we succeed up to a constant multi plicative factor. We discuss various pitfalls of the method and stake out a possible approach when applying it to the extreme horizons present in KN(a)dS. 1


Fields of current loops around black holes
Vlasáková, Zuzana ; Semerák, Oldřich (advisor) ; Svítek, Otakar (referee)
The magnetic field of a test circular current loop placed symmetrically around a Schwarzschild black hole has been determined several times in the literature and solutions has been expressed by different formulas. We compare these formulas analytically as well as numerically, and show, in particular, how they behave on the symmetry axis, in the equatorial plane and on the horizon. The problem is relevant for modelling accretion discs around black holes.


Covariant Loop Quantum Gravity
Irinkov, Pavel ; Hinterleitner, Franz (advisor) ; Svítek, Otakar (referee)
In this thesis we offer a broad introduction into loop quantum gravity against the backdrop of the quantum gravity research as a whole. We focus on both the canonical and covariant version of the theory. In the latter version we investigate the dynamics of some simple configurations in the simplified setting of PonzanoRegge model. We ascertain that the naïve approach to define a consistent dynamics, where the path integral's partition function is computed as a sum of amplitudes corresponding to all boundary and bulk states, fails in this case, on account of an appearance of divergences. This opens up space for the utilization of some more sophisticated methods.


Gravitational waves in cosmology
Kadlecová, Alžběta ; Svítek, Otakar (advisor) ; Žofka, Martin (referee)
In this work, we study the backreaction of highfrequency gravitational waves on cosmological backgrounds. To describe the wave, we use the Isa acson formalism, specifically the WKB approximation, which allows us to express the backreaction through an effective stressenergy tensor of the gra vitational wave. First, we consider the inhomogeneous cosmological model of Charach and Malin, which contains gravitational waves and a massless scalar field mini mally coupled to gravity. We show that although this is a spatially compacti fied solution, it is possible to add a highfrequency perturbation and solve Einstein's equations with the effective stressenergy tensor in a consistent way. The bacreaction is of the same order as the influence of the scalar field. Second, we add multiple incoherent highfrequency waves to the homoge neous Kasner background, and discuss the relation to the latetime limit of the Gowdy (vacuum Charach and Malin) model. 1
