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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.
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Semi-classical energy of elliptic Nambu-Goto string
Kozoň, Marek ; Zahn, Jochen (advisor) ; Scholtz, Martin (referee)
Po zhrnutí potrebných teoretických základov a predošlého výskumu v ob- lasti, prezentujeme semi-klasickú kvantovaciu schému pre uzavretú Nambuovu- Gotovu strunu. Týmto zovšeobecňujeme predošlú prácu, ktorá bola vykonaná pre otvorenú strunu a uzavretú strunu kruhového tvaru. Pomocou metód kvan- tovej teórie po©a v zakrivených priestoročasoch počítame strednú hodnotu vo©- ného Hamiltoniánu struny rotujúcej v dvoch na seba kolmých priestorových rovinách v priestoročase všeobecnej dimenzie. Táto hodnota je priamo úmerná kvantovej korekcii k celkovej energii struny, ktorá má formu tzv. Reggeovho interceptu. Výslednú hodnotu Reggeovho interceptu porovnávame s predošlým výskumom. Taktiež uvádzame porovnanie získaného spektra fyzikálnych stavov struny kvantovanej našou metódou so spektrom odvodeným pomocou kovariant- ného kvantovania. 1
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Twistor equation on isolated horizons
Matejov, Dávid ; Scholtz, Martin (advisor) ; Švarc, Robert (referee)
In the present work we investigate the solution of the univalent twistor equation on an isolated horizon that serves for the definition of the so-called Penrose mass. We start our discussion with the construction of adapted co- ordinates to the isolated horizon and summarizing the main results in this field that are needed for our work. We include a chapter devoted to the extre- mal isolated horizons and prove an important result concerning uniqueness of geometry therein. It is a generalization of the paper by Lewandowski and Pawlowski (Class. Quantum Grav. 31 (17), 2014), which states that the ex- tremal isolated horizons are necessarily isometric to the intrinsic geometry of the Kerr-Newmann black hole. Further we proceed to investigation of the twistor equation on the isolated horizon. We analyze conditions of integra- bility and derive the time dependent solution. Consequently we solve the 2-surface twistor equation and briefly discuss the general approach to the problem of defining the Penrose charge. 1
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Gravitational collapse of magnetized matter
Bucko, Jozef ; Ledvinka, Tomáš (advisor) ; Scholtz, Martin (referee)
The scope of interest of this work is to study the gravitational collapse of magnetized matter. We consider homogeneous, isotropic and ideally conduct- ing star threaded by the test magnetic field. This field is chosen to be a relativistic generalization of dipole field outside the star and homogeneous field under the stellar surface. Dynamics of the collapse is described by Oppenheimer-Snyder solution so we consider the framework in which the motion of stellar matter re- mains unaffected by the magnetic field. In Schwarzschild's coordinates we review the exact solution of Maxwell's equations for electric and magnetic field inside the star and find the evolution equations for both fields outside the star that have to be treated numerically. Then we introduce Novikov coordinates and find the respective fields evolution equations. We test the equations using known exact so- lution for time-dependent magnetic dipole. Finally, we adopt chosen coordinates to become singularity-avoiding and integrate the respective differential equations numerically. 1
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Termodynamika černých děr. Entropie a informace.
Liška, Marek ; Acquaviva, Giovanni (advisor) ; Scholtz, Martin (referee)
The aim of the thesis is to provide a review of black hole thermodynamics and its relation with concepts of entropy and physical information. We start by deriving the four laws of black hole thermodynamics in the context of classical general relativity. To supplement this, we use semiclassical limit of quantum mechanics to show that black holes radiate and have non-zero thermodynamic temperature. In the second part of the thesis we describe the concepts of the Shannon and von Neumann entropy and of physical information. Lastly, we discuss the use of these concepts in the context of black hole mechanics. 1
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Geometry of isolated horizons
Flandera, Aleš ; Scholtz, Martin (advisor)
While the formalism of isolated horizons is known for some time, only quite recently the near horizon solution of Einstein's equations has been found in the Bondi-like coordinates by Krishnan in 2012. In this framework, the space-time is regarded as the characteristic initial value problem with the initial data given on the horizon and another null hypersurface. It is not clear, however, what ini- tial data reproduce the simplest physically relevant black hole solution, namely that of Kerr-Newman which describes stationary, axisymmetric black hole with charge. Moreover, Krishnan's construction employs the non-twisting null geodesic congruence and the tetrad which is parallelly propagated along this congruence. While the existence of such tetrad can be easily established in general, its explicit form can be very difficult to find and, in fact it has not been provided for the Kerr-Newman metric. The goal of this thesis was to fill this gap and provide a full description of the Kerr-Newman metric in the framework of isolated horizons. In the theoretical part of the thesis we review the spinor and Newman-Penrose formalism, basic geometry of isolated horizons and then present our results. Thesis is complemented by several appendices.
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Twistors in relativistic field theories
Nárožný, Jiří ; Scholtz, Martin (advisor) ; Souček, Vladimír (referee)
In this thesis, we are concerning about the Twistor theory, field originally motivated purely physically, although these days fully developed into the many fields of mathem- atics and physics. With its complexion Twistor theory influences algebraic geometry, Clifford analysis as well as the String theory or Theory of quantum gravity. In the thesis we describe the origin of twistors projective or not. Mathematical background to the twistor theory is covered in the first chapter, where we study Clifford algebras and their representations. In the first part of the second chapter we are describing non-projective twistors as representation elements of certain Spin-group, and we find the connection with the standard definition of non-projective twistors as a kernel of the twistor operator. In the last part of the second chapter, we create a space of pro- jective twistors and show its certain properties, especially its correspondence with the complexified compactified Minkowski spacetime.
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Gravitational collapse of scalar field
Šípka, Martin ; Ledvinka, Tomáš (advisor) ; Scholtz, Martin (referee)
In this thesis we study a critical collapse of a scalar field. We follow the work of [1] and create detailed and easy to understand figures and images describing the results and the phenomena observed during our calculations. We also try to give a detailed look at numerical methods used along with two ways of convergence testing. 1
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Simulation of two-dimensional flow past obstacles using lattice-gas cellular automata
Tomášik, Miroslav ; Scholtz, Martin (advisor) ; Pavelka, Michal (referee)
Cellular automata constitutes a unique approach to the modeling of complex systems. The major phase of their development in continuum mechanics came in the late 80s, but the closer inspection of their macroscopic limit revealed that it does not accurately correspond to hydrodynamic equations. Besides the Lattice-Boltzmann model, various other approaches to improve LGCA have emerged. The main focus of our research is on the Pair-interaction cellular automaton. In this thesis, we propose the non-deterministic variant of this automaton, and we compare it with its predecessor on the simulations of the "exploding cube", Taylor- Green vortex and fully developed turbulence. The results for the non-deterministic automaton seem quiet reasonable, but derivation of the hydrodynamic equations is necessary to conclude in what extent it solves the problem with anisotropic viscosity.
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Investigation of chaotic dynamical systems using the methods of Riemannian geometry
Matejov, Dávid ; Scholtz, Martin (advisor) ; Ledvinka, Tomáš (referee)
In this thesis we investigate a chaos in dynamical systems described by the Hamilton function using a new geometric method. At first, necessary definitions and terms are introduced, like dynamical systems, Lyapunov ex- ponents and Poncaré sections. Subsequently we deal with the new geometric method (Horowitz et al., 2007), which predicts the stability of a system un- der consideration. An extensive part is devoted to the general relativity and spinor formalism, the Newman-Penrose formalism and the theory of optical scalars. In the computational part of the thesis, we investigate the pp-waves and chaotic behaviour of geodesics in this class of space-times. All calcula- tions are done in the Python programming language, so we the chapter on numerical calculations is included. 1
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