The study of black hole shadows provides a powerful tool for testing the predictions of general relativity and exploring deviations from the standard Kerr metric in the strong gravitational field regime. Here, we investigate the shadow properties of axisymmetric gravitational compact objects using a coordinate-independent formalism. We analyze black hole shadows in various spacetime geometries, including the Kerr, Taub-NUT, $$\gamma $$ , and Kaluza-Klein metrics, to identify distinctive features that can be used to constrain black hole parameters. To achieve a more robust characterization, we employ both Legendre and Fourier expansions, demonstrating that the Fourier approach may offer better coordinate independence and facilitate cross-model comparisons. Finally, we develop a machine learning framework based on neural networks trained on synthetic shadow data, enabling precise parameter estimation from observational results. Using data from observational astronomical facilities such as the Event Horizon Telescope (EHT), Keck, and the Very Large Telescope Interferometer (VLTI), we provide constraints on black hole parameters derived from shadow observations. Our findings highlight the potential of coordinate-independent techniques and machine learning for advancing black hole astrophysics and testing fundamental physics beyond general relativity.

China

Uzbekistan

Temurbek Mirzaev

Bobomurat Ahmedov

Cosimo Bambi

Exploring black hole shadows in axisymmetric spacetimes with coordinate-independent methods and neural networks

This study investigates the dynamics of electrically charged test particles around electrically charged Reissner–Nordström (RN) black holes (BHs) in scalar-tensor-vector gravity (STVG), known as modified gravity (MOG). Using Lagrangian and Hamiltonian formalisms, we derive the effective potential governing the motion of charged particles, taking into account interactions between the electrostatic field and the MOG field, and determine the innermost stable circular orbit (ISCO). The results indicate that an increase in the MOG parameter leads to shifting out the position of ISCOs, thereby altering the orbital dynamics of test particles compared to those predicted by general relativity (GR). We also study the radiation properties of the black hole accretion disc, including radiation flux, temperature distribution within the disc, and total radiation luminosity. We derive Keplerian and oscillation frequencies of charged particles around the charged MOG BH together with applications to quasi-periodic oscillations (QPOs) in the frame of relativistic precession (RP) model. The analysis incorporates the Markov Chain Monte Carlo (MCMC) statistical method to compare BH parameters with observed quasi-periodic oscillations (QPOs) data. The MCMC modeling results show a strong agreement with QPO observations from microquasars (GRO J1655−40, XTE J1550−564), intermediate-mass BHs (M82 X-1), and the supermassive BH Sgr A* at the Galactic center. This study confirms the astrophysical relevance of MOG theories and emphasizes the importance of further investigations into the effects of strong magnetic fields around BHs.

Isomiddin Nishonov

Sardor Murodov

Saeed Khan

Javlon Rayimbaev

Inomjon Ibragimov

Sardor Sabirov

QPOs from charged particles around charged black holes in STVG

We analyze the evolution of perturbations of a (charged) massive scalar field near a regular Simpson-Visser black hole, allowing for a non-zero external magnetic field. We show that the damping rate of the quasinormal frequencies is strongly suppressed by both the magnetic field and the mass term, with indications that arbitrarily long-lived modes, or quasi-resonances, may exist in the spectrum. In the time domain, the quasinormal ringing transitions into slowly decaying oscillatory tails, which are qualitatively distinct from the massive tails observed in the absence of a magnetic field. For nonzero multipole and azimuthal numbers, the power-law envelope characteristic of cases without a magnetic field transforms into an oscillatory envelope that cannot be easily fitted with a simple analytical formula.

Akbar Davlataliev

Bakhtiyor Narzilloev

Pakistan

Ibrar Hussain

Ahmadjon Abdujabbarov

Long-lived quasinormal modes and asymptotic tails of regular Schwarzschild-like black holes in the presence of a magnetic field

An exact analytical solution for the charged traversable wormhole solution has been found by solving minimally coupled Einstein-Maxwell-Scalar (EMS) field equations. It is shown that the spherically-symmetric solution for the charged wormhole of mass <math altimg='si9.svg'><mi>M</mi></math> and charge <math altimg='si8.svg'><mi>Q</mi></math> is represented by the following spacetime metric: <math altimg='si10.svg'><mrow><mi>d</mi><msup><mi>s</mi><mn>2</mn></msup><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mi>f</mi><mi>d</mi><msup><mi>t</mi><mn>2</mn></msup><mo linebreak='goodbreak'>+</mo><msup><mi>f</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup><mrow><mo stretchy='true'>[</mo><mi>d</mi><msup><mi>r</mi><mn>2</mn></msup><mo linebreak='badbreak'>+</mo><msup><mi>r</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>d</mi><msup><mi>&#952;</mi><mn>2</mn></msup><mo linebreak='badbreak'>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>&#952;</mi><mi>d</mi><msup><mi>&#981;</mi><mn>2</mn></msup><mo>)</mo></mrow><mo stretchy='true'>]</mo></mrow></mrow></math>, with <display><formula id='ueq0001'><math altimg='si11.svg'><mtable displaystyle='true'><mtr><mtd columnalign='right'><mrow><mi>f</mi><mo linebreak='goodbreak'>=</mo><msup><mrow><mo stretchy='true'>[</mo><mi>cosh</mi><mrow><mo stretchy='true'>(</mo><mfrac><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>&#8722;</mo><msup><mi>Q</mi><mn>2</mn></msup></mrow></msqrt><mi>r</mi></mfrac><mo stretchy='true'>)</mo></mrow><mo>+</mo><mfrac><mi>M</mi><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>&#8722;</mo><msup><mi>Q</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mi>sinh</mi><mrow><mo stretchy='true'>(</mo><mfrac><msqrt><mrow><msup><mi>M</mi><mn>2</mn></msup><mo>&#8722;</mo><msup><mi>Q</mi><mn>2</mn></msup></mrow></msqrt><mi>r</mi></mfrac><mo stretchy='true'>)</mo></mrow><mo stretchy='true'>]</mo></mrow><mrow><mo>&#8722;</mo><mn>2</mn></mrow></msup><mspace width='0.16em' /></mrow></mtd></mtr></mtable></math></formula></display>which covers the well-known Papapetrou &#8220;exponential&#8221; metric for the uncharged wormhole. It is also shown that the obtained result is the regular solution of EMS field equations. Along the solution we have presented overcharged naked singularity with <math altimg='si3.svg'><mrow><mi>Q</mi><mo linebreak='goodbreak'>&gt;</mo><mi>M</mi></mrow></math>. As a probe of the spacetime metric, the geodesic motion has been investigated. The innermost stable circular orbit of test particle has been studied.

Bobur Turimov

Czechia

Zdeněk Stuchlík

Exact charged traversable wormhole solution

In this paper, we study the periodic orbits, characterized by zoom-whirl behavior, around a Schwarzschild-like black hole (BH) embedded within a Dehnen-type dark matter (DM) halo. We demonstrate how the DM halo modifies the gravitational dynamics of the black hole, influencing the energy and angular momentum of timelike particle geodesics and enhancing their interaction with the BH. We determine the radii of the marginally bound orbits (MBOs) and innermost stable circular orbits (ISCOs), showing that the DM halo increases both. This provides a deeper understanding of how the DM alters the behavior, energy, and angular momentum of timelike particle geodesics. Furthermore, we explore the gravitational waveforms emitted by a timelike particle in periodic orbits around a supermassive black hole (SMBH) within this BH-DM system. Using a semi-analytical approach, we calculate particle trajectories and derive the corresponding waveforms, demonstrating that the DM halo modifies the zoom-whirl orbital behavior, leading to distinct changes in the waveform structure. Our findings suggest that future gravitational wave (GW) observations could constrain the properties of DM halos surrounding BHs, providing new insights into the gravitational wave signatures arising from the interaction between BH gravity and DM.

Mirzabek Alloqulov

Tursunali Xamidov

Azerbaijan

Sanjar Shaymatov

Gravitational waveforms from periodic orbits around a Schwarzschild black hole embedded in a Dehnen-type dark matter halo

In this paper, we investigate the dynamics of the neutral and charged test particles together with the radiation properties of an accretion disk around the charged hairy black hole. Firstly, we briefly overview the spacetime of the charged hairy BH and explore the horizon structure. Then we study the effect of the spacetime parameters on the motion of massive neutral and charged particles. Finally, we explore the radiation properties of the charged hairy BH using the Novikov–Thorne model. The important quantities are analyzed: the flux of electromagnetic radiation, the temperature of the accretion disk, and the differential luminosity. The parameters $$\alpha $$ and $\textit{l}$ are found to decrease the flux of electromagnetic radiation, while the BH charge $\textit{Q}$ increases it. Similarly, the disk temperature and differential luminosity decrease owing to the impact of the $$\alpha $$ and $\textit{l}$ parameters, and increase due to the BH charge $\textit{Q}$.

Chengxun Yuan

Chen Zhou

Particle dynamics and accretion disk radiation around charged hairy black holes

Testing gravity theories beyond General Relativity (GR) is crucial for a deeper understanding of fundamental physics, especially in the strong gravitational field regime around astrophysical compact objects. This work examines quasi-periodic oscillations (QPOs) generated by charged particles orbiting magnetized Bocharova–Bronnikov–Melnikov–Bekenstein (BBMB) black holes. We study charged particles’ circular orbits, deriving expressions for energy and angular momentum influenced by scalar and magnetic interactions, and investigate the innermost stable circular orbit (ISCO). Using the relativistic precession (RP) model, we derive the fundamental orbital, radial, and vertical frequencies of charged particle oscillations, highlighting deviations caused by conformal scalar and magnetic interactions. Our analysis shows that the conformally coupled scalar field significantly modifies the spacetime curvature, producing a repulsive gravitational effect that shifts stable orbits outward, thereby reducing the radial oscillation frequency. Using observational QPO data from well-known astrophysical sources-GRS 1915+105, GRO J1655-40, M82 X-1, and Sgr A*-we performed Markov Chain Monte Carlo (MCMC) simulations to constrain black hole mass, scalar coupling, magnetic interaction, and orbital radius parameters. Our results reveal significant deviations from standard General Relativity predictions, offering clear evidence of scalar and magnetic field influences. These findings indicate that scalar and magnetic fields produce observable shifts in QPO frequencies, providing promising observational signatures to test alternative gravitational theories in extreme astrophysical environments.

Saudi Arabia

Ashfaque Bokhari

QPOs tests and circular motions of charged particles around magnetized Bocharova–Bronnikov–Melnikov–Bekenstein black holes

In this study, we investigate the motion of magnetized particles around a Bocharova–Bronnikov–Melnikov–Bekenstein (BBMB) black hole in an external magnetic field, emphasizing the effects of an external magnetic field and a conformally coupled scalar field. We analyze the properties of circular orbits, the innermost stable circular orbit (ISCO), and the dynamics of high-energy particle collisions, focusing on the center-of-mass energy (CME) of colliding particles. We derive the equations governing the motion of magnetized particles in the BBMB spacetime and explore how the scalar coupling and the magnetic-dipole interaction parameters influence orbital stability and collision energetics. Our findings reveal that the ISCO radius is significantly modified by both scalar and magnetic interactions, leading to shifts in stability conditions and variations in the angular momentum requirements. The study also demonstrates that the critical angular momentum, which determines the transition between bound and unbound motion, is reduced compared to the Schwarzschild case due to the influence of the conformal scalar field. One key result is that the CME of colliding magnetized particles can be significantly enhanced in the BBMB spacetime. Increasing magnetic interaction and a stronger attractive scalar field lead to higher CME values, making the BBMB black hole a potential site for high-energy astrophysical processes. This suggests that external magnetic fields and scalar interactions play a crucial role in energy extraction mechanisms and the formation of ultrarelativistic particles.

Nuriddin Kurbonov

Circular orbits and collisions of particles with magnetic dipole moment around magnetized Bocharova–Bronnikov–Melnikov–Bekenstein black holes

In this study, we investigate the parameterized Konoplya–Rezzolla–Zhidenko (KRZ) black hole (BH) spacetime in the presence of an external asymptotically uniform magnetic field. We first examine the innermost stable circular orbit (ISCO) radii for both neutral and charged test particles, demonstrating that the deformation parameters, $$\delta _1$$ and $$\delta _2$$ , reduce the ISCO values. Subsequently, we assess the energy efficiency of the magnetic Penrose process (MPP) for an axially symmetric parameterized BH, analyzing the effects of the deformation parameters and the magnetic field on the energy extraction process. Our findings indicate that the rotational deformation parameter $$\delta _2$$ is crucial for the efficiency of energy extraction from the BH. The synergy between the rotational deformation parameter and the magnetic field significantly boosts the energy extraction efficiency, with values exceeding $$100\%$$ . Interestingly, for extremal BHs with negative $$\delta _2$$ values, the energy efficiency increases, in contrast to Kerr BHs where the MPP effect diminishes. Additionally, we explore the astrophysical implications of the MPP by deriving the maximum energy of a proton escaping from the KRZ parameterized BH due to the beta decay of a free neutron near the horizon. Our results show that negative $$\delta _2$$ values require stronger magnetic fields to achieve equivalent energy levels for high-energy protons, providing deeper insights into high-energy astrophysical phenomena around the parameterized BH.

India

Pankaj Sheoran

Astrophysical insights into magnetic Penrose process around parameterized Konoplya–Rezzolla–Zhidenko black hole

In this research, we explore the electromagnetic energy emitted by astrophysical black holes within the Kerr+PFDM (Kerr + perfect fluid dark matter) spacetime, a model comprising rotating black holes surrounded by dark matter. Our investigation focuses on black holes within X-ray binary systems, namely GRS 1915+105, GRO J1655-40, XTE J1550-564, A0620-00, H1743-322, and GRS 1124-683. Our findings indicate that the Kerr+PFDM spacetime can account for the radiative efficiency of these sources as determined through the continuum-fitting method (CFM). Additionally, employing the Blandford–Znajek mechanism, we demonstrate the ability to replicate the observed jet power. By combining the outcomes of both analyses for the selected objects, we establish more rigorous constraints on the spacetime parameters. Notably, our results reveal that, similar to the Kerr spacetime, the Kerr+PFDM spacetime cannot simultaneously account for the observed jet power and radiative efficiency of GRS 1915+105.

Observed jet power and radiative efficiency of black hole candidates in the Kerr+PFDM model