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Gravitational wave templates from Extreme Mass Ratio Inspirals
Skoupý, Viktor ; Loukes Gerakopoulos, Georgios (advisor) ; van de Meent, Maarten (referee) ; Wardell, Barry (referee)
Future space-based gravitational-wave detectors will require highly accurate gravi- tational wave templates for detecting extreme mass ratio inspirals and estimating their parameters. These templates must include the postadiabatic effects like the spin of the secondary body. Therefore, we investigate the influence of the secondary spin on the motion around a Kerr black hole, calculate the corresponding gravitational-wave fluxes to produce flux-driven inspirals and reveal the shifts of the gravitational-wave phases induced by the secondary's spin. In particular, this study begins by considering eccentric equatorial orbits, where we obtain the constants of motion and fundamental frequen- cies using the Mathisson-Papapetrou-Dixon equations. Next, we derive the linear-in-spin parts of these quantities. We introduce a new Teukolsky equation solver in the frequency domain to calculate the energy and angular momentum fluxes from these trajectories. We use the obtained fluxes to adiabatically evolve the orbital parameters and to find the spin-induced phase shifts. For off-equatorial orbits, a frequency-domain approach is employed to determine the trajectories in the linear-in-spin regime and to compute the re- spective fluxes. The agreement between the frequency-domain fluxes and those acquired using an existing...
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Orbital dynamics around a black hole surrounded by matter
Stratený, Michal ; Loukes Gerakopoulos, Georgios (advisor) ; Witzany, Vojtěch (referee)
This thesis studies the dynamics of geodesic motion within a curved spacetime around a Schwarzschild black hole, perturbed by a gravitational field of a far axisymmetric dis- tribution of mass enclosing the system. This particular spacetime can serve as a versatile model for a diverse range of astrophysical scenarios. At the beginning of the thesis, a brief overview of the theory of classical mechanical systems and properties of geodesic motion are provided. A brief introduction to the theory of integrability and non-integrability, along with essential tools for analysis of non-integrable systems, including Poincaré sur- face of section and rotation numbers, is provided as well. These methods are subsequently applied to the under study spacetime through numerical methods. By utilising the rota- tion numbers, the widths of resonances are calculated, which are then used in establishing the relation between the perturbation parameter and the parameter characterising the perturbed metric. 1
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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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Dynamical systems in cosmology
Knob, Lukáš ; Acquaviva, Giovanni (advisor) ; Loukes Gerakopoulos, Georgios (referee)
The main aim of this thesis is the analysis of different cosmological models from the standpoint of dynamical systems theory. We consider mostly spatially curved FLRW metric with different source terms, some of them possible candidates for dark matter and dark energy, particularly linear barotropic fluids, Chaplygin gas and canonical scalar field with exponential and general form of potential. We rewrite the cosmological equations as the system of the first order differential equations in dimensionless variables and study globally their phase space and the stability of the critical points. We also present few interesting features of models with interactions between two cosmic fluid constituents and mention dynamical properties of orthogonal Bianchi I models. 1
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Physics of extended objects in strong gravitational fields
Veselý, Vítek ; Žofka, Martin (advisor) ; Loukes Gerakopoulos, Georgios (referee)
We study several different models of extended bodies in gravitational fields. Firstly, we revisit the glider model of a dumbbell-like oscillating body. We develop an independent scheme to integrate the equations of motion. We study the radial fall of a Newtonian spring, calculate the position shifts of the spring and find the critical value of the spring constant which cannot overcome the tidal forces. We argue that the relativistic glider model is unphysical due to its behaviour in the critical regions. Secondly, we show that Dixon's theory of extended bodies predicts a geodesic motion of the centre of mass in maximally symmetric spacetimes. We prove that a system of test particles can be described by a conserved stress-energy tensor and we evaluate the position shifts of the glider model in the maximally symmetric spacetimes, showing its disagreement with Dixon's theory. We thus conclude again that the glider model must be rejected. And thirdly, we study a model of an extended body consisting of interacting particles, which is in accord with Dixon's theory. We calculate the position shifts for this model and show that the model does not predict any measurable swimming effect. Finally, we estimate the numerical error of the calculation by finding the position shifts of the model in maximally symmetric...
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Dynamics of spinning test particles in curved spacetimes
Zelenka, Ondřej ; Loukes Gerakopoulos, Georgios (advisor) ; Witzany, Vojtěch (referee)
The motion of a test particle in the Schwarzschild background models the merger of a compact object binary with extremely different masses known in the literature as Extreme Mass Ratio Inspiral. In the simplest geodesic approxima- tion, this motion is integrable and there is no chaos. When one takes the spin of the smaller body into account, integrability is broken and prolonged resonances along with chaotic orbits appear. By employing the methods of Poincaré surface of section, rotation number and recurrence analysis we show for the first time that there is chaos for astrophysically relevant spin values. We propose a uni- versal method of measuring widths of resonances in perturbations of geodesic motion in the Schwarzschild spacetime using action-angle-like variables. We ap- ply this novel method to demonstrate that one of the most prominent resonances is driven by second order in spin terms by studying its growth, supporting the expectation that chaos will not play a dominant role in Extreme Mass Ratio Inspirals. Last but not least, we compute gravitational waveforms in the time- domain and establish that they carry information on the motion's dynamics. In particular, we show that the time series of the gravitational wave strain can be used to discern regular from chaotic motion of the source. 1
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Model of relativistic spinning system
Slezák, Daniel ; Ledvinka, Tomáš (advisor) ; Loukes Gerakopoulos, Georgios (referee)
Contrary to massive point particles, a description of extended bodies dynamics inclu- des higher mass moments, the first of which is spin. In this manner, Mathisson- Papapetrou-Dixon (MPD) equations has to be used instead of the geodesic equation to capture the more complicated evolution of the system. In this work, an extended system is represented by a set of freely moving, occasionally colliding point particles. As an aid in the construction of the model, some of these particles carry negative mass so it is possible to enclose their trajectories by elastic collisions. We then define a system's representative quantities, such as mass, momentum and spin. However, their relativistic theory requires to solve mainly the problems of parallel transport and the choice of a reference frame. Finally - from the known movement of the in- dividual particles, we can show that the whole system obeys the MPD equations. For that we use the simplification of small spacetime curvature along with a more extensive use of parallel transport instead of stress-energy tensor dynamic equation, the significance of which we limit to the behaviour of the component particles.
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Chaotic motion in Johannsen-Psaltis spacetime
Zelenka, Ondřej ; Loukes Gerakopoulos, Georgios (advisor) ; Kopáček, Ondřej (referee)
The Johannsen-Psaltis spacetime is a perturbation of the Kerr spacetime de- signed to avoid pathologies like naked singularities and closed timelike curves. This spacetime depends not only on the mass and the spin of the central object, but also on extra parameters, making the spacetime deviate from Kerr; in this work we consider only the lowest order physically meaningful extra parameter. In this thesis we summarize the basics of the theory of regular and chaotic dynamics and we use numerical examples to show that geodesic motion in this spacetime can exhibit chaotic behavior. We study the corresponding phase space by using Poincaré sections and rotation numbers to show chaotic behavior both directly and indirectly (e.g. Birkhoff chains), and we use Lyapunov exponents to directly estimate the sensitivity to initial conditions for chaotic orbits. 1
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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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