Relativistický seminář
Ústavu teoretické fyziky

Příští semináře:

13. června 2023

!!! Od 10:40 !!!

Dancing with Gargantua: gravitational waves from extreme-mass-ratio inspirals

Dr. Gabriel A. Piovano

University College Dublin, School of Mathematics and Statistics

(seminář v angličtině)

Gravitational waves have revolutionized how we observe the Universe. However, current detectors are limited by the overwhelming seismic noise in the low-frequency range. Only space-borne observatories like the future LISA and Taiji missions can reach the rich gravitational wave spectrum in the milliHertz band. Among the sources detectable by space-borne interferometers, extreme-mass-ratio binaries are arguably the most interesting. These systems are composed of a massive black hole and a much smaller compact body. The latter falls into the heavier companion in a slow process called extreme-mass-ratio inspiral (EMRI), emitting gravitational waves with complex features. The properties of an EMRI binary could be recovered from a detected years-long signal with astounding precision, which might lead to exciting discoveries in astrophysics, gravitational, and particle physics. On the other hand, EMRIs are challenging sources to model, and there are still several open problems in data analysis for space detectors. This seminar aims to be an introduction to EMRIs, and is divided into three parts. In the first part, I will overview the theorical tools commonly employed to construct EMRI waveforms, focusing on black hole perturbation theory techniques. I will then introuduce Bayesian statistics applied to gravitational wave data analysis. Finally, I will talk about my current research on parameter estimation for spinning EMRI binaries.

13. června 2023

Gyroscopes orbiting gargantuan black holes: Incorporating secondary spin into large mass-ratio inspirals

Lisa V. Drummond

MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology

(seminář v angličtině)

Very large mass-ratio binary black hole systems are expected to radiate low-frequency gravitational waves detectable by planned space-based Laser Interferometer Space Antenna (LISA). We hope to use these systems to probe the spacetime in exquisite detail and make precision measurements of the larger black hole’s properties. Accurate models using general relativistic perturbation theory will allow us to realize the potential of these large mass-ratio systems. Such models must include post-geodesic corrections, which account for forces driving the smaller black hole away from a geodesic trajectory. An important post-geodesic effect is gravitational self-force, which describes the small body's interaction with its own spacetime curvature. This effect includes the backreaction due to gravitational-wave emission that leads to the inspiral of the small body into the black hole. When a spinning body orbits a black hole, its spin couples to the curvature of the background spacetime. This introduces a second post-geodesic correction called the spin-curvature force. In this talk, I will present our calculation of spinning-body inspirals and associated waveforms that include both spin-curvature forces and the leading backreaction of self forces. I will discuss what aspects of the self force have been neglected, and what must be done to include these aspects in the future. Finally, I will discuss how we use a near-identity transformation to eliminate dependence on the orbital phases, allowing for very fast computation of completely generic worldlines of spinning bodies.

Minulé semináře:

20. září 2022

Metric of an evaporating black hole

Dr. Shohreh Abdolrahimi

Department of Physics and Astronomy, California State Polytechnic University, Pomona

(seminář v angličtině)

We present an approximate time-dependent metric in ingoing Eddington-Finkelstein coordinates for an evaporating nonrotating black hole as a first-order perturbation of the Schwarzschild metric, using the linearized backreaction from a realistic approximation for the stress-energy tensor for the Hawking radiation in the Unruh quantum state.

21. září 2022

!!! STŘEDA 15:45 !!!

Three-dimensional gravity on spaces with multiple boundaries

Prof. Marc, Baron Henneaux

Theoretical and Mathematical Physics Unit, Free University of Bruxelles

(seminář v angličtině)

Three-dimensional gravity has no local dynamical degree of freedom. Yet, it is far from being trivial when the cosmological constant is negative. (i) It admits black hole solutions. (ii) It possesses remarkable asymptotic properties at infinity where an infinite-dimensional symmetry algebra emerges. These unique features make three-dimensional gravity a remarkable "theoretical laboratory" in which to explore in a simpler context various conceptual issues related to Einstein theory. The talk will first discuss general aspects of three-dimensional gravity. It will then focus on the question of (non-)factorization of the Hilbert space in the quantum theory when the spatial sections have multiple boundaries. (Prof. Henneaux also gives a lecture within the workshop Gravity&Prague 2022, on Friday, September 23, at 11:30.)

Prezentace:

27. září 2022

!!! od 10:40 !!!

Spacelike (cosmological or black hole) singularities

Prof. Marc, Baron Henneaux

Theoretical and Mathematical Physics Unit, Free University of Bruxelles

(seminář v angličtině)

The BKL analysis of the generic behaviour of the gravitational field near a spacelike singularity will be reviewed. Its modern reformulation in terms of hyperbolic billiards and Weyl groups of hyperbolic Kac-Moody algebras will be presented. The analysis will cover pure gravity in four spacetime dimensions, as well as higher dimensional models with matter fields (some of which are relevant for a holographic description of superconductivity).

Záznam:

11. října 2022

Universal Einstein-Maxwell solutions

Dr. Marcello Ortaggio

Department of Algebra, Geometry and Mathematical Physics, Institute of Mathematics of the Czech Academy of Sciences

(seminář v angličtině)

Certain classes of Einstein spacetimes are known to be "immune'' to higher-order corrections, i.e., they are simultaneous solutions of (virtually) any modification of Einstein's gravity in vacuum. In this talk we report on recent progress for the case of Einstein-Maxwell solutions, for which one has to consider also the backreaction of the electromagnetic field as well as corrections to the Maxwell equations. After giving a characterization of (at least some) universal Einstein-Maxwell solutions, we exemplify the obtained results in the case of particular theories such as ModMax or Horndeski electrodynamics.

Záznam:

18. října 2022

Stories from the realm of extended test bodies

Dr. Georgios Lukes-Gerakopoulos

Prague Relativistic Astrophysics group, Astronomical Institute of the Czech Academy of Sciences

(seminář v angličtině)

We shall discuss mainly results which have appeared in arXiv:1907.05659, arXiv:2104.11183 and arXiv:2206.11149. The first tackles the issue of whether an extended body can move on geodesic trajectories in a black hole spacetime background. The other two works look into equatorial circular orbits of extended bodies in the pole dipole approximation and show that in curved backgrounds the centroid choice is not a gauge.

Záznam:

25. října 2022

Resonances in gravitational-wave inspirals

Dr. Vojtěch Witzany

Relativity group @ University College Dublin

(seminář v angličtině)

Inspirals of compact objects due to gravitational radiation are the central sources for gravitational-wave detectors. The inspirals we have seen so far have had reasonably simple dynamics. However, upcoming gravitational-wave detectors will extend our reach and start seeing complex precessing and eccentric inspirals. An especially worrying feature of those will be resonances, moments when frequencies of the system become interlocked in integer ratios and the nature of the motion unexpectedly changes. I will discuss how these can be treated in a semi-analytical scheme and our progress in practically doing so.

Záznam:

1. listopadu 2022

Přednáška přes Zoom

The propagation time delay for pulsars orbiting a supermassive black hole

PD Dr. Eva Hackmann

The Center of Applied Space Technology and Microgravity (ZARM), University of Bremen

(seminář v angličtině)

Pulsar timing offers excellent capabilities to test the theory of gravity in the strong field regime. Particularly promising laboratories for such tests are pulsar – black hole binaries, and the search for pulsars closely orbiting the supermassive galactic center black hole is ongoing. Complementary to the post-Newtonian approach commonly employed in pulsar timing, here the test particle approximation provides another useful ansatz.
We present an analytical expression for the propagation time delay of lightlike geodesics in a Kerr spacetime, including the post-Newtonian Shapiro and frame dragging delays as well as effects due to the multipole structure of the Kerr black hole. As an specific example we discuss in detail the most extreme case of an equatorial egde on pulsar orbit and compare our results to the usual post-Newtonian expressions.

Záznam:

8. listopadu 2022

Přednáška přes Zoom

Generic orbits of spinning bodies around Kerr black holes

Lisa Valerie Drummond

MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology

(seminář v angličtině)

Accurate models of large mass-ratio black hole binary systems must include post-geodesic corrections which account for forces driving the small body away from the geodesic. The spin-curvature force, which arises due to the coupling of a test body’s spin to the spacetime curvature, is an example of a post-geodesic effect. We have developed a frequency-domain approach that precisely describes the orbits of spinning bodies around black holes. In this talk, I first show how we apply this approach to nearly-equatorial orbits (presented in arXiv:2201.1334). I then show how to use our method to characterize the behavior of fully generic spinning-body orbits, which are both inclined and eccentric and have the small body's spin arbitrarily oriented (described in detail in arXiv:2201.1335). The small body’s spin shifts the frequencies which affect orbital motion and these frequency shifts change accumulated phases which are direct gravitational-wave observables. A catalog of these frequencies and how they vary with the parameters describing orbits is therefore likely to be of use as waveform models for large mass ratio systems are developed. Next, I present fully generic spinning-body inspirals that include both the backreaction due to gravitational waves and spin-curvature forces, computed using an osculating element integration scheme (these inspirals omit some key elements which I will discuss in my talk). Finally, I present preliminary results (i) applying a near-identity transformation (NIT) which eliminates dependence on the orbital phases, allowing for very fast computation and isolation of the secular effects, and (ii) computing the corresponding gravitational waveforms.

Záznam:

10. listopadu 2022

First few overtones probe the event horizon geometry

Dr. Roman Konoplya

Institute of Physics, Silesian University in Opava

(seminář v angličtině)

It is broadly believed that quasinormal modes (QNMs) cannot tell the black-hole near-horizon geometry, because usually the low-lying modes are determined by the scattering of perturbations around the peak of the effective potential. Using the general parametrization of the black-hole spacetimes respecting the generic post-Newtonian asymptotic, we will show that tiny modifications of the Schwarzschild/Kerr geometry in a small region near the event horizon lead to almost the same Schwarzschild/Kerr fundamental mode, but totally different first few overtones. Having in mind that the first several overtones affect the quasinormal (QN) ringing at its early and intermediate stage [M. Giesler, M. Isi, M. Scheel, and S. Teukolsky, Phys. Rev. X 9, 041060 (2019)], we argue that the near-horizon geometry could in principle be studied via the first few overtones of the QN spectrum, which is important because corrections to the Einstein theory must modify precisely the near-horizon geometry, keeping the known weak field regime. We discuss the connection of this observation with the so called "overtones' instability" recently studied in [J. Jaramillo et. al., Phys. Rev. Lett. 128, 211102 (2022)].

Záznam:

15. listopadu 2022

Well-posed electromagnetic solutions in Robinson-Trautman geometry

Dr. Tayebeh Tahamtan

ÚTF

(seminář v angličtině)

Robinson-Trautman radiative solutions with the new nonlinear electrodynamics model are investigated. The results show that unlike the solutions of Maxwell electrodynamics in the Robinson-Trautman class, our new solutions are well-posed.

Záznam:

22. listopadu 2022

Generalizations of classical relativity theorems

Dr. Eleni-Alexandra Kontou

Department of Mathematics, Faculty of Natural, Mathematical & Engineering Sciences, King's College London

(seminář v angličtině)

Several classical relativity theorems, including the famous singularity theorems, have in their assumptions pointwise energy conditions. Those conditions bound the energy density (or similar quantities) on every spacetime point and are easily violated by quantum fields. One way to examine the applicability of those theorems in semiclassical gravity is to replace them with an averaged version, where the energy density is bounded on a segment of a causal geodesic. The index form method, used instead of the Raychaudhuri equation, provides a direct way of using those weakened conditions. In this talk I will apply this method to the Penrose singularity theorem and the Hawking area theorem and present progress and challenges for both.

Záznam:

29. listopadu 2022

Post-Newtonian theory for gravitational waves

Dr. Luc Blanchet, Directeur de Recherche

Group of Gravitation and Cosmology, Paris Institute of Astrophysics

(seminář v angličtině)

The post-Newtonian approximation to general relativity (expansion in small velocity and weak gravitational field) plays a paramount role in the discovery of gravitational waves by LIGO and Virgo. Indeed it is extremely efficient to describe the gravitational wave emission in the inspiral phase of compact binary systems, made of black holes or neutron stars. In this talk we review the latest developments on the post-Newtonian approximation pushed to very high order for gravitational waves.

Záznam:

6. prosince 2022

The Dynamics of Apparent Horizons in a Black Hole Collision (Zoom talk)

Robie A. Hennigar

Department of Quantum Physics and Astrophysics, Institute of Cosmic Sciences, University of Barcelona

(seminář v angličtině)

Apparent horizons are commonly used in numerical relativity to characterize the boundary of a black hole. While a theoretical understanding of the event horizon’s dynamics during a merger has been known for about half a century, significant progress on the corresponding problem for the apparent horizon has been occurring only very recently. In this talk, I will discuss the dynamical evolution of the apparent horizon as it is now understood. Time permitting, I will discuss how various behaviours for marginally outer trapped surfaces observed during the merger can be recovered by studying much simpler, isolated eternal black holes.

Záznam:

13. prosince 2022

The geometry of spacetime from quantum measurements

Tales Rick Perche

Perimeter Institute for Theoretical Physics / University of Waterloo

(seminář v angličtině)

General relativity describes spacetime as a 4 dimensional Lorentzian manifold. The spacetime metric allows one to compute space and time separations between events in spacetime. However, general relativity is a classical theory, and it is not expected to be valid in scales where quantum effects become relevant, such as for distances of the order of the Planck length. In regimes where quantum field theory is valid, one can instead define spacetime separations through correlation functions of a quantum field. In fact, it has been conjectured before that the geometry of spacetime might be emergent from entanglement present in quantum fields. In this description the metric becomes effective, and can be seen as a consequence of the presence of a quantum field in the background spacetime. Moreover, these correlations can be accessed, so that one can rebuild the geometry of spacetime simply by performing quantum measurements on a quantum field.

Záznam:

20. prosince 2022

Simulating particle acceleration in astrophysical shocks

Dr. Allard Jan van Marle

ELI Beamlines, Institute of Physics of the Czech Academy of Sciences

(seminář v angličtině)

Cosmic-rays are charged particles that have been accelerated to relativistic speeds. The general consensus is that the acceleration occurs in astrophysical shocks, with the particles gaining momentum as they repeatedly cross the shock front. In order to understand this process and determine which types of shock contribute to which part of the cosmic-ray spectrum, we need to run numerical simulations that incorporate both the large-scale shock structure and the microphysics of the interaction between charged particles and the local magnetic field. Here we introduce a combined particle-in-cell (PIC) and magnetohydrodynamics (MHD) code that combines the best aspects of both aspects into a single model. This allows us to run large-scale models of astrophysical shocks, incorporating both the particle acceleration process and the feedback of this process on the shock. From these models we can estimate how the characteristics of individual shocks influence their ability to accelerate particles and contribute to the cosmic-ray spectrum.

Záznam:

28. února 2023

Axially symmetric stationary perturbations of a Kerr black hole: Debye superpotential for charged ring or circular current, and further generalizations

Dr. David Kofroň

ÚTF

(seminář v angličtině)

We provide an explicit, closed and compact expression for the Debye superpotential of a circular EM source in a stationary axisymmetric spacetime. It is obtained by integrating the Green function of Teukolsky Master Equation (TME). The Debye potential itself is then, for a particular configuration, calculated in the same manner as the phi_0 field component is calculated from the Green function of the TME - by convolution of the Green function with sources. In such a way we provide an exact field of charged ring and circular current on the Kerr background, finalizing thus the work of Linet. We will also shortly discuss the recent progress (and complications) in extending these results to gravitational perturbations.

Záznam:

7. března 2023

Zoom talk:

Accretion disk structure around a uniformly accelerating black hole

Dr. Shokoufe Faraji

ZARM Institute, University of Bremen

(seminář v angličtině)

Generalisation of the relativistic accretion thick disc model is considered to the background of a spinning charged accelerating black hole described by the C-metric to study the effects of this background on the disc model. In this talk, a review of this metric will be presented and the main properties of the accretion disc model and its dependence on the initial parameters of the underlying spacetime will be discussed.

Záznam:

14. března 2023

Monarch migration of Carrollian particles on the black hole horizon

doc. David Kubizňák

ÚTF

(seminář v angličtině)

After discussing the basics of Carrollian physics, we shall revisit the motion of massless particles with anyonic spin in the black hole horizons. As recently shown, such particles can move within the horizon of the black hole due to the coupling of charges associated with a 2-parametric central extension of the 2-dimensional Carroll group to the magnetic field around black holes -- the so called "anyonic spin-Hall effect". We shall study several examples of such motions. Of these, the most interesting is the motion in a misaligned (asymptotically uniform) magnetic field around Kerr, which results in a time dependent motion of Carrollian particles that is reminiscent of "monarch migration".

Záznam:

21. března 2023

Gravitational self-force in a highly regular gauge

Dr. Samuel Upton

Astronomický ústav AV ČR

(seminář v angličtině)

Extreme-mass-ratio inspirals (EMRIs) will be an important gravitational-wave source for the Laser Interferometer Space Antenna (LISA), a future space-based gravitational-wave detector. In an EMRI, a stellar mass compact object, such as a black hole or neutron star, slowly spirals into a supermassive black hole while continually emitting gravitational waves. The primary way of modelling these systems is through a perturbative method known as gravitational self-force. To accurately model EMRI systems requires us to go to second order in perturbation theory, which creates a number of issues. In this talk, I will present an overview and current status of the field of gravitational self-force research and present work done to develop a class of highly regular gauges to ameliorate the strong divergences one finds near the worldline of the small object that hinder rapid numerical calculations.

Záznam:

28. března 2023

Relativistic positioning systems and automatic differentiation

Dr. Justin C. Feng

Center for Astrophysics and Gravitation @ Instituto Superior Técnico, University of Lisbon

(seminář v angličtině)

I discuss the framework of relativistic positioning systems for satellite navigation. I then discuss a new approach to the relativistic location problem (i.e. the determination of the position of a user from satellite data) in a generic spacetime geometry. This approach makes use of automatic differentiation. Time permitting, I will discuss in detail the potential applications of this powerful new tool in computational physics.

Záznam:

28. března 2023

!!! 10:40, v rámci Semináře ÚTF !!!

Efficient trajectory calculations for extreme mass ratio inspirals using near-identity (averaging) transformations

Dr. Philip Lynch

Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Department of Astrophysical and Cosmological Relativity

(seminář v angličtině)

Future space based gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA) will allow for the detection extreme mass ratio inspirals (EMRIs) which consist of a stellar mass compact object spiralling into a massive black hole (MBH) due to gravitational radiation reaction. These sources are of particular interest for their ability to accurately map the spacetime of the MBH, allowing for unprecedentedly accurate measurements of the MBH's mass and spin, and tests of general relativity in the strong field regime. To reach the science goals of the LISA mission, one requires waveform models that are (i) accurate to within a fraction of a radian, (ii) extensive in the source's parameter space and (iii) fast to compute, ideally in less than a second. We focus on the later criteria by utilising techniques that will speed up inspiral trajectory calculations as well as extending prior models to include the MBH's spin. We develop the first EMRI models that incorporate the spin of the MBH along with all effects of the gravitational self-force (GSF) to first order in the mass ratio. Our models are based on an action angle formulation of the method of osculating geodesics (OG) for generic inspirals in Kerr spacetime. For eccentric equatorial inspirals and spherical inspirals, the forcing terms are provided by an spectral interpolation of the first order GSF. For generic inspirals where sufficient GSF data is not available, we construct a toy model from the previous two models. However, the OG method is slow to evaluate due to the dependence of the equations of motion (EOM) on the orbital phases. Therefore, we apply a near-identity (averaging) transformation (NIT) to eliminate all dependence of EOM on the orbital phases while maintaining all secular effects to post-adiabatic order. This inspiral model can be evaluated in less than a second for any mass-ratio as we no longer have to resolve all ~105 orbit cycles of a typical EMRI.

4. dubna 2023

Exact solutions in 2+1 dimensional gravity

Mgr. Matúš Papajčík

ÚTF

(seminář v angličtině)

Exact solutions in the lower dimensional theory of gravity have found their use in many areas of theoretical physics from string theory and AdS/CFT correspondence to quantum gravity. They play an important role as toy models, therefore, their study is highly desirable. Keeping in mind that within 3D gravity virtually all spacetimes must belong to either the expanding Robinson-Trautman class or the non-expanding Kundt class of geometries, we investigate the exact solutions of the coupled system of Einstein-Maxwell equations for these geometries, allowing also for a non-vanishing cosmological constant. We discuss the found electrovacuum solutions and identify the special subclass of charged black hole spacetimes in 2+1 gravity.

Záznam:

11. dubna 2023

Seminář se nekoná

(seminář v angličtině)

18. dubna 2023

Zoom talk:

Toroidal structures in astrophysical objects: gravitational properties

Prof. Elena Bannikova

INAF Capodimonte Astronomical Observatory (Naples) / V. N. Karazin Kharkiv National University / Radio Astronomy Institute of NAS of Ukraine

(seminář v angličtině)

There are astrophysical objects possessing toroidal (ring) structures, for example, ring galaxies and dusty tori in active galactic nuclei. The masses of such toroidal-ring structures are substantial, which requires accounting of its gravitational properties. In my talk I will give a review of our results about the gravitational properties of a torus starting from its gravitational potential for the circular and elliptical cross-sections. I will consider the dynamics in a gravitational field of a central mass and a gravitating ring where there is a region of unstable equilibrium —the "Lagrangian circle" (LC). The closed circular orbits exist only to a certain radius corresponding to the "outermost stable circular orbit" (OSCO). In this case there is analogy with the ISCO in the relativistic case. The existence of region with non-circular orbits between LC and OSCO can explain the observed gap in the distribution of stellar density in ring galaxies. The effects of gravitational lensing show that in this lens system the formation of three Einstein rings is possible.

Záznam:

25. dubna 2023

Non-local scalar fields in static spacetimes

Dr. Ivan Kolář

ÚTF

(seminář v angličtině)

Non-local exponential operators with an infinite number of derivatives appear in modern physics in various contexts. They arise naturally in the effective models of string theories but they also play an important role in ghost-free UV completions of GR. In this talk, I will discuss the methods for solving linear non-local scalar field equations in (curved) static spacetimes. The methods are based on the heat kernels and their estimates in compact and non-compact manifolds. I will present several examples of static and time-dependent solutions and also discuss their regularity. The talk is based on arXiv:2201.09908.

Záznam:

2. května 2023

Comparing effective-one-body and gravitational self-force results as tools for third-generation gravitational wave detectors

Angelica Albertini

Astronomical Institute, Czech Academy of Sciences

(seminář v angličtině)

Gravitational self-force theory and the effective-one-body (EOB) formalism are two different approaches to the two-body problem in general relativity, both suited to describe black hole binary evolutions, but natively pertaining to different domains of validity. After briefly introducing the two formalisms, I will present a comparison between the related gravitational waveforms, both in the time domain and in the frequency domain, locating the region of largest agreement. I will then introduce some updates to the EOB model, paving the way for accurate waveform modeling for third-generation detectors.

Záznam:

9. května 2023

Seminář se nekoná. Místo toho se můžete podívat na MTW anniversary

(seminář v angličtině)

16. května 2023

Covariant relativistic stars

Dr. Sante Carloni

ÚTF

(seminář v angličtině)

We apply the 1+1+2 covariant semi-tetrad approach to derive in covariant form the Tolman-Oppenheimer-Volkoff equations. These equations naturally include anisotropic pressures and are easily extended to the multifluid case. The covariant TOV equations are then used to obtain new exact solutions using direct resolution and reconstruction techniques. Finally, we show that the well-known generating theorems for the solutions of the TOV equations can be represented by simple deformations and that these theorems can be used also to derive multifluid solutions from single fluid ones.

Záznam:

23. května 2023

Observers and flows in black hole space-times: The inside story

Prof. Oleg B. Zaslavskij

Department of Physics and Technology, V. N. Karazin Kharkiv National University

(seminář v angličtině)

We consider general radially moving frames realized in the background of nonextremal black holes having causal structure similar to that of the Schwarzschild metric. In doing so, we generalize the Lemaitre approach, constructing free-falling frames which are built from the reference particles with an arbitrary specific energy e_0 including e0<0 and a special case e0=0. The general formula of 3-velocity of a freely falling particle with the specific energy e with respect to a frame with e_0 is presented. We point out the relation between the properties of considered frames near a horizon and the Banados-Silk-West effect of an indefinite growth of energy of particle collisions. Using our radially moving frames, we consider also nonradial motion of test particles including the regions near the horizon and singularity. We also point out the properties of the Lemaitre time at horizons depending on the frame and sign of particle energy.

Záznam:

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• 2022: jaro
• 2021: jaro, podzim
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• 1997: podzim

Jiří Bičák                                                                                                  Oldřich Semerák