Silesian University in Opava

Abstract The WKB approach for finding quasinormal modes of black holes, suggested in Schutz and Will (1985 Astrophys. J. Lett . 291 L33–6) by Schutz and Will at the first order and later developed to higher orders (Iyer and Will 1987 Phys. Rev . D 35 3621; Konoplya 2003 Phys. Rev . D 68 024018; Matyjasek and Opala 2017 Phys. Rev . D 96 024011), became popular during the past decades, because, unlike more sophisticated numerical approaches, it is automatic for different effective potentials and mostly provides sufficient accuracy. At the same time, the seeming simplicity of the WKB approach resulted in appearance of a big number of partially misleading papers, where the WKB formula was used beyond its scope of applicability. Here we review various situations in which the WKB formula can or cannot bring us to reliable conclusions. As the WKB series converges only asymptotically, there is no mathematically strict criterium for evaluation of an error. Therefore, here we are trying to introduce a number of practical recipes instead and summarize cases in which higher WKB orders improve accuracy. We show that averaging of the Padé approximations, suggested first by Matyjasek and Opala (2017 Phys. Rev . D 96 024011), leads to much higher accuracy of the WKB approach, estimate the error and present the automatic code (The Mathematica ® package with the WKB formula of 13th order and Padé approximations ready for calculation of the quasinormal modes and grey-body factors, as well as examples of such calculations for the Schwarzschild black hole are publicly available to download from https://goo.gl/nykYGL ) which computes quasinormal modes and grey-body factors.

We consider a simple spherical model consisting of a Schwarzschild black hole of mass $M$ and a dark matter of mass $\Delta M$ around it. The general formula for the radius of black-hole shadow has been derived in this case. It is shown that the change of the shadow is not negligible, once the effective radius of the dark matter halo is of order $\sim \sqrt{3 M \Delta M}$. For this to happen, for example, for the galactic black hole, the dark matter must be concentrated near the black hole. For small deviations from the Schwarzschild limit, the dominant contribution into the size of a shadow is due to the dark matter under the photon sphere, but at larger deviations, the matter outside the photon sphere cannot be ignored.

Linear programming attracted the interest of mathematicians during and after World War II when the first computers were constructed and methods for solving large linear programming problems were sough

In Cardoso et al. [6] it was claimed that quasinormal modes which any stationary, spherically symmetric and asymptotically flat black hole emits in the eikonal regime are determined by the parameters of the circular null geodesic: the real and imaginary parts of the quasinormal mode are multiples of the frequency and instability timescale of the circular null geodesics respectively. We shall consider asymptotically flat black hole in the Einstein–Lovelock theory, find analytical expressions for gravitational quasinormal modes in the eikonal regime and analyze the null geodesics. Comparison of the both phenomena shows that the expected link between the null geodesics and quasinormal modes is violated in the Einstein–Lovelock theory. Nevertheless, the correspondence exists for a number of other cases and here we formulate its actual limits.

We show how to decoupling two spherically symmetric and static gravitational sources through the most general possible extension of the so-called Minimal Geometric Deformation-decoupling. As a test, we decouple the Einstein–Maxwell system and reproduce the Reissner–Nordstrom solution. We show the potential of this method to study i) the consequences of modified gravity on general relativity, ii) to investigate the conjectured dark matter, and iii) to study hairy black holes.

We investigate the extension of isotropic interior solutions for static self-gravitating systems to include the effects of anisotropic spherically symmetric gravitational sources by means of the gravitational decoupling realised via the minimal geometric deformation approach. In particular, the matching conditions at the surface of the star with the outer Schwarzschild space-time are studied in great detail, and we describe how to generate, from a single physically acceptable isotropic solution, new families of anisotropic solutions whose physical acceptability is also inherited from their isotropic parent.

Abstract Objective To determine whether a multicomponent intervention based on physical activity with technological support and nutritional counselling prevents mobility disability in older adults with physical frailty and sarcopenia. Design Evaluator blinded, randomised controlled trial. Setting 16 clinical sites across 11 European countries, January 2016 to 31 October 2019. Participants 1519 community dwelling men and women aged 70 years or older with physical frailty and sarcopenia, operationalised as the co-occurrence of low functional status, defined as a short physical performance battery (SPPB) score of 3 to 9, low appendicular lean mass, and ability to independently walk 400 m. 760 participants were randomised to a multicomponent intervention and 759 received education on healthy ageing (controls). Interventions The multicomponent intervention comprised moderate intensity physical activity twice weekly at a centre and up to four times weekly at home. Actimetry data were used to tailor the intervention. Participants also received personalised nutritional counselling. Control participants received education on healthy ageing once a month. Interventions and follow-up lasted for up to 36 months. Main outcome measures The primary outcome was mobility disability (inability to independently walk 400 m in <15 minutes). Persistent mobility disability (inability to walk 400 m on two consecutive occasions) and changes from baseline to 24 and 36 months in physical performance, muscle strength, and appendicular lean mass were analysed as pre-planned secondary outcomes. Primary comparisons were conducted in participants with baseline SPPB scores of 3-7 (n=1205). Those with SPPB scores of 8 or 9 (n=314) were analysed separately for exploratory purposes. Results Mean age of the 1519 participants (1088 women) was 78.9 (standard deviation 5.8) years. The average follow-up was 26.4 (SD 9.5) months. Among participants with SPPB scores of 3-7, mobility disability occurred in 283/605 (46.8%) assigned to the multicomponent intervention and 316/600 (52.7%) controls (hazard ratio 0.78, 95% confidence interval 0.67 to 0.92; P=0.005). Persistent mobility disability occurred in 127/605 (21.0%) participants assigned to the multicomponent intervention and 150/600 (25.0%) controls (0.79, 0.62 to 1.01; P=0.06). The between group difference in SPPB score was 0.8 points (95% confidence interval 0.5 to 1.1 points; P<0.001) and 1.0 point (95% confidence interval 0.5 to 1.6 points; P<0.001) in favour of the multicomponent intervention at 24 and 36 months, respectively. The decline in handgrip strength at 24 months was smaller in women assigned to the multicomponent intervention than to control (0.9 kg, 95% confidence interval 0.1 to 1.6 kg; P=0.028). Women in the multicomponent intervention arm lost 0.24 kg and 0.49 kg less appendicular lean mass than controls at 24 months (95% confidence interval 0.10 to 0.39 kg; P<0.001) and 36 months (0.26 to 0.73 kg; P<0.001), respectively. Serious adverse events occurred in 237/605 (39.2%) participants assigned to the multicomponent intervention and 216/600 (36.0%) controls (risk ratio 1.09, 95% confidence interval 0.94 to 1.26). In participants with SPPB scores of 8 or 9, mobility disability occurred in 46/155 (29.7%) in the multicomponent intervention and 38/159 (23.9%) controls (hazard ratio 1.25, 95% confidence interval 0.79 to 1.95; P=0.34). Conclusions A multicomponent intervention was associated with a reduction in the incidence of mobility disability in older adults with physical frailty and sarcopenia and SPPB scores of 3-7. Physical frailty and sarcopenia may be targeted to preserve mobility in vulnerable older people. Trial registration ClinicalTrials.gov NCT02582138 .

Properties of the Schwarzschild--de Sitter and Schwarzschild--anti-de Sitter spacetimes are characterized by three phenomena, namely, by the ``effective potential'' of the motion of test particles and photons, the photon escape cones, and the embedding diagrams of $t=\mathrm{const}$ sections of central planes of both the ordinary and optical reference geometry of these spacetimes. The phenomena are related to the corresponding phenomena of the Schwarzschild spacetime, and differences caused by the asymptotic structure of the spacetimes with a nonzero cosmological constant are discussed. The properties of the embedding diagrams of the optical geometry are related to the dynamical behavior of test particles. The limits of the embeddability of the optical geometry are given and compared with the limits on the outer radius of the interior solutions of Einstein's equations with a nonzero cosmological constant for static, spherically symmetric configurations of uniform density. It is shown that, contrary to the pure Schwarzschild case, these limits do not fully coincide for repulsive cosmological constants.

Based on the Newman-Janis algorithm, the Ay\'on-Beato-Garc\'{\i}a spacetime metric [Phys. Rev. Lett. 80, 5056 (1998)] of the regular spherically symmetric, static, and charged black hole has been converted into rotational form. It is shown that the derived solution for rotating a regular black hole is regular and the critical value of the electric charge for which two horizons merge into one sufficiently decreases in the presence of the nonvanishing rotation parameter $a$ of the black hole.

In all microquasars with double peak high frequency QPOs, the ratio of the frequencies is 3:2, which supports the suggestion that a non-linear resonance between two modes of oscillation in the accretion disk plays a role in exciting the observed modulations of the X-ray flux. We discuss evidence in favor of this interpretation and relate the black hole spin to the frequencies expected for various types of resonances that may occur in nearly Keplerian disks in strong gravity. For those microquasars where the mass of the central X-ray source is known, the black hole spin can be deduced from a comparison of the observed and expected frequencies.

We present a review of the influence of cosmic repulsion and external magnetic fields on accretion disks rotating around rotating black holes and on jets associated with these rotating configurations. We consider both geometrically thin and thick disks. We show that the vacuum energy represented by the relic cosmological constant strongly limits extension of the accretion disks that is for supermassive black holes comparable to extension of largest galaxies, and supports collimation of jets at large distances from the black hole. We further demonstrate that an external magnetic field crucially influences the fate of ionized Keplerian disks causing creation of winds and jets, enabling simultaneously acceleration of ultra-high energy particles with energy up to 10 21 eV around supermassive black holes with M ∼ 10 10 M ⊙ surrounded by sufficiently strong magnetic field with B ∼ 10 4 G. We also show that the external magnetic fields enable existence of “levitating” off-equatorial clouds or tori, along with the standard equatorial toroidal structures, if these carry a non-vanishing, appropriately distributed electric charge.

Abstract Recently a D -dimensional regularization approach leading to the non-trivial $$(3+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional Einstein–Gauss–Bonnet (EGB) effective description of gravity was formulated which was claimed to bypass the Lovelock’s theorem and avoid Ostrogradsky instability. Later it was shown that the regularization is possible only for some broad, but limited, class of metrics and Aoki et al. ( arXiv:2005.03859 ) formulated a well-defined four-dimensional EGB theory, which breaks the Lorentz invariance in a theoretically consistent and observationally viable way. The black-hole solution of the first naive approach proved out to be also the exact solution of the well-defined theory. Here we calculate quasinormal modes of scalar, electromagnetic and gravitational perturbations and find the radius of shadow for spherically symmetric and asymptotically flat black holes with Gauss–Bonnet corrections. We show that the black hole is gravitationally stable when ( $$-16 M^2&lt;\alpha \lessapprox 0.6 M^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>16</mml:mn> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>⪅</mml:mo> <mml:mn>0.6</mml:mn> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> ). The instability in the outer range is the eikonal one and it develops at high multipole numbers. The radius of the shadow $$R_{Sh}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>Sh</mml:mi> </mml:mrow> </mml:msub> </mml:math> obeys the linear law with a remarkable accuracy.

We investigate how a spherically symmetric fluid modifies the Schwarzschild vacuum solution when there is no exchange of energy-momentum between the fluid and the central source of the Schwarzschild metric. This system is described by means of the gravitational decoupling realised via the minimal geometric deformation approach, which allows us to prove that the fluid must be anisotropic. Several cases are then explicitly shown.

We study the motion of charged particles in the field of a rotating black hole immersed into an external asymptotically uniform magnetic field, focusing on the epicyclic quasicircular orbits near the equatorial plane. Separating the circular orbits into four qualitatively different classes according to the sign of the canonical angular momentum of the motion and the orientation of the Lorentz force, we analyze the circular orbits using the so-called force formalism. We find the analytical solutions for the radial profiles of velocity, specific angular momentum, and specific energy of the circular orbits in dependence on the black-hole dimensionless spin and the magnetic field strength. The innermost stable circular orbits are determined for all four classes of the circular orbits. The stable circular orbits with an outward-oriented Lorentz force can extend to radii lower than the radius of the corresponding photon circular geodesic. We calculate the frequencies of the harmonic oscillatory motion of the charged particles in the radial and vertical directions related to the equatorial circular orbits and study the radial profiles of the radial, ${\ensuremath{\omega}}_{\mathrm{r}}$; vertical, ${\ensuremath{\omega}}_{\ensuremath{\theta}}$; and orbital, ${\ensuremath{\omega}}_{\ensuremath{\phi}}$, frequencies, finding significant differences in comparison to the epicyclic geodesic circular motion. The most important new phenomenon is the existence of toroidal charged particle epicyclic motion with ${\ensuremath{\omega}}_{\mathrm{r}}\ensuremath{\sim}{\ensuremath{\omega}}_{\ensuremath{\theta}}\ensuremath{\gg}{\ensuremath{\omega}}_{\ensuremath{\phi}}$ that could occur around retrograde circular orbits with an outward-oriented Lorentz force. We demonstrate that for the rapidly rotating black holes the role of the ``Wald induced charge'' can be relevant.

We study the influence of the tidal charge parameter of the braneworld models on some optical phenomena in rotating black hole space–times. The escape photon cones are determined for special families of locally nonrotating, circular geodetical and radially free-falling observers. The silhouette of a rotating black hole, the shape of an equatorial thin accretion disk and the time delay effect for direct and indirect images of a radiating hot spot orbiting the black hole are given and classified in terms of the black hole rotational and tidal parameters. It is shown that increase of the negatively valued tidal parameter, with the rotational parameter fixed, generally strengthens the relativistic effects and suppresses the rotation-induced asymmetries in the optical phenomena.

eXTP is a science mission designed to study the state of matter under extreme conditions of density, gravity and magnetism. Primary goals are the determination of the equation of state of matter at supra-nuclear density, the measurement of QED effects in highly magnetized star, and the study of accretion in the strong-field regime of gravity. Primary targets include isolated and binary neutron stars, strong magnetic field systems like magnetars, and stellar-mass and supermassive black holes. The mission carries a unique and unprecedented suite of state-of-the-art scientific instruments enabling for the first time ever the simultaneous spectral-timing-polarimetry studies of cosmic sources in the energy range from 0.5-30 keV (and beyond). Key elements of the payload are: the Spectroscopic Focusing Array (SFA) - a set of 11 X-ray optics for a total effective area of ∼0.9 m² and 0.6 m² at 2 keV and 6 keV respectively, equipped with Silicon Drift Detectors offering &lt;180 eV spectral resolution; the Large Area Detector (LAD) - a deployable set of 640 Silicon Drift Detectors, for a total effective area of ∼3.4 m², between 6 and 10 keV, and spectral resolution better than 250 eV; the Polarimetry Focusing Array (PFA) – a set of 2 X-ray telescope, for a total effective area of 250 cm² at 2 keV, equipped with imaging gas pixel photoelectric polarimeters; the Wide Field Monitor (WFM) - a set of 3 coded mask wide field units, equipped with position-sensitive Silicon Drift Detectors, each covering a 90 degrees x 90 degrees field of view. The eXTP international consortium includes major institutions of the Chinese Academy of Sciences and Universities in China, as well as major institutions in several European countries and the United States. The predecessor of eXTP, the XTP mission concept, has been selected and funded as one of the so-called background missions in the Strategic Priority Space Science Program of the Chinese Academy of Sciences since 2011. The strong European participation has significantly enhanced the scientific capabilities of eXTP. The planned launch date of the mission is earlier than 2025.

To test the role of large-scale magnetic fields in accretion processes, we study the dynamics of the charged test particles in the vicinity of a black hole immersed into an asymptotically uniform magnetic field. Using the Hamiltonian formalism of the charged particle dynamics, we examine chaotic scattering in the effective potential related to the black hole gravitational field combined with the uniform magnetic field. Energy interchange between the translational and oscillatory modes of the charged particle dynamics provides a mechanism for charged particle acceleration along the magnetic field lines. This energy transmutation is an attribute of the chaotic charged particle dynamics in the combined gravitational and magnetic fields only, the black hole rotation is not necessary for such charged particle acceleration. The chaotic scatter can cause a transition to the motion along the magnetic field lines with small radius of the Larmor motion or vanishing Larmor radius, when the speed of the particle translational motion is largest and it can be ultra-relativistic. We discuss the consequences of the model of ionization of test particles forming a neutral accretion disc, or heavy ions following off-equatorial circular orbits, and we explore the fate of heavy charged test particles after ionization where no kick of heavy ions is assumed and only the switch-on effect of the magnetic field is relevant. We demonstrate that acceleration and escape of the ionized particles can be efficient along the Kerr black hole symmetry axis parallel to the magnetic field lines. We show that a strong acceleration of the ionized particles to ultra-relativistic velocities is preferred in the direction close to the magnetic field lines. Therefore, the process of ionization of Keplerian discs around the Kerr black holes can serve as a model of relativistic jets.

Abstract We employ the gravitational decoupling approach for static and spherically symmetric systems to develop a simple and powerful method in order to (a) continuously isotropize any anisotropic solution of the Einstein field equations, and (b) generate new solutions for self-gravitating distributions with the same or vanishing complexity factor. A few working examples are given for illustrative purposes.

Abstract\nWe study optical phenomena related to appearance of Keplerian accretion discs orbiting Kerr superspinars predicted by the string theory. The superspinar exterior is described by the standard Kerr naked singularity geometry breaking the black hole limit on the internal angular momentum (spin). We construct local photon escape cones for a variety of orbiting sources that enable to determine the superspinars silhouette in the case of distant observers. We show that the superspinar silhouette depends strongly on the assumed edge where the external Kerr spacetime is joined to the internal spacetime governed by the string theory and significantly differs from the black hole silhouette. The appearance of the accretion disc is strongly dependent on the value of the superspinar spin in both their shape and frequency shift profile. Apparent extension of the disc grows significantly with growing spin, while the frequency shift grows with descending spin. This behavior differs substantially from appearance of discs orbiting black holes enabling thus, at least in principle, to distinguish clearly the Kerr superspinars and black holes. In vicinity of a Kerr superspinar the non-escaped photons have to be separated to those captured by the superspinar and those being trapped in its strong gravitational field leading to self-illumination of the disc that could even influence its structure and causes self-reflection effect of radiation of the disc. The amount of trapped photons grows with descending of the superspinar spin. We thus can expect significant self-illumination effects in the field of Kerr superspinars with near-extreme spin a ~ 1.

We study the shadow of the rotating black hole with quintessential energy (i) in vacuum, (ii) in the presence of plasma with radial power-law density. For the vacuum case, the quintessential field parameter of the rotating black hole significantly changes the shape of the shadow. With increasing quintessential field parameter, the radius of the shadow also increases. With the increase of the radius of the shadow of the rotating black hole, the quintessential field parameter causes decrease of the distortion of the shadow shape: in the presence of the quintessential field parameter, the shadow of the fast rotating black hole becomes too close to the circle. We assume the distant observer of the black hole shadow to be located near the so-called static radius where the gravitational attraction of the black hole is just balanced by the cosmic repulsion. The shape and size of the shadow of quintessential rotating black hole surrounded by plasma depends on (i) plasma parameters, (ii) black hole spin and (iii) quintessential field parameter. With the increase of the plasma refraction index, the apparent radius of the shadow increases. However, for the large values of the quintessential field parameter, the change of the black hole shadow shape due to the presence of plasma is not significant, i.e. the effect of the quintessential field parameter dominates over the plasma effect.