## Overview of our research activities in 2015

- Introduction
- Mathematical Relativity
- Black Holes: Physical and Mathematical Aspects
- Black Holes: Astrophysical Aspects
- Astrophysical Processes and Compact Objects
- Gravitational lensing
- Star Formation
- References

### Introduction

In the Introduction of the report last year we briefly wrote about two volumes published by Springer Verlag in summer 2014 (see refs. [1], [2]) which are based on the proceedings of conference “Relativity and Gravitation – 100 years after Einstein in Prague” we organized in Prague’s Carolinum. We mentioned that the reviews of the volumes are expected to appear during 2015. These very favourable reviews, written by leading scientists indeed appeared in [3], [4]; for a nice domestic review, see [5].

There were continuing numerous interactions between the members of the Center on the occasion of local seminars organized by all four groups involved in the Center but also abroad like in the 14th Marcel Grossmann Meeting in Rome in July.

As in our report last year we divide it into several subsections.

### Mathematical Relativity

As included in the plan for 2015, J. Bičák and his PhD student J. Schmidt extended and finished the work on the uniqueness of the energy-momentum tensors in linearized Einstein’s theory and in massive gravity. In the procedure the tensors are not derived from a variational principle, only field equations are assumed as a consequence of which the tensors are conserved. The work [6] will appear in these days/appeared very recently. It contains signicantly more new results than known a year ago. In the case of the linear gravity we found a 4-parametric system of conserved second-rank tensors which contains a unique symmetric tensor. This turns out to be the linearized Landau-Lifshitz pseudotensor. Still more interesting is the case of massive gravity where we show how one can arrive not only at the “generalized Landau-Lifshitz” tensor but also to a new simpler symmetric tensor not involving higher derivatives which appear in the expressions derived by variational methods in case of higher-spin fields. For the first time in our group the software CADABRA used in the field theory for tensor manipulations was employed in this work very efficiently.

In our “traditional topic” of boost roration symmetric radiative spacetimes representing uniformly accelerated objects, in particular in case of the C-metric representing accelerated black holes, a remarkable progress has been achieved by D. Kofroň in [7]. The separability of the equations for massless test fields of arbitrary spin at the C-metric background was demonstrated, the equation analogous to the well-known Teukolsky master equation for Kerr-Newmann metric was derived for the non-rotating C-metric. In another work, Kofroň found an explicit coordinate transformation between algebraically adapted and boost-rotation symmetric coordinates for the charged rotating C-metric (submitted to Classical and Quantum Gravity).

Another field of research pursued traditionally in our group are Robinson–Trautman radiative spacetimes. In the last year, T. Tahamtan and O. Svítek have derived and analyzed an explicit Robinson–Trautman solution with minimally coupled free scalar field. It was shown that the solution contains curvature singularity which is initially naked but later surrounded by horizon. The existence the quasilocal horizon in later retarded times was proved using sub- and supersolution method combined with growth estimates. The solution is in general of algebraic type II. Interestingly, the asymptotic behavior analysis developed by Chrusciel for vacuum solution still applies in case with scalar field. These results have been interpreted by using no-hair theorem and cosmic censorship hypothesis [8].

Tahamtan and Svitek also analyzed Robinson Trautman solutions with electromagnetic field satisfying the field equations of various types of nonlinear electrodynamis. In all cases considered the electromagnetic field singularity is removed while the gravitational one persists. Unfortunately, it was found that the nonlinear electrodynamics models resolving curvature singularity in spherical symmetry could not be generalized to Robinson Trautman geometry. This research has been submitted to PRD.

The collaboration of J. Podolský and R. Švarc with mathematicians from the Vienna University resulted in [9], where the existence, uniqueness and C1-regularity of geodesics in nonexpanding impulsive gravitational waves with cosmological constant were proved using the Filippov solution concept to ODEs. As a natural extension of these results geodesics in Robinson-Trautman type expanding impulses were also studied. The analogous theorems were found which may serve as a rigorous basis for the description of geodesic refraction induced by the impulse generated by a pair of snapped cosmic strings [10].

M. Ortaggio, together with J. Podolský and M. Žofka, studied a large class of exact higher dimensional Robinson-Trautman spacetimes coupled to p-form Maxwell fields. These may represent static black holes dressed with an electric and a magnetic field and, for particular values of p, non-static solutions gaining (or losing) mass by receiving (or emitting) electromagnetic radiation. [11].

P Krtouš and his PhD student I. Kolář studied the Kerr–NUT–(A)dS and related spacetimes. In these spacetimes the tower of Killing–Yano forms and Killing tensors exist which encode the hidden symmetries. The Killing tensors allow to define conserved particle observables and commuting wave operators. In [12] we have found the most general form of weak (test) electromagnetic field compatible with the complete integrability of a charged particle motion and with the commutativity and separabilty of the charged wave operators. If one of the Killing tensors is interpreted as a metric of an alternative geometry, the particle motion remains integrable, however, the wave operators cease to commute.

In [13], Krtouš and Kolář constructed the class of metrics in higher spacetime dimensions which enable the separability of the Klein–Gordon equation. This class includes the off-shell Kerr-NUT–(A)dS metrics. This represents the generalization of the classical Carter results in four dimensions. In addition, the class contains also new Einstein–Kähler metrics of Euclidian signature. Thanks to a larger freedom in higher dimensions these new Euclidian metrics can be combined through the warped product with the Kerr–NUT–(A)dS metric to form a new solution of the Einstein equation in higher dimensions.

A. Pravdová with M. Kuchynka studied general geometric properties of Weyl type N and III and Ricci type N spacetimes in arbitary dimension. It has been shown that in this case the geometry provides an "interaction" between the Weyl and the Ricci tensors - these tensors have to be aligned, irrespective of the gravitational theory used and the aligned null vector is always geodetic. For the type N, a generalization of the Goldberg-Sachs theorem, which so far has been studied only for the vacuum case in higher dimensions, has been also provided. Their results were submitted for publication in October 2015 –see [14].

T. Málek in collaboration with S. Hervik (University of Stavanger, Norway) studied universal spacetimes in neutral signature. Universal spacetimes solve, by definition, all theories with the Lagrangian described by any polynomial curvature invariant. In the case of Lorentzian signature, the only presently known examples of universal spacetimes belong to the Kundt class. In this paper, the authors have found a class of universal spacetimes of neutral signature which are not Kundt spacetimes. The paper is expected to be submitted for a publication in the 1st quarter of 2016.

Two following contributions [15], [16] corresponding to the talks given at the 14th Marcel Grossmann meeting in Rome will appear probably in February 2016.

Our work involved also mathematical cosmology. In particular, O. Svítek, together with his PhD student P. Kašpar, generalized Buchert's averaged equations to the LRS (locally rotational symmetric) class II dust models in the sense that all Einstein equations are averaged, not only their trace part. The backreaction on expansion and shear scalars was investigated in an approximate Lemaitr-Tolman-Bondi model [17].

### Black Holes: Physical and Mathematical Aspects

J. Bičák and his PhD student F. Hejda extended substantially their analysis from 2014 of stationary spacetimes representing rotating, charged black holes in strong axisymmetric magnetic fields. They investigated the properties of extremal black holes by constructing simpler spacetimes that exhibit the geometries near their degenerate horizons. Starting from the newly discovered symmetry arguments they found that the near-horizon geometries of extremal magnetized Kerr-Newman black holes can be characterized by just one dimensionless parameter: "effective Kerr-Newman mixing angle". Their analysis has been related also to very recent results by other authors. The Meissner effect of magnetic field expulsion from extremal black holes was so demonstrated in a lucid way. The results are described in a rather extensive work [18].

M. Scholtz collaborating with our former PhD student N. Gürlebec (ZARM, University of Bremen, Germany) arrived at new results on the Meissner effect for black holes, namely, the existence of the Meissner effect for arbitrary stationary, axially symmetric black holes. Their results will be soon submitted for publication.

O. Semerák together with his PhD student M. Basovník computed and plotted simple invariant quantities representing gravitational potential, acceleration and curvature, for a symmetric binary of extremally charged black holes, described by the Majumdar-Papapetrou metric, and for the (originally Schwarzschild) black hole surrounded and deformed by the Bach-Weyl ring, in both cases mainly focusing on the region inside black-hole horizons. The results have been submitted for publication (in October already) [19,20].

M. Žofka with his student Ryzner investigated the (electro-)geodesic structure of the Majumdar-Papapetrou metrics with two static charged black holes in equilibrium, imparting thus the spacetime axial symmetry. They studied electrogeodesics both in- and off the equatorial plane and explored the stability of circular trajectories via geodesic deviation equation. In contrast to the classical Newtonian situation, there are regions of spacetime admitting two different angular frequencies for a given radius of the circular electrogeodesic [22].

J. Bičák together with A. Anabalon (University of Valparaiso, Chile) during his visit in Prague in June 2015 extended their analysis of perturbations of hairy black holes [22] to include spherical perturbations with in general anti-de Sitter asymptotics and summarized their results in the subsequent talks at the 14th Marcel Grossmann meeting which will be published in [23]. The problem of spherical perturbations has been analyzed later in depth by Anabalon and his collaborators in [24] where the support from AE Center grant is also acknowleged.

In the Silesian University group Z. Stuchlík and collaborators studied alternate models, namely the Bardeen and Ayón-Beáto–García (ABG) spacetimes of both the black hole and no-horizon type that are solutions of the combined Einstein equations and equations of non-linear electrodynamics. The structure of circular geodesics and photon trajectories were investigated.. The results were used to construct Keplerian disc images in the vicinity of the compact objects. The presence of an additional image (a “ghost image”) was uncovered. Its origin is directly linked with the regularity of the Bardeen and ABG no horizon spacetimes and with the de Sitter character at their central region. Ghost image imprints its presence in the distribution of frequency shift in Keplerian disc image, the width of spectral line, and in its profile. The group also continued the research of the Kehagias-Sfetsos spacetime, which is a solution of Hořava-Lifshitz gravity, in particular the properties of the toroidal perfect fluid configurations and collisional processes. Rotating regular black hole spacetimes were constructed and their properties studied: in particular the collisional and Penrose processes leading to ultrahigh energy, or quasinormal modes of the scalar and gravitational test field. Similar effects were studied in the field of Gauss-Bonnet 6D black holes. The results obtained are described in a number of publications which appeared in 2015 – see [25-32].

### Black Holes: Astrophysical Aspects

L. Šubr together with J. Haas continued to work on N-body modelling of eccentric stellar discs around supermassive black holes. In particular, they found that a non- negligible subset of the individual orbits in the disc undergo various modes of the Kozai-Lidov oscillations, showing a remarkable robustness of this classical mechanism. Some orbits reach so very high eccentricities which bring them very close to the central SMBH where they get tidally disrupted. The global angular momentum transfer throughout the disc was also studied. This process is capable to modify eccentricity distribution of the stellar orbits, pushing some of them to very high values, helping them to reach the Kozai-Lidov resonance. The results are summerized in [33].

In the Astronomical Institute of the Czech Academy of Sciences, two Ph.D. Dissertation Theses (by J. Hamerský and M. Horký) were successfully completed and defended in connection with this project during 2015. Properties of accretion tori around black holes were explored. They are believed to account for variety of features of mass inflow and release of radiation on diverse scales near stellar-mass black holes as well as supermassive black holes. As the stationary torus is perturbed, it starts to oscillate and once some part of the torus overflows the closed equipotential surface, this material becomes accreted or ejected. Oscillations reveal the spacetime properties as well as the intrinsic characteristics of the torus model, and these were studied using general relativistic magnetohydrodynamic simulations and assuming axial symmetry and 2-D approximation. The impact of the presence of large scale magnetic fields was also discussed.

J. Čechura et al. [34] have developed a new code for the three-dimensional time-dependent raditation hydrodynamic simulation of the stellar wind in interacting binaries to improve models of accretion in high-mass X-ray binaries involving often stellar mass black holes. They used the code to test the influence of various parameters on the structure and properties of circumstellar matter. The code takes into account a number of phenomen (acceleration of the wind, Coriolis force, gas pressure, radiative pressure etc). The parameters of the Cygnus X-1 (the first system involving a black-hole candidate discovered) were used to test the properties of the system as a proto-typical example of High-mass X-ray binaries (HMXBs) and to calculate synthetic Doppler tomograms of predicted emission in low/hard and high/soft X-ray states. P. Hadrava et al continued this research by additional spectroscopic observations of the Cyg X-1 and by interpreting the spectra of this object using their method based on the comparison of observed and simulated Doppler tomograms (see [35]). Spectroscopic peculiarities in a specific binary system were also investigated in [36].

In Opava group M. Kološ, Z. Stuchlík and A. Tursunov continued in the research of string loop oscillations and in the investigations of the influence of an ambient magnetic field on charged particles motion around black holes. They calculated oscillation frequencies of such particles and compared them with those coming from real astrophysical sources. With an intention to find an explanantion for relativistic jets, they also studied the acceleration of escaping particles along the black hole background symmetry. The results have been summarized in [37] and [38].

### Astrophysical Processes and Compact Objects

In Opava group the work on dielectric charged toroidal structures of polytropic fluid was continued. The focus was on the structures centered in the equatorial plane, but also on those `levitating’ above and under equatorial plane of a compact object endowed with a dipole-type magnetic field. By a topological study, various classes of the tori (some of them exhibiting even three cusps) were exhibited. The pressure, density, charge and temperature profiles of the constructed tori were studied to correspond to astrophysical limits. The summarizing paper is now being sent to Physical Review D. A new research area 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 was also initiated [39].

In Opava also high frequency QPO’s (quasi-periodic oscillations) in accretion discs around compact objects were further studied--ndash;- the standard resonant models related to oscillating toroidal structures to explain twin HF QPO’s were applied in three microquasars. In parallel, investigations of the resonant switch model were continued and applied to an explanation of twin HF QPO’s in the neutron star source 4U636-53, testing various equations of state in Hartle-Thorne models of neutron stars. Strong limits on the mass, spin and quadrupole moment in the source were obteined. See [40-42].

In the Astronomical Institute CAS M. Horký has continued to explore stability of weakly collisional plasmas. In the case of collisional plasmas, the E x B terms appear and the problem becomes more complex than in the collisionless case. In the recent work, Horký et al. address this problem by 3D electrostatic Particle-In-Cell (PIC) numerical simulations to include elastic collisions and charge exchange collisions for ions, i.e., collision types with largest collision cross-sections, and elastic collisions for electrons. The results are analyzed in terms of velocity distribution functions, amplitudes of potential fluctuations, and electrostatic wave spectra to show that the type of ion-neutral collisions play a substantial role in the plasma stability. Results from numerical simulations show that there is a strong dependence of dynamic parameters on the type of ion-neutral collisions [43].

In Opava group kinetic theory of collisional processes and variational principles in general-relativistic kinetic theory were studied. such as the synchronous Lagrangian variational principles, particle dynamics in the presence of electromagnetic radiation-reaction, collisional invariants for the Master kinetic equation, axiomatic foundations of entropic theorems for hard-sphere systems and global validity of the Master kinetic equation for hard-sphere systems. The results are contained in [44-48]

Last but not least let us mention that V. Karas from the Astronomical Institute CASdelivered an opening talk at the meeting on eXTP (X-ray Timing and Polarimetry) future satellite proposal meeting in Beijing (26-29 November 2015). Participation at this event was planned within the project but we received external funding from conference organizers.

### Gravitational lensing

D. Heyrovský with his student L. Ledvina focused on quasar microlensing using their previously derived analytical model of peaks in the Fe K-alpha line profile generated by a microlensing caustic crossing the inner accretion disk. They mapped maximum possible peak strength generated by caustics passing through a given point of the plane-of-the-sky map of the accretion disk. Such a peak occurs at an energy corresponding to the total energy-shift (g-factor) contour passing through the point. These maps for a range of disk inclinations showed that the pattern of weak / strong regions becomes more pronounced and more structured with increasing inclination. The strongest peaks occur at positions at which the simplest quadratic expansion of the contours fails, i.e., at inflection points of the g-factor contours and at the g-factor maximum. The accuracy of the derived analytical shape of the peak was demonstrated by fits to microlensed line profiles. For most positions the relative error is lower than 1% within a g-factor interval larger than +- 0.05 around the peak. Initial tests of running computations were performed on CUDA-supporting graphics cards. The parallelized computations look promising as per speed, but run into problems with available memory. We plan to run further tests and implement GPU-based computations in the more massive upcoming simulations.

Lukáš Ledvina presented our results at three conferences in 2015: Week of Doctoral Students (June, Prague), CPP15 - Cologne-Prague Meeting in Prague (November, Prague), and GR 100 Years (December, Lisabon). We published our results in [49], and finalized a paper for submission to The Astrophysical Journal.

### Star Formation

In the Astronomical Institute of CAS Wunsch et al developed an algorithm for identification of filaments in 3D datacubes from hydrodynamic simulations of star forming clouds. At first, it removes large scale structures in the density field using the FFT filter; then, it runs the clump finding code DENDROFIND [50] on it; and finally, it connects the obtained clumps into filaments using several criteria based on density gradients and angles between neighbour objects. Using this algorithm, several radiation hydrodynamic simulations by Dale et al [51] were analyzed and properties of identified filaments determined. The filaments in simulations have typically much shallower radial profile, similarly to the observed filaments, than what is predicted by a model of an isothermal filament in the hydrostatic equilibrium. On the other hand, filaments in the simulations exhibit various widths in range 0.1 pc to several pc, while the Herschel observations show that interstellar filaments have always a constant width 0.1 pc [52]. The results were presented at the meeting "Structure of Filaments in molecular clouds and their relation to the star-formation processes" held in Munich, March 23-25.

### References

- Bičák J., Ledvinka T. (Eds.): General Relativity, Cosmology and Astrophysics, Fundamental Theories of Physics, Vol. 177 (Springer 2014)
- Bičák J., Ledvinka T. (Eds.): Relativity and Gravitation: 100 Years after Einstein in Prague, Springer Proceedings in Physics, Vol. 157 (Springer 2014)
- Andersson L.: „Book Reviews: 100 years after Einstein’s stay in Prague edited by Jiří Bičák and Tomáš Ledvinka“ in Classical and Quantum Gravity + (2015) (http://cqgplus.com/2015/09/09/book-reviews)
- Barbour J.: General Relativity and Gravitation 47, 74 (2015)
- Novotný J.: Československý Časopis pro fyziku 65, 200-201 (2015)
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- Podolský J., Sämann C., Steinbauer R., Švarc R.: Classical and Quantum Gravity 32, 025003 (2015)
- Podolský J., Sämann C., Steinbauer R., Švarc R.: in preparation
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