## Overview of our research activities in 2014

- Introduction
- Mathematical Relativity
- Black Holes: Physical and Mathematical Aspects
- Black Holes: Astrophysical Aspects
- Particles around Naked Singularities and Generalized Compact Objects
- Gravitational lensing
- Star Formation and other Astrophysical Processes
- References

### Introduction

We start this report by briefly describing the work which had been initiated more than two years ago and was only finished effectively in the summer 2014. As we mentioned in our proposal of the AE Center, a number of people involved in the Center participated as both scientists and organizers of the international conference "Relativity and Gravitation — 100 years after Einstein in Prague" held in June 2012 in Prague's Carolinum. The conference involved more than 200 scientists from 31 countries. A number of them are leading world experts in their fields. Texts written by the majority of plenary speakers as well as by the authors of contributed talks and posters have been published in two volumes by Springer Verlag in summer 2014 (see refs. [1] and [2]). They were still under preparation last year and since during this period our work was supported by the Albert Einstein Center already, a number of contributions as well as both fairly detailed prefaces by the editors, J. Bičák and T. Ledvinka, conclude with acknowledgments of the AE Center (see refs. [3-10]). These two volumes have already been reviewed very favorably in the “Observatory magazine” Vol. 134, December 2014. Reviews in other leading journals are expected to appear during the first months of 2015.

A number of members of the Center participated actively also in the latest “Ragtime” conference organized by the Opava group in Prague in autumn 2014, during which there was a general meeting of all members of the AE Center. There were numerous interactions between the members on the occasion of “local” seminars organized by all four groups involved in the Center.

### Mathematical Relativity

Within the GERG conference in Warsaw in summer 2013 — the main conference on general relativity and gravitation organized every 3 years — J. Bičák, together with Prof. J. Tafel from the Warsaw University, organized a 4-day workshop on exact solutions of Einstein's field equations and their interpretation; several relativists involved in the Center also participated. An overview of the workshop, written in 2014, is contained in [11].

Proceeding to specific topics, let us start with our work on mathematical relativity at the Mathematical Institute of the Czech Academy of Sciences. The fall-off behavior of test electromagnetic fields in higher dimensions was studied. Various possible boundary conditions result in different characteristic fall-offs. The peeling-off of radiative fields differs from the standard four-dimensional case [12]. In another work, general properties of an extension of the Kerr-Schild ansatz were analyzed. Conditions for geodeticity of the null congruence of the Kerr-Schild vector field were obtained together with the relation between the admissible Weyl types and the geometrical properties of the spacetime [13]. In another work in which efforts of researchers from two different groups associated with the Center were involved, a family of new exact solutions to the Einstein equations coupled to p-form fields in arbitrary dimensions were constructed, in particular, new solutions describing electromagnetic radiation in n=2p even dimensions [14].

In collaboration with relativists from University of Vienna we were interested in the motion of test particles in nonexpanding impulsive gravitational waves. Geodesics in impulsive pp-waves in the distributional form have been analyzed rigorously using a rather heavy machinery of nonlinear generalized functions. However, there exists an alternative approach based on a Lipschitz continuous form of the pp-wave and other nonexpanding metrics. By using an enhanced solution concept to ordinary differential equations due to Filippov we proved the existence, uniqueness and C1-regularity of geodesics in the whole class of nonexpanding impulsive gravitational waves propagating on a cosmological background of (anti-)de Sitter universe. This result justifies the key ingredient of “C1-matching procedure” based on the Penrose junction condition for constructing geodesics in these spacetimes [15].

In collaboration with A. Zelnikov (U of Alberta, Edmonton, Canada), we have studied geometry of minimal surfaces with fixed spherical boundaries at infinity in anti-de Sitter (AdS) spacetimes [16]. Their area is related through the AdS/CFT correspondence to entanglement entropy of the conformal field associated with spherical domains at AdS infinity. We have shown that there are two different minimal surfaces for two sufficiently close spherical domains and no non-trivial surface for domains far away, which suggests a phase change behavior. Our results have been used, for example, in [arXiv:1411.1787] for computation of the quark-antiquark potential. We have given also a dynamical interpretation of the minimal surface spanned on domains following boost Killing vectors in AdS spacetime.

P. Krtouš and his student I. Kolář studied properties of the Killing tensors of rank two related to hidden symmetries. They found general form of the electromagnetic field compatible with the hidden symmetries and algebraic conditions on the commutativity of the field operators built from the Killing tensors (see [17]). A publication is under preparation.

Within our “traditional topic” of boost-rotation symmetric radiative spacetimes representing uniformly accelerated objects, D. Kofroň investigated the Meissner effect of expulsion of magnetic fields from the vicinity of charged, rotating, and accelerating extreme black holes immersed in a strong magnetic field with a symmetry compatible with these spacetimes. The results are visualized through an appropriate transformation from the standard coordinates used to represent the metric describing uniformly accelerated, charged, and rotating black holes (a “generalized” C-metric). This coordinate transformation can be written down explicitly. The Meissner effect itself is studied in the weak-field limit as well as in the strong-field regime. D. Kofroň talked about these results at the “Ragtime” conference and soon they will be summarized in a publication.

In the field of Mathematical Cosmology, we developed a new approach to averaging based on the Cartan scalars. We applied this theory to two different Lemaitre-Tolman-Bondi (LTB) models. In the first one, the correlation term behaves as a positive cosmological constant, in the second example, the leading correlation term behaves as spatial curvature [18]. The backreaction on expansion and shear scalars was investigated in an approximate LTB model using an averaging method developed for LRS class II dust models [19]. The behavior of the density contrast in the quasi-spherical Szekeres spacetime was studied and its analytical behavior was exhibited. An initial-data formulation for inhomogeneity modelling was presented and conditions for avoiding the shell crossing singularity were derived. In the special case of a trivial curvature function, these conditions are preserved by evolution [20]. This work was done in collaboration with two PhD students who defended their theses in 2014.

In collaboration with T. Tahamtan, O. Svítek analyzed the persistence of curvature singularities using quantum theory. Quantum test particles obeying the Klein-Gordon and Chandrasekhar-Dirac equation were used to probe the classical timelike naked singularity and the singularity was shown to persist effectively. Loop quantization was then used to resolve the singularity hidden beneath the horizon [21].

### Black Holes: Physical and Mathematical Aspects

Following our original plan we studied exact models of black holes immersed in strong magnetic fields (J. Bičák and his PhD student F. Hejda). In particular, we concentrated on extremal black holes and found simpler spacetimes and their geometries near degenerate horizons. By discovering effective parameters describing near-horizon geometries of black holes in strong magnetic fields, we were able to demonstrate the Meissner effect of magnetic field expulsion from extremal black holes. The first results are published in [22], an extensive work is near completion and will be submitted soon. In another work associated with these spacetimes, we analyze the motion of charged particles and study the effects of possible acceleration close to the holes’ horizons.

J. Bičák, in collaboration with A. Anabalon and J. Saavedra (from Valparaiso, Chile), studied spherically symmetric, planar, and hyperbolic hairy black holes in asymptotically flat or asymptotically (anti-) de Sitter spacetimes. There is a scalar field (”hair”) extending beyond the holes’ horizon. They have shown that these objects are stable under all perturbations with odd parity. In particular, it was demonstrated that there is no obstruction to the existence of rotating hairy black holes (in contrast to recent claims in the literature) and, interestingly, these black holes can behave in a way that strongly departs from the standard rotating (Kerr) black hole in vacuum [23].

We have been also studying spacetimes of black holes deformed by the presence of an additional source such as another black hole, a thin disc or a ring. Following the original plan, O. Semerák together with his PhD student M. Basovník computed and plotted invariant quantities representing the gravitational potential, field, and curvature for the binary Majumdar-Papapetrou metric and for a black hole surrounded by the Bach-Weyl ring. They are preparing two papers for publication. Preliminary results were presented by M. Basovník at the Week of Doctoral Students 2014 organized at the Faculty of Mathematics and Physics, Charles University in Prague.

O. Semerák's student J. Pejcha numerically checked and compared several different formulas, which had been suggested in the literature for the magnetic field of a current loop placed symmetrically around a Schwarzschild black hole. The formulas were confirmed to yield almost identical results, although certain differences of course occur due to the necessary truncation of the series in terms of which they are expressed. After J. Pejcha succesfully defended his bachelor thesis, the topic will now be further studied by another student F. Vlk.

### Black Holes: Astrophysical Aspects

One of main topics of the teams at the Astronomical Institute of ASCR and at Institute of Physics of the Silesian University in Opava was modelling the relevant physical processes by taking into account the electro-magnetic effects near black holes. They studied the recently developed model of a dielectric, charged, perfect fluid toroidal structure (disk) where the disk encircles a non-rotating charged black hole immersed in a large-scale, asymptotically uniform magnetic field [24]. The existence of orbiting structures in permanent rigid rotation in the equatorial plane and of charged clouds hovering near the symmetry axis was demonstrated. We constrained the range of parameters that allow stable configurations and derived the geometrical shape of equipressure surfaces. This analytical study suggests that the regions of stability may be relevant for trapping electrically charged particles and dust grains in some areas of the black hole magnetosphere, being thus important in some astrophysical situations. Regarding magnetic fields around astrophysical black holes, we also studied the role of frame dragging near a rotating black hole in oblique magnetic fields. While test-particle motion is strictly regular in the classical black-hole spacetime — both with and without the effects of rotation and/or electric charge — gravitational perturbations and imposed external electromagnetic fields may lead to chaos [25].

Famously, the first black-hole candidate identified was the source Cygnus X-1. J. Čechura and P. Hadrava developed a novel interpretation method for observations of high-mass X-ray binaries (HMXBs) based on a combination of spectroscopic data and numerical results from a radiation hydrodynamic model of stellar wind in HMXBs. Using an indirect imaging method of Doppler tomography, they calculated synthetic tomograms of the predicted emission in Low/Hard and High/Soft X-ray states and compared them with tomograms produced using phase-resolved optical spectra of Cygnus X-1, the prototype of HMXBs [26,27].

In another line of research on astrophysical black holes, a novel formalism was developed to investigate the role of the spin angular momentum of astrophysical black holes in the behavior of low-angular momentum general relativistic accretion. A metric-independent analysis of axisymmetric general relativistic flow was proposed, and, consequently, the space and time dependent equations describing the general relativistic hydrodynamic accretion flow in the Kerr metric were formulated [28].

We should also mention the results of our efforts to model and assess the capabilities of the Large Observatory for X-ray Timing (LOFT) in the context of relativistic astrophysics and, in particular, of physics around supermassive black holes (a “white paper” is under preparation). The exceptional ability of LOFT to obtain the spectrum of an active galaxy nucleus in a short integration time will allow direct studies of spectral energy evolution during strong individual flares. The project enables Czech astronomers and their students to intensify the current collaboration on the analysis and interpretation of data from ground-based and space observatories. The co-PI of the LOFT proposal is a researcher at the Astronomical Institute in Prague and all three groups in the Center benefit from this effort.

### Particles around Naked Singularities and Generalized Compact Objects

Analyzing test-particle motion, we studied the properties of Kehagias-Sfetsos spacetime, originating from the Hořava gravity theory. We considered circular geodesics and collisions of test particles, focusing on the naked singularity of the spacetime [29]. We established the existence of an antigravity sphere where the particle can be at rest. There are two regions where separated marginally stable orbits exist. We found orbits where the angular frequency is maximal [30].

We also started investigating the properties of regular black holes or "no-horizon" spacetimes that were found in the case of spherically symmetric spacetimes by Bardeen or by Ayon-Beato and Garcia. These regular spacetimes have no physical singularity inside, due either to a modification of Einstein equations or to nonlinear electrodynamics coupled to the Einstein gravity. We derived one possible form of a rotating regular spacetime based on the Ayon-Beato and Garcia solution [31].

We studied ultra-high-energy collisions in Kerr naked singularity spacetimes (“superspinars”) and concluded that in the case of elastic collisions, the efficiency is largest for near-extreme superspinars. We used our previous results on circular geodesics to study optical phenomena in these spacetimes. We constructed a frequency shift map of Keplerian discs and the corresponding profiles of spectral lines, which enable us to distinguish between standard black-hole and naked singularity spacetimes. There is an important special kind of electromagnetic radiation related to the strong gravity of black holes, neutron stars and exotic objects such as superspinars, namely, the high frequency quasiperiodic oscillations (HF QPOs) of X-ray radiation coming from microquasars and binary systems containing neutron stars. We have developed a new model for HF QPOs in such systems, based on the so-called resonant switch of twin oscillations. We have tested this model on 4U 1636-53, a source where the observed HF QPOs data give a clear signal of a possible resonant switch and we demonstrated that our model significantly improves the data fit [32]. We have shown that the observed data for three well-known microquasars (GRS 1915+105, XTE 1550-564, GRO 1655-40 exhibiting twin HF QPOs with a frequency ratio 3:2) could be explained by a simple resonance model based on radial and vertical epicyclic oscillations, if we assume a near-extreme naked-singularity Kerr field [33].

### Gravitational lensing

In our study of quasar X-ray microlensing, we concentrated on the effect a fold caustic has on the observed profile of the Fe K-alpha line as the caustic crosses the quasar accretion disk. As our main result of this work in progress we demonstrated the connection between features of the line profile and the caustic position with respect to the emission map of the disk. We used a fully relativistic model of a thin accretion disk in the equatorial plane of a Kerr black hole. Local emission from a point on the disk is described by a narrow K-alpha iron line superimposed on a power-law continuum. Photon deflection in the Kerr spacetime, as well as gravitational and Doppler shifts were taken into account. The combined effect generates a broad K-alpha line in disk-integrated spectra. The constructed emission map serves as the background source for microlensing by the stellar population of a foreground galaxy. Microlensing occurs as the caustic network generated by the gravitational field of the stars moves transversally with respect to the quasar. For a given emission map and a given caustic position, we compute the specific flux by convolving the emission map with the caustic amplification pattern. We generated sequences of spectra for caustics crossing the inner disk in different directions and studied the associated spectral changes.

Concentrating on the most prominent microlensing-generated spectral features, we found that for certain caustic positions the spectrum exhibited peaks, for others it had step-like edges at specific energies. We developed a simple model, which demonstrated the required positions of the caustic and the respective energy contours. It also yielded analytical shapes of the corresponding spectral features. L. Ledvina included results of the described research in his Master's thesis [34], which he defended successfully. He continues his work on the project as a PhD student now. The described results will be submitted for publication in early 2015.

### Star Formation and other Astrophysical Processes

Concerning the subject of Giant Star-forming Regions and Massive Star Clusters, we considered the strong evolution experienced by matter reinserted by massive stars both in giant star-forming regions driven by a constant star formation rate and in massive and coeval superstar clusters. In both cases we took into account the changes in the number of massive stars due to stellar evolution and their effect on the number of ionizing photons, and the integrated mechanical luminosity of the star-forming regions. The latter is at all times compared with the critical luminosity that defines the lower mechanical luminosity limit above which the matter reinserted via strong winds and supernova explosions suffers frequent and recurrent thermal instabilities that reduce its temperature and pressure and inhibit its exit as part of a global wind. As the evolution proceeds, more unstable matter accumulates and the unstable clumps grow in size. We present the results of several calculations of this positive star formation feedback scenario promoted by strong radiative cooling and mass loading [35].

In another line of research, M. Horký has worked on a numerical study of the stability of electromagnetic fields in weakly collisional plasmas. One paper was recently submitted for publication and another is expected to be finalized in 2015, during which the modelling of electromagnetic and radiation processes will be one of the main topics of our work.

Finally in the area of the “Large-scale magnetic fields in astrophysics”, we studied charged toroidal structures of magnetized plasmas in the framework of kinetic theory. We developed a model of kinetic equilibria of collisionless plasmas in the presence of non-stationary electromagnetic fields and derived a covariant formulation of spatially non-symmetric kinetic equilibria for magnetized astrophysical plasmas in strong gravity. The study of possible transition of gas to plasma kinetic equilibria in the presence of gravitation in axisymmetric systems is astrophysically relevant [36-39].

Since thin magnetized toroidal structures in the fields of black holes or neutron stars can be modeled by infinitesimally thin, axisymmetric, current-carrying relativistic string loops, we also studied the acceleration of an electric string loop of a toroidal form in the vicinity of a Schwarzschild black hole embedded in an external asymptotically uniform magnetic field as a possible astrophysical model of a relativistic jet. We demonstrated that the interaction between the electric current of the string loop and the external magnetic field can lead to efficient ultra-relativistic acceleration. The black hole combined with the strong external magnetic field can accelerate the toroidal string loop structure up to the velocities close to the speed of light. For black holes with an external magnetic field, the motion of charged test particles is generally of chaotic character but in the vicinity of the local minima of the effective potential it corresponds to a linear harmonic or quasi-harmonic oscillation with frequencies close to those of the harmonic motion. An analogous situation occurs for the motion of axisymmetric string loops in the field of black holes. The quasi-harmonic oscillations of the string loop toroidal structure can be relevant for the explanation of the HF QPOs observed in the three microquasars GRS 1915+105, XTE 1550-564, and GRO 1655-40 [40,41].

### 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)
- Bičák, J., Einstein in Prague: Relativity then and now, in [1] pp. 33-63
- Bičák, J., Katz, J., Ledvinka, T., & Lynden-Bell, D., On the effects of rotating gravitational waves, in [2] pp. 255-260
- Bičák, J., & Kofroň, D., Variations on spacetimes with boost-rotation symmetry, in [2] pp. 261-265
- Haláček, J., & Ledvinka, T., Analytical conformal compactification of Schwarzschild spacetime, in [2] pp. 279-282
- Kašpar, P., Vrba, D., & Svítek, O., Averaging inside the LRS family, in [2] pp. 431-434
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