We succeed to find compact analytical expressions which allow to easily extract the black hole spin from observations of its shadow, without need to construct or model the entire curve of the shadow. The deformation of Kerr black hole shadow can be characterised in a simple way by oblateness (the ratio of the horizontal and vertical angular diameters which are supposed to be measured by an observer). The deformation is significant in case the black hole is nearly extreme and observer is not so far from the equatorial plane. In this approximation, we present: (i) the spin lower limit via oblateness, (ii) the spin via oblateness and viewing angle, in case the latter is known from other observations.
We have studied the shadow of a disformal Kerr black hole with an extra deformation parameter, which belongs to non-stealth rotating solutions in quadratic Degenerate Higher Order Scalar Tensor (DHOST) theory. Our result show that the size of the shadow increases with the deformation parameter for the black hole with arbitrary spin parameter. However, the effect of the deformation parameter on the shadow shape depends heavily on the spin parameter of black hole and the sign of the deformation parameter. The change of the shadow shape becomes more distinct for the black hole with the more quick…
We have investigated the shadow of a Konoplya-Zhidenko rotating non-Kerr black hole with an extra deformation parameter. The spacetime structure arising from the deformed parameter affects sharply the black hole shadow. With the increase of the deformation parameter, the size of the shadow of black hole increase and its shape becomes more rounded for arbitrary rotation parameter. The D-shape shadow of black hole emerges only in the case $a<\frac{2\sqrt{3}}{3}M$ with the proper deformation parameter. Especially, the black hole shadow possesses a cusp shape with small eye lashes in the cases wit…
Analyzing the Influence of Geometrical Deformation on Photon Sphere and Shadow Radius: A New Analytical Approach -Stationary, and Axisymmetric Spacetime
We consider a black hole mimicker given by an exact solution of the stationary and axially symmetric field equations in vacuum known as the $δ$-Kerr metric. We study its optical properties based on a ray-tracing code for photon motion and characterize the apparent shape of the shadow of the compact object and compare it with the Kerr black hole. For the purpose of obtaining qualitative estimates related to the observed shadow of the supermassive compact object in the galaxy M87 we focus on values of the object's spin $a$ and inclination angle of observation $θ_0$ close to the measured values. …
In this article, we provide a review of the current state of the research of the black hole shadow, focusing on analytical (as opposed to numerical and observational) studies. We start with particular attention to the definition of the shadow and its relation to the often used concepts of escape cone, critical impact parameter and particle cross-section. For methodological purposes, we present the derivation of the angular size of the shadow for an arbitrary spherically symmetric and static space-time, which allows one to calculate the shadow for an observer at arbitrary distance from the cent…
Cosmic expansion influences the angular size of black hole shadow. The most general way to describe a black hole embedded into an expanding universe is to use the McVittie metric. So far, the exact analytical solution for the shadow size in the McVittie metric, valid for arbitrary law of expansion and arbitrary position of the observer, has not been found. In this paper, we present the first analytical solution for angular size of black hole shadow in McVittie metric as seen by observer comoving with the cosmic expansion. We use a method of matched asymptotic expansions to find approximate sol…
We consider the equivalence of quasinormal modes and geodesic quantities recently brought back due to the black hole shadow observation by Event Horizon Telescope. Using WKB method we found an analytical relation between the real part of quasinormal frequencies at the eikonal limit and black hole shadow radius. We verify this correspondence with two black hole families in $4$ and $D$ dimensions, respectively.
Can the observation of the "shadow" allow us to distinguish a black hole from a more exotic compact object? We study the motion of photons in a class of vacuum static axially-symmetric space-times that is continuously linked to the Schwarzschild metric through the value of one parameter that can be interpreted as a measure of the deformation of the source. We investigate the lensing effect and shadow produced by the source with the aim of comparing the expected image with the shadow of a Schwarzschild black hole. In the context of astrophysical black holes we found that it may not be possible …
We study the shadow of the Cardoso-Pani-Rico (CPR) black hole for different values of the black hole spin $a_*$, the deformation parameters $ε_3^t$ and $ε_3^r$, and the viewing angle $i$. We find that the main impact of the deformation parameter $ε_3^t$ is the change of the size of the shadow, while the deformation parameter $ε_3^r$ affects the shape of its boundary. In general, it is impossible to test the Kerr metric, because the shadow of a Kerr black hole can be reproduced quite well by a black hole with non-vanishing $ε_3^t$ or $ε_3^r$. Deviations from the Kerr geometry could be constrain…
The spherically symmetric deformation of the Schwarzschild solution owing to the quantum corrections to gravity is known as Kazakov-Solodukhin black-hole metric. Neglecting non-spherical deformations of the background the problem was solved non-perturbatively. Here we analyze the basic characteristics of this geometry, such as: quasinormal modes and grey-body factors of fields of various spin and shadow cast by this black hole. The WKB approach and time-domain integration method, which we used for calculation of quasinormal modes, are in a good concordance. The analytical formula for quasinorm…
Motivated by recent work on rotating black hole shadow [Phys. Rev. D101, 084029 (2020)], we investigate the shadow behaviors of rotating Hayward-de Sitter black hole for static observers at a finite distance in terms of astronomical observables. This paper uses the newly introduced distortion parameter in [arXiv:2006.00685] to describe the shadow's shape quantitatively. We show that the spin parameter would distort shadows and the magnetic monopole charge would increase the degree of deformation. At the same time, the distortion could be relieved because of the cosmological constant and the di…
Within 5-10 years, very-long baseline interferometry (VLBI) facilities will be able to directly image the accretion flow around SgrA$^*$, the super-massive black hole candidate at the center of the Galaxy, and observe the black hole "shadow". In 4-dimensional general relativity, the no-hair theorem asserts that uncharged black holes are described by the Kerr solution and are completely specified by their mass $M$ and by their spin parameter $a$. In this paper, we explore the possibility of distinguishing Kerr and Bardeen black holes from their shadow. In Hioki & Maeda (2009), under the assumpt…
This paper investigates the shadow of a charged rotating non-commutative black hole. For this purpose, we first formulate the null geodesics and study the effects of non-commutative charge on the photon orbit. We then explore the effect of spin, angle of inclination as well as non-commutative charge on the silhouette of the shadow. It is found that shape of the shadow deviates from the circle with the decrease in the non-commutative charge. We also discuss observable quantities to study the deformation and distortion in the shadow cast by the black hole which decreases for small values of non-…
We investigate the shadow cast by a rotating braneworld black hole, in the Randall-Sundrum scenario. In addition to the angular momentum, the tidal charge term deforms the shape of the shadow. For a given value of the rotation parameter, the presence of a negative tidal charge enlarges the shadow and reduces its deformation with respect to Kerr spacetime, while for a positive charge, the opposite effect is obtained. We also analyze the case in which the combination of the rotation parameter and the tidal charge results in a naked singularity. We discuss the observational prospects correspondin…
Here we construct approximate analytical forms for the metric coefficients and fields representing the scalarized Einstein-Maxwell black holes with various couplings of the scalar field, once the parameters of the system are fixed. By increasing approximation order, one can obtain the analytic representation with any desired accuracy, what was tested via calculations of shadows for these black holes by using approximate analytical and accurate numerical metric functions. We share the Mathematica code which allows one to find an appropriate analytical form of the metric for any couplings and va…
In this paper, the shadow casted by the rotating black hole inspired by noncommutative geometry is investigated. In addition to the dimensionless spin parameter $a/M_{0}$ with $M_{0}$ black hole mass and inclination angle $i$, the dimensionless noncommutative parameter $\sqrt{\vartheta}/M_{0}$ is also found to affect the shape of the black hole shadow. The result shows that the size of the shadow slightly decreases with the parameter $\sqrt{\vartheta}/M_{0}$, while the distortion increases with it. Compared to the Kerr black hole, the parameter $\sqrt{\vartheta}/M_{0}$ increases the deformatio…
The shadow of a black hole is usually calculated, either analytically or numerically, on the assumption that the black hole is eternal, i.e., that it existed for all time. Here we ask the question of how this shadow comes about in the course of time when a black hole is formed by gravitational collapse. To that end we consider a star that is spherically symmetric, dark and non-transparent and we assume that it begins, at some instant of time, to collapse in free fall like a ball of dust. We analytically calculate the dependence on time of the angular radius of the shadow, first for a static ob…
In this paper, we discuss how the shadow of a Kerr black hole depends on the motion of the observer. In particular, we derive an analytical formula for the boundary curve of the shadow for an observer moving with given four-velocity at given Boyer--Lindquist coordinates. We visualize the shadow for various values of parameters.
We investigate the shadow cast by a sort of new regular black holes which are characterized by an asymptotically Minkowski core and the Sub-Planckian curvature. Firstly, we extend the metric with spherical symmetry to the one of rotating Kerr-like black holes and derive the null geodesics with circular orbit near the horizon of the black hole, and then plot the shadow of black holes with different values of the deviation parameter. It is found that the size of the shadow shrinks with the increase of the deviation parameter, while the shape of the shadow becomes more deformed. In particular, by…
Analytical approach for calculating shadow of dynamical black hole
Herzen state Pedagogical University of Russia（俄罗斯赫尔岑国立师范大学）；Center for Theoretical Physics, Khazar University（哈扎尔大学理论物理中心）；SPB branch of SAO RAS（俄罗斯科学院天体物理研究所圣彼得堡分部）；Physics Department, Eastern Mediterranean University（东地中海大学物理系）
In an earlier paper we have analytically determined the photon regions and the shadows of black holes of the Plebański class of metrics which are also known as the Kerr--Newman--NUT--(anti-)deSitter metrics. These metrics are characterized by six parameters: mass, spin, electric and magnetic charge, gravitomagnetic NUT charge, and the cosmological constant. Here we extend this analysis to the Plebański--Demiański class of metrics which contains, in addition to these six parameters, the so-called acceleration parameter. All these metrics are axially symmetric and stationary type D solutions to …
Shadow and Thermodynamics of deformed Schwarzschild-AdS black hole with a Cloud of Strings embedded in Perfect Fluid Dark Matter
The Assam Royal Global University（阿萨姆皇家全球大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）；Università di Roma “La Sapienza”（罗马第一大学）；Al-Farabi Kazakh National University（阿尔法拉比哈萨克国立大学）
Geodesics and shadows of the spindle-deformed Kerr black hole
Key Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier and Interdisciplinary Science, Shandong University（粒子物理与粒子辐照教育部重点实验室，前沿交叉科学研究院，山东大学）；School of Mathematics and Physics, Qingdao University of Science and Technology（青岛科技大学数理学院）；Department of Physics, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Institute of Interdisciplinary Studies, Hunan Normal University（湖南师范大学物理系，教育部低维量子结构与量子控制重点实验室，交叉科学研究院）；Hunan Research Center of the Basic Discipline for Quantum Effects and Quantum Technologies, Hunan Normal University（湖南师范大学量子效应与量子技术基础学科研究中心）
We analytically investigate the influence of a cosmic expansion on the shadow of the Schwarzschild black hole. We suppose that the expansion is driven by a cosmological constant only and use the Kottler (or Schwarzschild-deSitter) spacetime as a model for a Schwarzschild black hole embedded in a deSitter universe. We calculate the angular radius of the shadow for an observer who is comoving with the cosmic expansion. It is found that the angular radius of the shadow shrinks to a non-zero finite value if the comoving observer approaches infinity.
Recently, an analytical solution in Einstein-aether theory was presented explicitly in isotropic coordinates. It is characterized by a mass parameter $m$ and a combined coupling constant $c_{14}$. By a coordinate transformation, we verify this solution is equivalent to the previously known static aether solution. Assuming photons couple minimally to aether and gravity, we investigate the photon sphere and shadow of the solution by varying $c_{14}$. Results are compared with those of the Schwarzschild black hole.
In this paper we derive exact analytical formulas for the evolution of the photon sphere and for the angular radius of the shadow in a special Vaidya spacetime. The Vaidya metric describes a spherically symmetric object that gains or loses mass, depending on a mass function $m(v)$ that can be freely chosen. Here we consider the case that $m(v)$ is a linearly increasing or decreasing function. The first case can serve as a simple model for an accreting black hole, the second case for a (Hawking) radiating black hole. With a linear mass function the Vaidya metric admits a conformal Killing vecto…
Weak field deflection angle and analytical parameter estimation of the Lorentz-violating Bumblebee parameter through the black hole shadow using EHT data
The null geodesics of the regular and rotating magnetically charged black hole in a non-minimally coupled Einstein-Yang-Mills theory surrounded by a plasma medium is studied. The effect of magnetic charge and Yang-Mills parameter on the effective potential and radius of photon orbits has investigated. We then study the shadow of a regular and rotating magnetically charged black hole along with the observables in presence of the plasma medium. The presence of plasma medium affects the apparent size of the shadow of a regular rotating black hole in comparison to vacuum case. Variation of shadow …
We study the equivalence principle, regarded as the cornerstone of general relativity, by analyzing the deformation observable of black hole shadows. Such deformation can arise from new physics and may be expressed as a phenomenological violation of the equivalence principle. Specifically, we assume that there is an additional background vector field that couples to the photons around the supermassive black hole. This type of coupling yields impact on the way the system depends on initial conditions, and affects the black hole shadow at different wavelengths by a different amount, and therefor…
Geodesics and Shadows in the Kerr-Bertotti-Robinson Black Hole Spacetime
Beijing Normal University（北京师范大学）；Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University（北京师范大学多尺度自旋物理教育部重点实验室）；Tsung-Dao Lee Institute, Shanghai Jiao-Tong University（上海交通大学汤川研究所）；Institute of Fundamental Physics and Quantum Technology, & School of Physical Science and Technology, Ningbo University（宁波大学基础物理与量子技术学院、物理科学与技术学院）；School of Physics, Peking University, & Center for High Energy Physics（北京大学物理学院、高能物理中心）
Cosmic expansion is expected to influence on the size of black hole shadow observed by comoving observer. Except the simplest case of Schwarzschild black hole in de Sitter universe, analytical approach for calculation of shadow size in expanding universe is still not developed. In this paper we present approximate method based on using angular size redshift relation. This approach is appropriate for general case of any multicomponent universe (with matter, radiation and dark energy). In particular, we have shown that supermassive black holes at large cosmological distances in the universe with…
Recently, numerical solutions to the field equations of Einstein-scalar-Gauss-Bonnet gravity that correspond to black-holes with non-trivial scalar hair have been reported. Here, we employ the method of the continued-fraction expansion in terms of a compact coordinate in order to obtain an analytical approximation for the aforementioned solutions. For a wide variety of coupling functionals to the Gauss-Bonnet term we were able to obtain analytical expressions for the metric functions and the scalar field. In addition we estimated the accuracy of these approximations by calculating the black-ho…
We introduce the notion of a local shadow for a black hole and determine its shape for the particular case of a distorted Schwarzschild black hole. Considering the lowest-order even and odd multiple moments, we compute the relation between the deformations of the shadow of a Schwarzschild black hole and the distortion multiple moments. For the range of values of multiple moments that we consider, the horizon is deformed much less than its corresponding shadow, suggesting the horizon is more `rigid'. Quite unexpectedly we find that a prolate distortion of the horizon gives rise to an oblate dis…
We study analytically the shadow cast by the rotating black hole in the perfect fluid dark matter. The apparent shape of the shadow depends upon the black hole spin $a$ and the perfect fluid dark matter intensity parameter $k$ ($k>0$). In general, the shadow is a perfect circle in the non-rotating case ($a=0$) and a deformed one in the rotating case ($a\neq 0$). The deformation gets more and more significant with the increasing $a$, similar to the Schwarzschild and Kerr black holes. In addition, there exists a reflection point $k_0$. When $k<k_0$, the size of the shadow decreases with the incr…
Shadow curves and quasinormal modes for rotating black holes surrounded by dark matter, radiation and dust
Universitat de Barcelona（巴塞罗那大学）；Institut de Ciències del Cosmos, Universitat de Barcelona（巴塞罗那大学宇宙科学研究所）；Institució Catalana de Recerca i Estudis Avançats (ICREA)（加泰罗尼亚高级研究院）
We study shadow aspects of superentropic black holes in four dimensions. Using Hamilton-Jacobi formalism, we first get the null geodesic equations. In the celestial coordinate framework relying on fixed positions of observers, we investigate the shadow behaviors in terms of the mass and the cosmological scale variation parameters. Among others, we obtain ellipse shaped geometries contrary to usual black hole solutions. Modifying the ordinary relations describing geometrical observables, we discuss the size and the shape deformation parameters of these non-trivial geometric forms. Due to horizo…
The Event Horizon Telescope collaboration has revealed the first direct image of a black hole, as per the shadow of a Kerr black hole of general relativity. However, other Kerr-like rotating black holes of modified gravity theories cannot be ignored, and they are essential as they offer an arena in which these theories can be tested through astrophysical observation. This motivates us to investigate asymptotically de Sitter rotating black holes wherein interpreting the cosmological constant $Λ$ as the vacuum energy leads to a deformation in the vicinity of a black hole -- new Kerr--de Sitter s…
A brief illustrative discussion of the shadows of black holes at local and cosmological distances is presented. Starting from definition of the term and discussion of recent observations, we then investigate shadows at large, cosmological distances. On a cosmological scale, the size of shadow observed by comoving observer is expected to be affected by cosmic expansion. Exact analytical solution for the shadow angular size of Schwarzschild black hole in de Sitter universe was found. Additionally, an approximate method was presented, based on using angular size redshift relation. This approach i…
The shadow of a rotating black hole with nonvanishing gravitomagnetic charge has been studied. It was shown that in addition to the angular momentum of black hole the gravitomagnetic charge term deforms the shape of the black hole shadow. From the numerical results we have obtained that for a given value of the rotation parameter, the presence of a gravitomagnetic charge enlarges the shadow and reduces its deformation with respect to the spacetime without gravitomagnetic charge. Finally we have studied the capture cross section for massive particles by black hole with the nonvanishing gravitom…
Observational Constraints on Kazakov-Solodukhin Quantum-Deformed Black Holes from M87$^*$ and Sgr A$^*$ Shadows
University of KwaZulu-Natal（夸祖鲁-纳塔尔大学）；Khazar University（哈萨尔大学）；University Hassan II（哈桑二世大学）；Ibn Tofail University（伊本·图菲利大学）；Abdus Salam International Centre for Theoretical Physics（阿卜杜斯·萨拉姆国际理论物理中心）；Central Asian University（中亚大学）；University of Tashkent for Applied Sciences（塔什干应用科学大学）；Kimyo International University in Tashkent（塔什干化学国际大学）；Samarkand State University of Architecture and Construction（撒马尔罕建筑与建设国立大学）；Urgench State Pedagogical Institute（乌尔根奇国立师范学院）
We study the shadow cast by the H$D$ Schwarzschild-Tangherlini black hole, and analytically calculate the influence of extra dimensions on the shadow of a black hole. A black hole casts a shadow as an optical appearance because of its strong gravitational field which is known to be a dark zone covered by a circle for a Schwarzschild black hole. We demonstrate that the null geodesic equation can be integrated by Hamilton-Jacobi approach, which enables us to investigate the shadow cast by the H$D$ Schwarzschild-Tangherlini black holes. Interestingly, it turns out that, for fixed values of the ma…
Shadows of multi-black holes have structures distinct from the mere superposition of the shadow of a single black hole: the eyebrow-like structures outside the main shadows and the deformation of the shadows. We present analytic estimates of these structures using the static multi-black hole solution (Majumdar-Papapetrou solution). We show that the width of the eyebrow is related with the distance between the black holes and that the shadows are deformed into ellipses due to the presence of the second black holes. These results are helpful to understand qualitatively the features of the shadow…
We consider a simple model of a black hole surrounded by a toroidal structure and experiencing deformation due to magnetic field and the tidal force. For this system we construct the shadows of Preston-Poisson black hole, apply it to supermassive black hole $M87^*$ and compare it with the shadows of corresponding Schwarzschild and Kerr black holes. We find that for large deviation parameters $B$ and $E$ the shadow of Preston-Poisson black hole is clearly distinguishable from Kerr black hole provided the angular resolution of measurements is of order $μ\mathrm{arcsec}$.
We analytically investigate the shadows cast by rotating black holes in the asymptotically safe gravity (ASG) by deriving complete null geodesics and observables using the Hamilton-Jacobi equation and Carter separable method. It turns out that the apparent shape and size of the shadow depend on the ASG parameters ($ζ, γ$) in addition to other black hole parameters ($M, a$). The size of black hole shadows monotonically decreases and shadows get more distorted with increasing values of ASG parameters when compared with the Kerr black hole shadows. In turn, we use shadow observables to estimate b…
Investigating the Connection between Generalized Uncertainty Principle and Asymptotically Safe Gravity in Black Hole Signatures through Shadow and Quasinormal Modes
