Albert Einstein Center
for Gravitation and Astrophysics

Overview of our research activities in 2016


Since three years elapsed after the formation of the Center the report mentions also some results from years 2014 and 2015 but the main emphasis is on the last year 2016. We refer to individual yearly reports for more details. However, all works published by members of the Center and acknowledging the support from the Center from 2014 until now are available in the part f of the complete report and online at

Since the year 2016 was very special for general relativity, gravitation and astrophysics (first direct observation of gravitational waves, 100 years from the death of Ernst Mach), at the end of our report we mention our activities bringing these events to the community of Czech physicists and to the general public.

We start again by the two volumes [1], [2] published by Springer, which were initiated more than two years ago and finished only in summer 2014. People involved in the Center participated both scientifically and also in the organization of the international conference "Relativity and Gravitation--100 years after Einstein in Prague" held in Prague Carolinum and visited by more than 200 scientists from 31 countries. The volumes, edited by J. Bičák and T. Ledvinka, appeared in summer 2014. They contain 15 contributions by the members of the Center, a number of them acknowledging support by the Center. These two volumes were very positively reviewed in the "Observatory magazine”, Classical and Quantum Gravity and in General Relativity and Gravitation.

A number of people involved in the project participated in the 6th Central European Relativity Seminar (CERS) organized by the Center at the Malá Strana building of the Faculty of Mathematics and Physics on January 28-30, 2016. More details can be found on the website of the Center here Relativists from 5 European countries were participating.

Last three years, the relativity group in the Silesian University in Opava organized the traditional RAGTIME autumn workshop—the last one in Opava in October 2016 was the 18th workshop already. The 16th RAGTIME took place in Prague in 2014. Usually about 25-30 people discuss hot topics in relativistic astrophysics, with the participants being commonly from the Opava group but several of them are also visiting from the Prague group at the Astronomical Institute; and there are always a few foreign participants as well. The proceedings of the RAGtime workshop are published typically every third year. For more details, see and

As we shall see below, there exist not only mutual interactions between the groups participating in the Center (traditionally in local seminars or conferences abroad) but new direct collaborations on papers have now also arisen between (i) members of the group of the Astronomical Institute CAS in Prague and the group of relativistic astrophysics in Opava, and (ii) members of the group in the Mathematical Institute CAS in Prague and mathematical relativists at the Institute of Theoretical Physics of Charles University. As in the last year, we divide our report into several subsections.

Mathematical Relativity

J. Bičák and his PhD student J. Schmidt from the Faculty of Nuclear Sciences and Physical Engineering published at the beginning of 2016 a comprehensive work on uniqueness of the energy-momentum tensors in linearized Einstein’s theory and in massive gravity [3]. The procedure only assumes field equations that result in the conservation of the tensors. In the case of linear gravity, a 4-parametric system of conserved second-rank tensors was found that contains a unique symmetric tensor, which turned out to be the so-called linearized Landau-Lifshitz pseudotensor—the authors obtained it by a completely different procedure and its unique role has not been known. In the case of massive gravity, we arrived not only at the “generalized Landau-Lifshitz” tensor but also found a new simpler symmetric tensor. For the first time in our group the software CADABRA used in the field theory for tensor manipulations was employed in this work by J. Schmidt efficiently.

This work was noticed by Lars Andersson, the present head of the mathematical relativity section of the Albert Einstein Institute in Golm-Potsdam. Andersson, with several of his coworkers from different institutions, tackles a very complicated fundamental problem of the stability of rotating (Kerr) black holes. For this reason they found various types of conserved quantities in both linear electromagnetism and gravity that involve higher derivatives of the fields and that could be useful in the stability problems. During the stay of both JB and JS in Golm in the autumn we started to work on the classification of these conserved quantities and plan to proceed in this now.

However, during our stay we also progressed in a different problem—finding a generalization of the superpotential called Katz-Bicak-Lynden-Bell superpotential in literature [4] and used in exact general relativity to find energy and other quantities relevant for perturbations of a given background spacetime. The generalization concerns the so-called Horndeski’s generalizations of general relativity by adding also scalar fields of various general characteristics but leading to field equations of an order not higher than 2nd derivatives. This type of theories has become of great interest recently in connection with a possible explanation of dark energy. We also employed a special sector of these theories in our work on perturbations of black holes with scalar hair [5,6]. The calculations become unmanageable without using CADABRA and the resulting expression becomes quite complicated. However, we proceeded quite significantly thanks to our “continuous interaction” in Golm, and we hope to submit the paper for publication within several months.

In our “traditional topic” of boost-rotation symmetric radiative spacetimes representing uniformly accelerated objects and including particularly the C-metric representing accelerated black holes, a remarkable progress has been achieved by D. Kofroň in [7] who proved the possibility to separate the equations for massless test fields of arbitrary spin on the C-metric background when the black holes do not rotate. Subsequently, Kofroň has shown that the separability is still achievable when the accelerated black holes are charged and rotating. The paper demonstrating this appeared in [8] in 2016.

More recently, Kofroň investigated the possible nature of conical singularities in the C-metric (causing the acceleration), by considering the Bonnor rocket (the axially symmetric solution containing null radiation with an arbitrary angular distribution) with the null radiation focused along the axis. In the limit, the C-metric is recovered and thus also the stress energy tensor and physical interpretation of the conical singularities elucidated.

The importance of the C-metric lies, among others, in the fact that it contains regions with radiation. Other exact radiative spacetimes also belong to our traditional themes, in particular the Robinson-Trautman and Kundt solutions. In the last two years, T. Tahamtan and O. Svítek derived and analyzed an explicit Robinson–Trautman solution with a minimally coupled free scalar field. The solution contains a curvature singularity, which is initially naked but later it is covered by a horizon [9]. This study was extended by investigating physical characteristics like Bondi mass and specific subcases of the solution [10]. Another generalization of the Robinson–Trautman family was found by coupling it to an electromagnetic field satisfying nonlinear field equations. The solutions were generated from spherically symmetric ones, in all cases the electromagnetic field singularity is removed while the gravitational one persists. [11].

O. Svítek and T. Tahamtan also analyzed the behavior of simple scalar-field spacetimes during ultrarelativistic boost using two methods. In the first one the fundamental object is the metric while in the second one it is the energy momentum tensor. The results show that the scalar field is completely removed during the boost. The paper received Editor's choice status by the journal General Relativity and Gravitation [12].

T. Tahamtan continued collaboration with her previous colleagues at EMU and published a paper investigating colliding plane waves in the f(R) theory of gravity [13].

The collaboration between J. Podolský, R. Švarc and the DIANA group from the University of Vienna resulted in a rigorous description of geodesics in spacetimes of constant curvature with expanding impulsive gravitational waves [14]. The global uniqueness and C1 regularity was proved and the matching conditions of geodesics crossing the impulsive surface were justified in a precise way.

A new collaboration was established between the team members from the Mathematical Institute (V. Pravda, A. Pravdová) and the Institute of Theoretical Physics (J. Podolský, R. Švarc). They studied various properties of exact solutions to quadratic gravity in four dimensions. All Einstein spacetimes are solutions to this theory but other type solutions also exist. They have shown that in contrast to the Einstein gravity, the simplest types of the Ricci tensor imply that quadratic gravity solutions belong to the Kundt radiative class. The authors also proved that all Robinson-

Trautman spacetimes are conformal to Kundt spacetimes and they employed conformal transformations to construct new explicit non-Einstein exact solutions to quadratic gravity. The results have been submitted to Phys. Rev. D [15].

A. Pravdová and M. Kuchynka studied the geometric properties of spacetimes of Weyl types N and III and Ricci type N typical for radiative spacetimes in higher dimensions [16]. They proved that the Weyl and Ricci tensors are necessarily aligned and the corresponding aligned null direction is geodetic (a purely geometrical result). For the Weyl and Ricci type N, a generalization of the Goldberg-Sachs theorem, studied so far only for the vacuum case in higher dimensions, has also been provided.

T. Málek in collaboration with S. Hervik from University of Stavanger, Norway, studied universal spacetimes (vacuum solutions to all theories with the Lagrangian given by any polynomial curvature invariant) of neutral signature. In the case of the Lorentzian signature, the only known examples belong to the Kundt class. The authors have found a class of neutral signature that are not Kundt spacetimes. The paper is expected to be submitted for publication at the beginning of 2017.

Going over to the mathematical aspects of cosmology (reviewed over the last three years), the main contributions were due to O. Svítek and his students. With P. Kašpar they presented a new approach to averaging based on Cartan scalars. The theory was applied to two different Lemaitre-Tolman-Bondi (LTB) models. In one the correlation term behaves as a positive cosmological constant, in the second example it behaves as spatial curvature [17]. They also generalized Buchert's averaged equations to a special class of anisotropic dust models (all Einstein equations are averaged, not only their trace part) and studied the backreaction on expansion and shear scalars in an approximate LTB model [18]. With another PhD student (D. Vrba) Svítek studied the behavior of the density contrast in quasi-spherical Szekeres spacetime and derived its analytical behavior [19]. Both PhD students successfully defended their PhD theses.

In 2016, M. Žofka continued his work on a cosmological generalization of the Bonnor-Melvin spacetime. He gave a seminar on the topic at the Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava. He published a paper [20] investigating the properties of a static, cylindrically symmetric Majumdar-Papapetrou-type solution of Einstein-Maxwell equations. This particular solution describes the field of an infinitely thin source extending to infinity along the axis of the spacetime and generating both gravitational and electric fields. The paper discusses its singularities, algebraic type, asymptotic properties, weak-field limit, structure of electrogeodesics, and the mass and charge of the source. It further provides an interpretation of the spacetime and discusses the parameter appearing in the metric.

Black Holes and Gravitational Waves: Physical and Mathematical Aspects

O. Semerák and his PhD. student M. Basovník studied the response of black-hole space-times to the presence of additional strong sources of gravity, mainly focusing on the deformation induced inside the horizon. Restricting to static and axially symmetric (electro-)vacuum exact solutions of Einstein equations, we first considered the Majumdar–Papapetrou solution for binary extreme black holes in [21]. Although "the other" black hole is a very strong source, it turned out that the horizon interiors are not deformed significantly; this is likely connected with the extreme character of the given horizons. In the second paper [22], we thus considered a black hole which is far from being extreme, namely a Schwarzschild black hole surrounded by a concentric thin ring described by the Bach–Weyl solution. Extending the metric inside the black hole along null geodesics tangent to the horizon, we found that the black-hole interior is very strongly influenced by the external source. Already distinct on the level of potential and acceleration, this is still more pronounced on the level of curvature: for a sufficiently massive and/or nearby (small) ring, the Kretschmann scalar even becomes negative in certain toroidal regions mostly touching the horizon from inside. In the literature such regions have been interpreted as those where magnetic-type curvature dominates, but here we deal with space-times which do not involve rotation and the negative value is achieved due to the electric-type components of the Riemann/Weyl tensor. The Kretschmann scalar also charts rather non-trivial landscapes outside the horizon.

In numerical relativity T. Ledvinka with his PhD student A. Khirnov investigated the possibility of modifying the slicing conditions for collapsing gravitational waves by introducing a new type of gauge-source functions. These are given as solutions of elliptic equations, but their function-space is much smaller than that of the usual "grid variable" and so the solution can be found efficiently during the simulation. The numerical simulations will be used to study in detail the collapse of the so-called Brill-wave initial data into a Schwarzschild black hole.

With his student V. Bára, T. Ledvinka investigated classical superradiant electromagnetic fields and compared rotational electromagnetic superradiation of a Kerr black hole with a classical model inspired by a membrane-paradigm approach. A publication is under preparation.

T. Ledvinka and J. Bičák, together with prof. D. Lynden-Bell and his student W. Barker from University of Cambridge, studied rotation/dragging of inertial frames by rotating material shells and by rotating gravitational waves. It was interesting to compare these gravomagnetic effects caused by different-types of sources both analytically and by producing illustrative videos. The resulting work was submitted for publication in Classical and Quantum Gravity now at the beginning of 2017 [23].

J. Bičák and F. Hejda analyzed in 2014-2015 black holes in strong magnetic fields [24]. Recently they considered the motion of particles in black hole spacetimes with magnetic fields in particular from the perspective of possible large accelerations produced near extremal horizons. Their new work considers unification of two mechanisms caused by either angular momentum or by charge. Among a number of various effects they discovered the possibility of a “mega”–effect of producing large acceleration in which a critically charged particle must have an enormous angular momentum and a rotating black hole carries just a small charge. The final paper [25] was accepted recently for publication and the first proofs of the article were already sent to the editors.

Black Holes: Astrophysical Aspects

For a number of years L. Šubr has dedicated his research to important problems of stellar dynamics in the vicinity of supermassive black holes. By means of direct N-body modelling he studied dynamics of flattened structures of stars orbiting around supermassive black holes (SMBH). Having focused on stellar discs formed by nested eccentric orbits, together with J. Haas, they identified a rich family of Kozai-Lidov resonances which may bring stars to extremely eccentric orbits [26]. They have estimated the rates of tidal disruption of stars for several different configurations and have shown that they highly exceed the values expected from two-body relaxation. Two main simplifications of their models consist in (i) treating gravity of the SMBH in the Newtonian approximation and (ii) estimating the influence of the spherical component of the nuclear star cluster (NSC) by means of a smooth analytic potential. In 2016 they have improved performance of the N-body integrator so that it now enables them to study reliably systems of ~100000 stars, i.e., including the spherical component of the NSC.

Regarding the center of our Galaxy we should also mention the thesis „Coherent dusty and gaseous structures near the Galactic centre“ by L. Štofanová from the Astronomical Institute of the Charles University (with supervisor V. Karas, and M. Zajaček, from Koeln and Bonn as a consultant). The introductory part of this thesis is a brief review of the history of the Galactic center research and its discovery in radio wavelengths. The main body of the thesis is focused on a simplified model of the bow-shock structures that are generated by stars moving supersonically with respect to the ambient medium. The study is devoted to the question how these structures vary along the orbit. To this end, four different models are considered. The profiles for the tangential velocity in the shell and the surface mass density of the bow-shock shell along the stellar orbit are given here for all considered models.

J. Schee and Z. Stuchlík studied the accretion and optical effects in Bardeen and Ayón-Beáto – García (ABG) spacetimes with a regular black hole and no horizon, namely the spectral line profiles and the effect of `ghost image’ on the profiled line shape. As a reaction to the research of Ghasemi-Nodehi and Bambi, they also discussed the silhouette and spectral line profiles in special modification of the Kerr geometry generated by a quintessential field in order to demonstrate the influence of the quintessential field on specified optical phenomena—see [27], [28].

With other colleagues they also analyzed the quasi-normal perturbative field modes of static braneworld black holes in the framework of Randall-Sundrum model to search for quantitative traces of the tidal charge in braneworld black hole ringing. They gave an overview of the Keplerian accretion efficiency in the Kerr-Newman braneworld spacetimes and found a special extraordinary new class of naked singularity spacetimes demonstrating the so-called mining instability related to the (formally) unlimited accretion efficiency [29], [30].

In addition, they extended their studies to rotating wormholes, focusing on the shadow created also under the influence of a plasma environment. They introduced a new class of rotating black holes surrounded by a quintessential energy field and studied the shadow of such quintessential black holes including the plasma effects and particle collisions with related energy extraction and particle acceleration. The role of the plasma environment in the gravitational lensing by the regular black holes was also investigated—see [31]- [34].

D. Pugliese and Z. Stuchlík continued the investigation of the so-called ringed accretion discs, i.e., complex multi-ring structures that could represent a remnant of several accretion regimes on a supermassive black hole. They analyzed the possibility that several instability points may be formed in the orbiting matter around a supermassive Kerr black hole in the ringed accretion disc model, proving that the number of the instability points is limited and depends on the dimensionless spin of the rotating attractor [35].

A. Tursunov, Z. Stuchlík and M. Kološ studied the dynamics of charged test particles in the vicinity of a black hole immersed in an asymptotically uniform magnetic field and discussed the consequences of a model of ionization of test particles forming a neutral accretion disc. They showed that a strong acceleration of the ionized particles to ultra-relativistic velocities is preferred in the direction close to the magnetic field lines, representing an alternative to the Blandford-Znajek process of jet production. They also explored circular orbits and related quasiharmonic oscillatory motion of charged particles around a weakly magnetized rotating black hole—see [36], [37].

Z. Stuchlík and M. Kološ studied high frequency QPO’s (quasi-periodic oscillations) in accretion discs around compact objects [38], [39]. The frequencies of three quasi-periodic oscillation modes observed simultaneously in the accreting black hole GRO J1655-40 were compared with the predictions of different theoretical models. The controversial estimates of the black hole mass and the dimensionless spin in the microquasar GRO J1655-40 implied by strong gravity effects related to the timing and spectral measurements were discussed. The controversy can be cured if we abandon the assumption of the occurrence of twin HF QPOs and the simultaneously observed LF QPO at a common radius. Using realistic neutron star equations of state, they also studied ways to constrain the models of twin peak quasi-periodic oscillations observed in the LMXB 4U 1653 [40].

Astrophysical Processes and Compact Objects

Two research groups involved in the Center, one in the Astronomical Institute CAS and the other at the Institute of Physics of the SU have continued a collaboration project to investigate the combined effects of self-gravity and electrical charge in fluid accretion discs. In their paper published in a recent issue of the leading Astrophysical Journal Supplement Series [41] the authors present an analytical approach to searching for equilibria of a self-gravitating charged fluid embedded in a spherical gravitational and dipolar magnetic fields of a central mass. A new scheme was proposed as in the context of gaseous/dusty tori surrounding supermassive black holes in galactic nuclei. While the central black hole dominates the gravitational field and remains electrically neutral, the surrounding material has a non-negligible self-gravitating effect on the torus structure. By charging mechanisms it also acquires a non-zero electric charge density. Using this approach the authors discuss the impact of self-gravity on the conditions for the existence of the equilibrium and the morphology and typology of the tori. By comparison with a previous work (without self-gravity, see [42]) the authors demonstrate that the conditions can be very different. Although the main aim of the new paper is to discuss a framework for the classification of electrically charged, magnetized, self-gravitating tori, the authors also mention potential astrophysical applications to vertically stratified fluid configurations. To put this work into an astrophysical context, the authors identify the central object with an idealization of a non-rotating magnetic neutron star. The astrophysical applications will be also the subject of future work. We continued our investigation of dielectric charged toroidal structures of polytropic fluid. We focused on the structures centered in the equatorial plane, and also on those `levitating’ above equatorial plane of a compact object endowed with a dipole-type magnetic field. The main part of the work was represented by the topological study, showing up various classes of the tori (some of them exhibiting even three cusps, forming this way the so-called coupled rings). To contextualize the study we went through a physical discussion as well, fitting the pressure, density, charge and temperature profiles of the constructed tori so that they correspond to astrophysical limits. This work has been complemented by a Newtonian study of the role of self-gravity on these charged fluid toroidal structures.

The effect of the repulsive cosmological constant on the possible relativistic polytropes was discussed and it was demonstrated that the general relativistic polytropic configurations cannot exceed the static radius related to their external spacetime [43].

The Opava group continued the studies on kinetic theory and its relation to both general relativity and quantum theory. The Hamilton structure of Schrodinger dynamic system and generalized Lagrangian path representation of quantum mechanics were investigated. A new approach to general relativity was studied based on non-local point transformations. Kinetic theory of non-interacting particles has been studied for the Kerr spacetime in relation to the Carter constant of motion reflecting the hidden symmetry of the spacetime—cf. [44-47].

Physics of compact objects in binary and multiple stellar systems was investigated by Hadrava and Čechura. In particular, the X-ray and optical behavior of high-mass X-ray binaries was studied by means of numerical modelling in [48] and in combination with observations of the black-hole candidate Cyg X-1 in [49] MNRAS 450, 2410. General aspects of interpretation of the spectroscopic observations were studied in [50] and in a chapter in the monograph "Astronomy at high angular resolution" [51].

Gravitational lensing

D. Heyrovský and his student L. Ledvina continued to concentrate on gravitational lensing of quasar X-ray regions. Due to the small size of the emitting region, the crossing of a microlensing caustic provides an opportunity to resolve the structure of the innermost accretion disk. They explored the effect on the iron K-alpha line, the most prominent spectral feature originating in this region. Simulations started by using a fully relativistic model of a thin accretion disk in the equatorial plane of a maximally rotating (extreme) Kerr black hole. Local emission from any point on the disk consists of a narrow K-alpha line superimposed on a power-law continuum. Taking into account photon deflection, gravitational and Doppler shifts, emission maps for an asymptotic observer were generated and used as a background source which is differentially amplified by a linear microlensing fold caustic crossing the disk. The results showed profound variations in the iron K-alpha line profile during the microlensing caustic crossing. In addition to overall time-dependent flux amplification, new peaks and step-like edges appeared, moving across the profile, and disappearing as the crossing progressed. These features were closely associated with the pattern of the total energy-shift (g-factor) contours at the position of the caustic. The authors constructed a simple analytical model of a small circular part of the disk with a Taylor-expanded g-factor map and demonstrated that the peaks and edges appear at energies corresponding to g-factor contours that are tangent to the caustic at its current position. Analytical shapes were constructed and used to fit the features in numerically computed microlensed spectra. The results show that observed line profiles are directly linked to the emission-map details at the position of the caustic. A sequence of observed profiles thus can be used to resolve the emission map of the innermost accretion disk.

Concentrating on the microlensing-generated peaks, the results imply that for a given point of the disk a peak is generated at the energy corresponding to the local g-factor contour if the caustic passes through the point oriented so that the contour is internally tangent. Peak-strength maps were computed for different inclinations of the disk—the obtained pattern of weak / strong regions becomes more pronounced with increasing inclination. In order to perform these demanding computations, a parallelized code running on CUDA-supporting graphics cards (GPUs) was developed.

The study was extended to disks around non-maximally rotating black holes in which the diversity of microlensing-generated spectral features is higher since the horizon does not coincide with the innermost stable circular orbit. The authors presented their results at five conferences. For more details, see [52], [53], [54].

Star formation

In addition to the basic work [55] and the results we reported in 2015, in particular on the algorithm (developed by R. Wunsch) for identification of filaments in 3D datacubes from hydrodynamic simulations of star forming clouds, we now briefly report on the successfully defended PhD thesis by F. Dinnbier (supervised by R. Wunsch) on “Propagating star formation”. Massive stars are powerful energetic sources shaping their surrounding interstellar medium (ISM), which is often swept up into a cold dense shell. If the shell fragments and forms a new generation of massive stars, the stars may form new shells, and this sequence repeats recursively leading to propagating star formation. Using three dimensional hydrodynamic simulations, the fragmentation of the shell was studied in order to estimate masses of stars formed in the shell. The main results are as follows: by comparing numerical calculations with several analytical theories, one can constrain the parameter space. A new qualitatively different mode of fragmentation was discovered: the coalescence driven collapse. Thirdly, the conjecture that layers tend to self-organize and form regular patterns was studied and no evidence for this behavior found. An analytic estimate for fragment properties was given, which in contrast to previous works, suggests that fragment masses are typically too low to form massive stars needed for self-propagating star formation.


A number of researchers participating in the Albert Einstein Center have been traditionally involved in activities aiming to bring the results to a wider public, both within the physics community and to the general public. As mentioned in the Introduction of this report, the year 2016 was special. In February 2016 exactly 100 years elapsed from the death of Ernst Mach who influenced strongly Einstein in his development of ideas leading to the final formulation of general relativity. J. Bičák wrote an article on Mach and Machian effects in general relativity [56] based on his invited lecture at a conference in Brno.

It was also in February 2016 that the LIGO collaboration announced the first direct observation of gravitational waves. T. Ledvinka and J. Bičák described these results in [57]. Both papers mentioned above acknowledge support from the Albert Einstein Center. The Center is mentioned also in the paper by J. Podolský [58]. We organized special lectures on the discovery of gravitational waves: for students and staff at the Faculty of Mathematics and Physics and for the general public in the central building of the Academy of Sciences through the courtesy of the Learned Society of Czech Republic. The Albert Einstein Center was mentioned on posters inviting to these lectures.


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