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Rotating thin disc around a Schwarzschild black hole: properties of perturbative solution
Kotlařík, Petr ; Semerák, Oldřich (advisor) ; Ledvinka, Tomáš (referee)
In 1974, Will presented a solution for the perturbation of a Schwarzschild black hole due to a slowly rotating and light thin disc given in terms of a multipole expansion of the perturbation series. In a recently submitted paper, P. Čížek and O. Semerák generalized this procedure to the perturbation by a slowly rotating finite thin disc, using closed forms of Green functions rather than the multipole expansion. The method is illustrated there, in the first perturbation order, on the constant-density disc. In this thesis, we summarize, check and plot some of the obtained properties, and show how the presence of the disc changes the geometry of a horizon and the position of significant circular orbits. 1
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Astrofyzikální procesy v blízkosti kompaktních objektů
Sochora, Vjačeslav ; Karas, Vladimír (advisor) ; Schee, Jan (referee) ; Semerák, Oldřich (referee)
Title: Astrophysical processes near compact objects: studying extremal en- ergy shifts from accretion rings Author: Vjačeslav Sochora Department: Academy of Sciences of the Czech Republic, Astronomical In- stitute Supervisor: doc. RNDr. Vladimír Karas, DrSc.; Academy of Sciences of the Czech Republic, Astronomical Institute Abstract: The X-ray emission from inner regions of an accretion disk around black holes provides wealth of information about matter in extreme con- ditions. A spectral profile of radiation from a narrow circular ring has a characteristic double-horn profile. Red and blue peaks of the profile are close to the extremal values of the energy shift. We describe a useful approach to calculate the extremal energy shifts in the regime of strong gravity. We dis- cuss if the radial structure of the disk emission could be reconstructed using extremal energy shifts of the individual rings. For this purpose, we simulate artificial data from a bright active galactic nucleus and show that the re- quired sensitivity and energy resolution can be reached with the proposed LOFT mission. Keywords: black hole physics, accretion disks, galactic nuclei
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Geodesic chaos in a perturbed Schwarzschild field
Polcar, Lukáš ; Semerák, Oldřich (advisor) ; Kopáček, Ondřej (referee)
We study the dynamics of time-like geodesics in the field of black holes perturbed by a circular ring or disc, restricting to static and axisymmetric class of space-times. Two analytical methods are tested which do not require solving the equations of motion: (i) the so-called geometric criterion of chaos based on eigenvalues of the Riemann tensor, and (ii) the method of Melnikov which detects the chaotic layer arising by break-up of a homoclinic orbit. Predictions of both methods are compared with numerical results in order to learn how accurate and reliable they are.
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Centre of the Kerr and Appell space-times
Jurčík, Róbert ; Semerák, Oldřich (advisor) ; Žofka, Martin (referee)
One of the most important solutions of Einstein equations is the Kerr metric. At the very centre of this space-time, there lies a ring curvature singularity. The singularity encircles a surface which joins together two asymptotically flat sheets of the manifold. The surface is intrinsically flat and is standardly interpreted as a planar disc. However, an article has been recently published which claims that the central surface is actually a dicone, with vertex (vertices) on the symmetry axis. In this thesis we analyse various geometric characteristics of the surface, in order to check which of the pictures is more adequate. We also examine the same surface of the Appell space-time which has the same spatial structure as the Kerr one. 1
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Homoclinic Chaos in Black-hole Fields
Hájková, Tereza-Marie ; Semerák, Oldřich (advisor) ; Suková, Petra (referee)
The existence of homoclinic orbits is an intrinsic feature of static black hole's space- time. The regularity of the geodesic motion around these black holes can therefore be quickly disrupted due to the small changes in the original space-time. The nature of the dynamics around a perturbed orbit depends on the manner of intersection of the surrounding stable and unstable manifolds. If they intersect transversally, the homoclinic orbit splits into chaotic layers. In this thesis, the mathematical formulation of chaotic dynamical systems and main properties of the geodesic motion in circular space-times are discussed. Thereupon, the space-time around a static black hole is reproduced by classical approximations by using Paczyński-Wiita and logarithmic pseudo-Newtonian potentials. By means of the effective potential method, the homoclinic orbits are found for these potentials. In addition, the analysis of the general circular space-time is done and the equations of geodesic motion in axially symmetric space-times are examined. Finally, the motion in a Schwarzschild space-time with a static axially symmetric external source is inspected. 1
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Shape of the Kerr gravitational field
Tynianskaia, Valeriia ; Semerák, Oldřich (advisor) ; Švarc, Robert (referee)
Kerr metric is one of the most well-known and useful exact solutions of Einstein equations. We study various geometric properties of the Kerr spacetime in order to gain intuition for its spatial shape. In the review part we summarize basic features of the Kerr geometry, we write down Carter equations for geodesic motion in the Kerr spacetime, and we introduce kinematic characteristics of time-like and light-like congruences, such as expansion, shear and twist. In the second part of the thesis we calculate scalars for acceleration, expansion, shear and twist - and plot the corresponding "equipotential" surfaces - for several privi- leged congruences, namely the Carter observers, the static observers, the zero-angular- momentum observers, the principal null congruence and the recently found non-twisting null congruence(s). We also draw surfaces radially equidistant from the horizon and sur- faces spatially orthogonal to the PNC and to the twist-free congruences, as well as the surfaces of constant energy and redshift for the important time-like congruences. 1
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Study of geodesic chaos by fractal methods
Sychrovský, David ; Semerák, Oldřich (advisor) ; Čížek, Martin (referee)
We study the dynamics of free test particles in a field of Schwarzschild black hole surrounded by an external exact thin axisymmetric solutions of Einstein's equations. Specifically, we use the Bach-Weyl ring and two member of the inverted Morgan-Morgan family of solutions as the additional sources. The fractal basin boundary and other meth- ods are used to detect and quantify chaos in time-like geodesic motion of the particles, primarily by computing box-counting dimension of said basin boundary. Our results mainly consist of the dependence of the chaoticity of these systems on mass and radius of the additional source as well as conserved energy and angular momentum of the test particles. We compare our results to literature and expand on them. 1
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Magnetic field of current loops around black holes
Vrba, Šimon ; Semerák, Oldřich (advisor) ; Kofroň, David (referee)
The magnetic field of a testing current loop in the equatorial plane around a Schwarzschild and Kerr black holes is visualized. In particular, the cases of an extreme black hole and of a Kerr naked singularity are analyzed. The simplest models of massless thin and thick current disks around a Schwarzschild black hole are presented. 1
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Stationary fields in black-hole space-times
Čížek, Pavel ; Semerák, Oldřich (advisor)
Motivated by modelling of astrophysical black holes surrounded by accretion structures, as well as by theoretical interest, we study two methods how to ob- tain, within stationary and axisymmetric solutions of general relativity, a metric describing the black hole encircled by a thin ring or a disc. The first is a suitable perturbation of a Schwarzschild black hole. Starting from the seminal paper by Will (1974), we showed that it is possible to express the Green functions of the problem in a closed form, which can then be employed to obtain, e.g., a reason- able linear perturbation for a black hole surrounded by a thin finite disc. In the second part we tackle the same problem using the Belinskii-Zakharov generating algorithm, showing/confirming that in a stationary case its outcome is unphysi- cal, yet at least obtaining a modest new result for the (static) "superposition" of a Schwarzschild black hole with the Bach-Weyl ring. 1
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