Radius Of Shadow Research Articles

Null geodesics, QNMs, emission energy and thermal fluctuation of charged T-duality black hole with simple logarithmic correction

This research investigates the properties of charged T-duality black holes within the context of string theory. The investigation focuses on the implications of T-duality, electric charge, and thermodynamic parameters on the curvature and behavior of a black hole. The findings demonstrate that physical variables like μ and H have significant impacts on the position of the event horizon, photon radius as well as shadow radius. The study calculates the values of quasinormal modes and observes through the graphical behavior. Additionally, the paper investigates the entropy and free energies of the system by considering simple logarithmic correction and also demonstrates how the T-duality parameter enhances the thermodynamic stability of the Bardeen black hole geometry. We explore the emission energy of the considered black hole. Overall, this research contributes to our understanding of black holes and their role in the universe within the context of string theory.

Shadows and weak gravitational lensing for black holes within Einstein-Maxwell-scalar theory* *Supported by the National Natural Science Foundation of China (11675143), and the National Key Research and Development Program of China (2020YFC2201503). M.A and B.A wish to acknowledge the support from Research Grant F-FA-2021-432 of the Ministry of Higher Education, Science and Innovations of the Republic of Uzbekistan.

In this study, we investigated the optical properties of charged black holes within the Einstein-Maxwell-scalar (EMS) theory. We evaluated the shadow cast by these black holes and obtained analytical solutions for both the radius of the photon sphere and that of the shadow. We observed that black hole parameters γ and β both influence the shadow of black holes. We also found that the photon sphere and shadow radius increase as a consequence of the presence of the parameter γ. Interestingly, the shadow radius decreases first and then remains unchanged owing to the impact of the parameter β. Finally, we analyzed the weak gravitational lensing and total magnification of lensed images around black holes. We found that the charge of the black holes and the parameter β both have a significant impact, reducing the deflection angle. Similarly, the same behavior for the total magnification was observed, also as a result of the effect of the charge of the black holes and the parameter β.

Investigating the shadows and strong gravitational lensing of modified Bardeen black holes

This study is dedicated to exploring the distinctive observational features affected by the modified Bardeen black hole through meticulous analysis of its shadow and strong gravitational lensing observations. We thoroughly investigated the impact of the black hole’s parameters q, g and μ on its shadow radius through numerical simulations and graphical representations. Utilizing the Event Horizon Telescope (EHT) data, we constrained the modified Bardeen black hole parameter μ of the modified Bardeen black hole within specific ranges: 0.24 ≤ μ ≤ 1.16 for M87*; and 0.036 ≤ μ ≤ 0.51 for Sgr A *, while maintaining the fixed values g = 0.2 and q = 0.3. This restriction of the modified Bardeen black hole parameter μ by the EHT findings illustrates the viability of modified Bardeen black holes as astrophysical candidates. Additionally, we study strong gravitational lensing and its various observables for the modified Bardeen black hole, comparing its behavior to other astrophysical black holes such as the Schwarzschild (μ = 0 = q) and Bardeen (μ = 0) black holes. By examining the astrophysical ramifications through strong gravitational lensing, considering supermassive black holes at the center of nearby galaxies, we uncovered that the modified Bardeen black hole exhibits distinct characteristics, offering a quantitative distinction from other black holes such as the Schwarzschild and Bardeen black holes. These findings in astrophysical consequences provide a promising pathway to differentiate the modified black hole from its counterparts in the realm of general relativity.

Dynamics of holographic images of scalar-tensor-vector gravity-AdS black holes* *Supported by the National Natural Science Foundation of China (11675140, 11705005, 12375043), the Innovation and Development Joint Foundation of Chongqing Natural Science Foundation (CSTB2022NSCQ-LZX0021), and the Basic Research Project of Science and Technology Committee of Chongqing (CSTB2023NSCQ-MSX0324).

Using AdS/CFT correspondence, we analyze the holographic Einstein images via the response function of the complex scalar field as a probe wave on an AdS Schwarzschild scalar-tensor-vector gravity (STVG) black hole (BH). We find that the amplitude of the response function decreases with increasing values of coupling parameter α and increases with decreasing temperature T. The frequency ω of the wave source also plays a significant role in wave periods; as we increase the values of ω, the periods of waves decrease, indicating that the total response function closely depends on the wave source. Further, we investigate the optical appearance of the holographic images of the BH in bulk. We found that the holographic ring always appears with surrounding concentric stripes when the observer is located at the north pole, and an extremely bright ring appears when the observer is at the position of the photon sphere of the BH. This ring changes into a luminosity-deformed ring or a bright light spot as the observational angle changes. The corresponding brightness profiles show that the luminosity of the ring decreases and the shadow radius increases with increasing values of α. The relation between temperature T and the inverse of the horizon is discussed; T is small at the beginning of the horizon and then increases as the horizon radius increases. This effect can be used to distinguish the STVG BH solution from other BH solutions. Moreover, these significant features are also reflected in the Einstein ring and corresponding brightness profiles. In addition, we compare the results obtained by wave optics and geometric optics, which align well, implying that the holographic scheme adopted in this study is valid.

Euler–Heisenberg black hole surrounded by perfect fluid dark matter

A generation method of new metric in the case of static spherically symmetric space-time is derived. Using this approach, we construct a metric which describes Euler–Heisenberg black hole surrounded by perfect fluid dark matter and investigate its optical and thermodynamic properties. We found that radius of shadow will increase with the increase of dark matter effect, and more strong dark matter will diminish the light intensity of accretion disk generally. Moreover, in thermodynamics, when quantum electrodynamic parameter is positive, there will be a critical value of dark matter parameter, which determine the number of black hole’s critical points.

Apparent and emergent dark matter around a Schwarzschild black hole

Inspired by the two different dark matter frameworks that were studied recently: one that arises from the non-local effects of entanglement entropy as emergent gravity (characterized by the parameter ξ(M), and zero-point scale length l), and one from dark energy viewed as a superconducting medium (characterized by η(M), and screening length parameter λG), two black hole solutions spherically surrounded with these dark matter models were derived. The effect of these two frameworks on SMBH was analyzed through the resulting deviations in the null regions and the black hole shadow. In addition, constraints to the parameters ξ and η (at 3σ level) were found using the available EHT data for Sgr. A* and M87*. These constraints allow one to deduce the effective mass M, which causes uncertainties in the measurement. On the other hand, if the effective mass is known, one can also deduce the constants associated with ξ, and η. The former framework also introduces an Appell function, a hypergeometric function of two variables that separately allows the analysis of macroscopic and (hypothetical) microscopic black holes. This first framework was found to decrease the radii of the null regions respective to the Schwarzschild counterpart. The shadow radius, however, behaves reversibly. The result of the numerical analysis for the latter framework revealed increased values for the photonsphere and shadow radii. Remarkably, the study also showed that for SMBHs, the amplifying effects of λG are stronger than the scale length l. Finally, results of constraints, as an example, for the upper bound in ξ for M87* indicated that the effective mass causing the deviation was around 2.4×1020M⊙ given that the observed Milgrom’s constant is 5.4×10−10m/s2.

Exploring non-commutativity as a perturbation in the Schwarzschild black hole: quasinormal modes, scattering, and shadows

In this work, by a novel approach to studying the scattering of a Schwarzschild black hole, the non-commutativity is introduced as perturbation. We begin by reformulating the Klein–Gordon equation for the scalar field in a new form that takes into account the deformed non-commutative spacetime. Using this formulation, an effective potential for the scattering process is derived. To calculate the quasinormal modes, we employ the WKB method and also utilize fitting techniques to investigate the impact of non-commutativity on the scalar quasinormal modes. We thoroughly analyze the results obtained from these different methods. Moreover, the greybody factor and absorption cross section are investigated. Additionally, we explore the behavior of null geodesics in the presence of non-commutativity. Specifically, we examine the photonic, and shadow radius as well as the light trajectories for different non-commutative parameters. Therefore, by addressing these various aspects, we aim to provide a comprehensive understanding of the influence of non-commutativity on the scattering of a Schwarzschild-like black hole and its implications for the behavior of scalar fields and light trajectories.

Optical features of rotating quintessential charged black holes in de-Sitter spacetime

One of the most important and actual issues in relativistic astrophysics is testing gravity theories and obtaining constraint values for the parameters of black holes using observational data. In this research, we aimed to explore the optical features of a Kerr–Newman black hole model in the presence of a quintessential field, which may be a candidate for a dark-energy model with a nonzero cosmological constant. First, we obtain the equations of motion for photons using the Hamilton–Jacobi formalism. We also study the horizons and shapes of the apparent regions of the photon region around the said black hole. In various scenarios, we investigate shadows cast by the black hole using celestial coordinates. Furthermore, we analyze the effects of the quintessential field and black hole charge on the shadow radius and distortion. Furthermore, we look into the constraints on the spin and charge of supermassive black holes M87∗ and Sagittarius A∗ for different values of the quintessential field using their shadow size measured by the Event Horizon Telescope Collaboration. Finally, we study the effects of the quintessential field, black hole spin and charge on its energy emission rate by Hawking radiation and compare our results with those of the available literature.

Effect of magnetized plasma on shadow and gravitational lensing of a Reissner–Nordström black hole

We explore the influence of the axion–plasmon on the optical properties of the charged black hole and show how the axion–plasmon coupling modifies the motion of photons around the charged black hole. We then explore in details observational effects such as the black hole shadow, the gravitational deflection angle, Einstein rings and shadow images obtained by radially infalling gas on a black hole within a plasma medium. An important finding is that the intensity of the electromagnetic radiation increases with the increase of charge and the size of the black hole shadow decreases with increase of the electric charge for a fixed axion–plasmon coupling values when observed from sufficiently large distance. Overall, for a constant value of charge the optical appearance of the black hole shadow depends on the surrounding plasma model and the largest shadow radius if found for the case of no plasma, while the smallest shadow radius is found for the case of homogeneous plasma.

Infinitely degenerate slowly rotating solutions in f(R) gravity

This work tests the no-hair conjecture in f(R) gravity models. No-hair conjecture asserts that all black holes in general relativity coupled to any matter must be Kerr–Newman type. However, the conjecture fails in some cases with non-linear matter sources. Here, we address this by explicitly constructing multiple slow-rotating black hole solutions, up to second order in rotational parameter, for a class of f(R) models ( f(R)=(α0+α1R)p,p>1 ). Such an f(R) includes all higher-powers of R. We analytically show that multiple vacuum solutions satisfy the field equations up to the second order in the rotational parameter. In other words, we show that the multiple vacuum solutions depend on arbitrary constants, which depend on the coupling parameters of the model. Hence, our results indicate that the no-hair theorem for modified gravity theories merits extending to include the coupling constants. The uniqueness of our result stems from the fact that these are obtained directly from metric formalism without conformal transformation. We discuss the kinematical properties of these black hole solutions and compare them with slow-rotating Kerr. Specifically, we show that the circular orbits for the black holes in f(R) are smaller than that of Kerr. This implies that the inner-most stable circular orbit for black holes in f(R) is smaller than Kerr’s; hence, the shadow radius might also be smaller. Finally, we discuss the implications of our results for future observations.

General approach on shadow radius and photon spheres in asymptotically flat spacetimes and the impact of mass-dependent variations

Recent observations of black hole shadows have revolutionized our ability to probe gravity in extreme environments. This manuscript presents a novel analytic method to calculate, in leading-order terms, the key parameters of photon sphere and shadow radius. This method offers advantages for complex metrics where traditional approaches are cumbersome. We further explore the impact of black hole mass on the photon sphere radius, providing insights into black hole interactions with their surroundings. Our findings contribute significantly to black hole physics and gravity under extreme conditions. By leveraging future advancements in observations, such as the next-generation Event Horizon Telescope (ngEHT), this work paves the way for even more precise tests of gravity near black holes.

Analyzing the influence of geometrical deformation on photon sphere and shadow radius: A new analytical approach — Spherically symmetric spacetimes

In this paper, we introduce a new approach to study the behavior of the photon sphere and shadow radius. Our method uses extended gravitational decoupling and reveals two important analytic results. First, the additional matter field alters the photon sphere radius: it increases if g′(rph(0))>0 and decreases if g′(rph(0))<0 (where g′ represents the derivative of a specific metric function evaluated at the original photon sphere radius). Second, the presence of the matter field can modify the black hole shadow size. If grph(0)>0, the shadow shrinks, while it grows for grph(0)<0. These findings provide a deeper insight into how matter distribution influences the characteristics of black holes and their observable features. Through a systematic framework and various illustrative examples, our investigation not only clarifies these fundamental aspects but also significantly enhances the theoretical framework of black hole astrophysics.

Hotspots and photon rings in spherically symmetric space–times

ABSTRACT Future black hole (BH) imaging observations are expected to resolve finer features corresponding to higher order images of hotspots and of the horizon-scale accretion flow. In spherical space–times, the image order is determined by the number of half-loops executed by the photons that form it. Consecutive-order images arrive approximately after a delay time of ≈π times the BH shadow radius. The fractional diameters, widths, and flux-densities of consecutive-order images are exponentially demagnified by the lensing Lyapunov exponent, a characteristic of the space–time. The appearance of a simple point-sized hotspot when located at fixed spatial locations or in motion on circular orbits is investigated. The exact time delay between the appearance of its zeroth and first-order images agrees with our analytic estimate, which accounts for the observer inclination, with $\lesssim 20~{{\ \rm per\ cent}}$ error for hotspots located about ≲ 5M from a Schwarzschild BH of mass M. Since M87⋆ and Sgr A⋆ host geometrically thick accretion flows, we also explore the variation in the diameters and widths of their first-order images with disc scale-height. Using a simple ‘conical torus’ model, for realistic morphologies, we estimate the first-order image diameter to deviate from that of the shadow by $\lesssim 30~{{\ \rm per\ cent}}$ and its width to be ≲ 1.3M. Finally, the error in recovering the Schwarzschild lensing exponent (π), when using the diameters or the widths of the first and second-order images is estimated to be $\lesssim 20~{{\ \rm per\ cent}}$. It will soon become possible to robustly learn more about the space–time geometry of astrophysical BHs from such measurements.

Quasi-normal modes, emission rate, and shadow of charged AdS black holes with perfect fluid dark matter* *Supported by the Doctoral Foundation of Zunyi Normal University of China (BS [2022] 07, QJJ-[2022]-314), and the National Natural Science Foundation of China (12265007). Additionally, S. H. Dong supported by 20230316 and 20240220-SIP-IPN, Mexico, and began this work with permission from IPN for a

In this study, we comprehensively investigated charged AdS black holes surrounded by a distinct form of dark matter. In particular, we focused on key elements including the Hawking temperature, quasi-normal modes (QNMs), emission rate, and shadow. We first calculated the Hawking temperature, thereby identifying critical values such as the critical radius and maximum temperature of the black hole, essential for determining its phase transition. Further analysis focused on the QNMs of charged AdS black holes immersed in perfect fluid dark matter (PFDM) within the massless scalar field paradigm. Employing the Wentzel-Kramers-Brillouin (WKB) method, we accurately derived the frequencies of these QNMs. Additionally, we conducted a meticulous assessment of how the intensity of the PFDM parameter α influences the partial absorption cross sections of the black hole, along with a detailed study of the frequency variation of the energy emission rate. The pivotal role of geodesics in understanding astrophysical black hole characteristics is highlighted. Specifically, we examined the influence of the dark matter parameter on photon evolution by computing the shadow radius of the black hole. Our findings distinctly demonstrate the significant impact of the PFDM parameter α on the boundaries of this shadow, providing crucial insights into its features and interactions. We also provide profound insights into the intricate dynamics between a charged AdS black hole, novel dark matter, and various physical phenomena, elucidating their interplay and contributing valuable knowledge to the understanding of these cosmic entities.

Nonlinearly charged black holes: Shadow and thin-accretion disk

In this paper, we explore the effect of the parameters of non-linear electrodynamics (NLED) and magnetic charges on various aspects of black holes (BH) arising in the particular NLED theory. More precisely, we look into the behavior of photons around the BH and produce shadow by considering a thin accretion disk model. We initially examine the overall behavior of the photon sphere and the corresponding shadow silhouette under the effects of these parameters. Using the EHT data for Sgr. A* and M87*, we provide constraints on the magnetic charge qm. Our results indicate that M87* gives better constraints, and as the value of the NLED coupling parameter β is increased, the constrained range for qm widens. At lower values of qm, we find that the shadow radius is close to the observed value. Then, we study different other aspects, such as the energy emitted by the accretion disk, the temperature of the disk around the BH, and the nature of light it gives off. We also look at how the black hole shadow appears in different situations. Finally, we investigate how the magnetic charge affects all the above mentioned aspects when we take into account NLED along with gravity. This study helps us understand the complex relationship between magnetic charge and its effect on various aspects related to a BH.

Exploring antisymmetric tensor effects on black hole shadows and quasinormal frequencies

This study explores the impact of antisymmetric tensor effects on spherically symmetric black holes, investigating photon spheres, shadows, emission rate and quasinormal frequencies in relation to a parameter which triggers the Lorentz symmetry breaking. We examine these configurations without and with the presence of a cosmological constant. In the first scenario, the Lorentz violation parameter, denoted as λ, plays a pivotal role in reducing both the photon sphere and the shadow radius, while also leading to a damping effect on quasinormal frequencies. Conversely, in the second scenario, as the values of the cosmological constant (Λ) increase, we observe an expansion in the shadow radius. Also, we provide the constraints of the shadows based on the analysis observational data obtained from the Event Horizon Telescope (EHT) focusing on Sagittarius A* shadow images. Additionally, with the increasing Λ, the associated gravitational wave frequencies exhibit reduced damping modes.

Geodesic structure, shadow and optical appearance of black hole immersed in Chaplygin-like dark fluid

In this study, we focus on a black hole immersed in a cosmological Chaplygin-like dark fluid (CDF), characterized by the equation of state p = -B/ρ and an additional parameter q influencing the energy density of the fluid. We investigate the geodesic structure, shadow, and optical appearance of such a black hole. Through analysis on the effective potential and the epicyclic frequencies, it is found that the existence of innermost/outermost stable circular orbits for a timelike particle is governed by the CDF parameters. The behaviors of the orbital conserved quantities and Keplerian frequency are also examined. Due to the existence of pseudo-cosmological horizon, the determination of the shadow radius depends significantly on the position of the observer. By placing the static observer at an approximately flat position between the event and pseudo-cosmological horizons, we constrain the CDF parameters using EHT observations. We investigate the effect of CDF on the shadows and optical images of the black hole, surrounded by various profiles of accretions. For the thin disk accretion, the light trajectories are categorized into direct emission, lensing ring, and photon ring based on impact parameters. Due to the existence of outermost stable circular orbits, outer edges could exist in the direct and lensing ring images. The observed brightness is mainly due to direct emission, with a minor contribution from the lensing ring, while the contribution from the photon ring is negligible due to extreme demagnetization. In the case of spherical accretion, we consider both static and infalling accretion models. The images obtained under infalling accretion are slightly darker than those under static accretion, attributed to the Doppler effect. Throughout the study, we analyze the influence of the parameters B and q on the results.

Quazinormal modes and greybody factor of black hole surrounded by a quintessence in the S-V-T modified gravity as well as shadow

The purpose of this study is to investigate the quasinormal modes (QNMs), greybody factors (GFs) and shadows in a plasma of a black hole (BH) surrounded by an exotic fluid of quintessence type in a scalar-vector-tensor modified gravity. The effects of a quintessence scalar field and the modified gravity (MOG) field on the QNM, GF, and shadow are examined. Using the sixth-order WKB approach, we investigate the QNMs of massless scalar and electromagnetic perturbations. Our findings show that as the quintessence and the MOG parameter (ϵ and α) increase, the oscillation frequencies decrease significantly. Gravitational wave damping, on the other hand, decreases with increasing ϵ and α. In addition, we obtain an analytical solution for the transmission coefficients (GF) and demonstrate that more thermal radiation reaches the observer at spatial infinity as both the ϵ and α parameters increase. We also investigate the effect of the plasma background on the BH shadow and show that as the plasma background parameter increases, the shadow radius slightly shrinks. Nevertheless, the shadow radius increases as α and ϵ increase. Particularly intriguing is the fact that increasing ϵ has a greater impact on the shadow radius than increasing α, indicating that the quintessence parameter has a greater impact than the MOG parameter.

Signatures of the accelerating black holes with a cosmological constant from the Sgr [formula omitted] and [formula omitted] shadow prospects

Recently, the Event Horizon Telescope (EHT) achieved the realization of an image of the supermassive black hole Sgr A⋆ showing an angular shadow diameter D=48.7±7μas and the fractional deviation δ=−0.08−0.09+0.09(VLTI),−0.04−0.10+0.09(Keck), alongside the earlier image of M87⋆ with angular diameter D=42±3μas, deviation δ=−0.01−0.17+0.17 and deviations from circularity estimated to be ΔC≲10%. In addition, the shadow radii are assessed within the ranges 3.38≤rsM≤6.91 for M87⋆ and 3.85≤rsM≤5.72 as well as 3.95≤rsM≤5.92 for Sgr A⋆ using the Very Large Telescope Interferometer (VLTI) and Keck observatories, respectively. These values are provided with 1-σ and 2-σ measurements. Such realizations can unveil a better comprehension of gravitational physics at the horizon scale. In this paper, we use the EHT observational results for M87⋆ and Sgr A⋆ to elaborate the constraints on parameters of accelerating black holes with a cosmological constant. Concretely, we utilize the mass and distance of both black holes to derive the observables associated with the accelerating black hole shadow.First, we compare our findings with observed quantities such as angular diameter, circularity, shadow radius, and the fractional deviation from the M87⋆ data. This comparison reveals constraints within the acceleration parameter and the cosmological constant. In fact, the acceleration parameter A is found to be in the interval [0,0.475−0.0125+0.0125], the cosmological constant spans the range of Λ∈[−14.5801−1.1051×10−6+1.1051×10−6,(1.0086×10−6)−1.5396×10−4+9.2121×10−5]. Then by considering the Sgr A⋆ data, the acceleration scheme persists with the same interval range, while, the cosmological constant shifts very slightly to the following domain [−14.7266−1.1051×10−6+1.1051×10−6,(4.6200×10−5)−1.5888×10−4+6.7890×10−5] for the VLTI observatory. Moreover, within the Keck measurements, one can find Λ∈[−14.7266−1.1051×10−6+1.1051×10−6,(2.4149×10−5)−1.5094×10−4+7.2836×10−5]. Besides, for the others parameters, both black holes reveal the range of the spin parameter a in [0,1] and electrical/magnetic charges are {e,g}∈[0,0.09].Further, only the cosmological constant intervals are altered by rsM measurements, precisely within the M87⋆ data, the upper bound shifts to (Λ1-σ=1.1361×10−4,Λ2-σ=1.7822×10−4), the VLTI gives (Λ1-σ=1.5457×10−4,Λ2-σ=1.8208×10−4), while the Keck results uncover Λ2-σ=1.8015×10−4.Lastly, one cannot rule out the possibility of the negative values for the cosmological constant on the emergence of accelerated black hole solutions within the context of minimal gauged supergravity. This intriguing possibility raises the prospect of such solutions being plausible candidates for astrophysical black holes.

The optical features of noncommutative charged 4D-AdS-Einstein–Gauss–Bonnet black hole: shadow and deflection angle

In this paper, we have explored the optical characteristics, namely the shadow and the deflection angle, inherent to the solution of a 4D-AdS-Einstein–Gauss–Bonnet black hole. This solution, which finds its inspiration in noncommutative geometry, had previously been established in our previous work. The radius of the shadow was determined using the Hamilton-Jacobi method and the Carter separation. Our results revealed that the presence of noncommutativity in the background of spacetime impacts the variation of the shadow radius. More specifically, we have demonstrated that an increase in the parameter θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document} induces a decrease in the radius of the shadow. In a similar way, analogous observations have been made by studying the variation of the electric charge Q. The noncommutative parameter θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document} and the electric charge Q have been constrained regarding the EHT observation data of the M87* and Sgr A* black holes. Furthermore, the angle of deflection, which is the outcome of lensing by the black hole, has been derived following the Ishihara et al. approach for a receiver and source positioned at finite distances from the black hole in an asymptotically non-flat spacetime. The impact of the noncommutative parameter θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta $$\\end{document} and the charge Q of the black hole are hence analyzed, and our results depict that these parameters have a significant influence on the angle at which light is deflected by the gravitational field of the black hole.