Black hole images: A Review

The shape and size of black hole shadows depend on the black hole parameters and the observer inclination. For a Schwarzschild black hole, the shadow is a perfect disk for the observer with arbitrary inclination Bardeen (1973); Chandrasekhar (1984). For a rotating Kerr black hole, the shadow also presents a circular silhouette for the observer located on its rotation axis. However, for the observer in the equatorial plane, the shadow gradually becomes a “D”-shaped silhouette with increasing black hole spin Bardeen (1973); Chandrasekhar (1984). For a Konoplya-Zhidenko rotating non-Kerr black hole with an extra deformation parameter described the deviations from the Kerr metric, the “D”-shaped shadow could disappear and the special cuspy shadow could emerge in a certain range of parameter values for the equatorial observer Wang et al. (2017). This is also true for black holes with Proca hair Cunha et al. (2017a); Sengo et al. (2022). Thus, the dependence of shadows on black hole parameters could provide a potential tool to identify black holes in nature. It also triggers the further study of black hole shadows in various theories of gravity Amarilla et al. (2010); Dastan et al. (2022); Long et al. (2020); Stepanian et al. (2021); Prokopov et al. (2022); Younsi et al. (2021).

Black hole shadow also depends on the (non-)integrability of motion equation of photon travelling in the background spacetime. The completely integrable systems are such kind of dynamical systems where the number of the first integrals is equal to its degrees of freedom. Generally, in static spacetimes of black holes with spherical symmetry, such as in the Schwarzschild black hole spacetime, the dynamical system of photons is completely integrable since it possesses three independent first integrals, i.e., the energy $E$, the $z$-component of the angular momentum $L_{z}$ and the Carter constant $Q$ Carter (1968). This ensures the motion of photons is regular so that black hole shadow has the same shape as the photon sphere

surface. In these completely integrable systems, the black hole shadow can be calculated by analytical methods Bardeen (1973); Chandrasekhar (1984). However, as the dynamical system of photons is not completely integrable, the null geodesic equations are not be variable separable because there is no existence of a Carter-like constant apart from the usual two integrals of motion $E$ and $L_{z}$. This implies that the motion of photons could be chaotic and sharply affect the shadow so that its shape is different from that of the photon sphere surface Cunha et al. (2015); Vincent et al. (2016); Cunha et al. (2016); Wang et al. (2018a, b, 2020). In particular, due to chaotic lensing, there are eyebrowlike shadows with the self-similar fractal structure near the primary shadow, as shown in Fig.2. In addition, the black hole shadows in the nonintegrable cases can be only obtained by numerical simulations with the so-called “ray-tracing” codes Bohn et al. (2015); Cunha et al. (2015); Wang et al. (2018a); Riazuelo (2018).

The photon sphere plays an important role in the formation of black hole shadows. Actually, the photon sphere is composed of unstable photon circular orbits around black holes. The unstable photon circular orbits in the equatorial plane are determined by the effective potential and its derivatives Claudel et al. (2001); Virbhadra and Ellis (2002), i.e., $V_{\rm eff}(r)=0$, $V_{{\rm eff}}(r)_{,r}=0$ and $V_{{\rm eff}}(r)_{,rr}<0$. These unstable orbits can also be obtained by a geometric way with Gauss curvature and geodesic curvature in the optical geometry Qiao (2022). For a static four dimensional spacetime, its optical geometry restricted in the equatorial plane is defined by ${\rm d}t^{2}\equiv g^{\rm OP}_{ij}{\rm d}x^{i}{\rm d}x^{j}$, $i=r,\phi$. The geodesic curvature and Gauss curvature can be expressed as Qiao (2022)

The geodesic curvature $\kappa_{\rm geo}=0$ gives the radius of the circular photon orbit and the positive (negative) of Gauss curvature $\kappa_{\rm Gau}$ determines that the circular photon orbit is stable (unstable).

Fundamental photon orbits   The unstable photon circular orbits in the equatorial plane are often called light rings. Light rings also affect dynamical properties of ultracompact objects Cunha et al. (2017a). For the ultracompact objects without horizon Cunha et al. (2017b), light rings often come in pairs, one stable and the other unstable. The existence of a stable light ring always implies a spacetime instability Cunha et al. (2022). In the Schwarzschild spacetime, light rings are the only bound photon orbits. In the rotating Kerr spacetime, there are two light rings located in the equatorial plane, one for co-rotating photon and one for counter-rotating photon with respect to the black hole. Moreover, there also exist the non-planar bound photon orbits with constant $r$ and motion in $\theta$, known as spherical orbits ( see also Fig. 2 in Cunha et al. (2017a)). These spherical orbits are unstable and completely determine the Kerr black hole shadow.

Fundamental photon orbits are the generalization of light rings and spherical orbits in usual stationary and axisymmetric spacetimes Cunha et al. (2017a). The definition of fundamental photon orbits is given in Cunha et al. (2017a). According to the features of orbits, the fundamental photon orbits can be categorized as $X^{n_{r}\pm}_{n_{s}}$, where $X=\{O,C\}$, and $n_{r},n_{s}\in\mathbb{N}_{0}$.

The orbit $O$ is open and it can reach the boundary of the effective potential. The orbit $C$ is closed and it can not reach the boundary. The sign $+(-)$ denotes the even (odd) parity of the orbit under the $\mathbb{Z}_{2}$ reflection symmetry around the equatorial plane. $n_{r}$ is the number of distinct $r$ values at where the orbit crosses the equatorial plane. For the light rings lied in the equatorial plane, their $n_{r}=0$ because such special kind of orbits never cross the equatorial plane. $n_{s}$ is the number of self-intersection points of the orbit. In Fig.3, some fundamental photon orbits and their classification are illustrated in the $(r,\theta)$-plane.

By making use of the fundamental photon orbits, P. V. P. Cunha et al. Cunha et al. (2017a) explained the formation of the cuspy silhouette of a Kerr black hole with Proca hair shown in Fig.4(a). To clearly demonstrate the formation mechanism of the black hole shadow, ten fundamental photon orbits are selected out and marked by “A1-A4, B1-B3, C1-C3”. Then, the distribution of $\Delta\theta\equiv|\theta_{\text{max}}-\frac{\pi}{2}|$ and $r_{\text{peri}}$ with the impact parameter $\eta$ are presented for each fundamental photon orbit, where $\theta_{\text{max}}$ is the maximal/minimal angular coordinate at the spherical orbit and $r_{\text{peri}}$ is the perimetral radius as a spherical orbit crosses the equatorial plane. The right panel of Fig.4(b) shows the spatial trajectories of the ten fundamental photon orbits in Cartesian coordinates, which move around the black hole. The orbits A1 and C3 are the unstable prograde and retrograde light rings respectively shown as two black circles on the equatorial plane. Other

fundamental photon orbits are non-planar bound photon orbits crossing the equatorial plane. The continuum of fundamental photon orbits can be split into one stable branch (the red dotted line) and two unstable branches (the green and blue lines). The swallow-tail shape pattern related to the fundamental photon orbits in the $\eta-\Delta\theta$ plane yields a jump occurred at A4 and C1 in the $\eta-r_{\text{peri}}$ plane. The discontinuity originating from this jump, i.e., $r_{\text{peri}(\text{C1})}>r_{\text{peri}(\text{A4})}$, induces the emergence of the cuspy shadow. The unstable fundamental photon orbits (C1-B3) non-related to the shadow could be associated to a set of lensing patterns attached to the shadow edge, called “eyelashes” in Fig.4(a). In the black hole spacetimes where there exists a second pair of light rings, the fundamental photon orbits can be classified into two fully disconnected branches: the shadow related branch and the non-shadow related one. If the shadow related branch is connected, the shadow edge will be smooth with no cusp. But the eyelashes, caused by the non-shadow related unstable branch, appear to be disconnected from the shadow, forming a pixelated banana-shaped strip in the lensing image as shown in Fig.2. This feature has been dubbed “ghost shadow” in Sengo et al. (2022). This mechanism is also applied to explain the formation of the cuspy shadow in the Konoplya-Zhidenko rotating non-Kerr black hole spacetime Wang et al. (2017). It is further confirmed that the unstable fundamental photon orbits play an important role in determining the boundary of shadow and the patterns of the shadow shape. In terms of a toy model Qian et al. (2022), the feature of cuspy shadow can be derived by employing the Maxwell construction for phase transition in a two-component system. In addition, the shadows for the black holes with general parameterized metrics have been studied by using the fundamental photon orbits Li et al. (2021); Bambi (2017); Ghasemi-Nodehi et al. (2015); Kumar and Ghosh (2020); Medeiros et al. (2020); Carson and Yagi (2020). These studies could also be beneficial to test the Kerr hypothesis through black hole shadows.

Invariant phase space structures The invariant phase space structures are very important for dynamical systems because they remain invariant under the dynamics. There are several types of invariant structures including fixed points, periodic orbits and invariant manifolds. The simplest is the fixed points. These phase space structures are applied extensively to design space trajectory for various of spacecrafts Mingotti et al. (2009); Koon et al. (2001); Henon (1997); Richardson (1980). Since black hole shadow depends on the dynamics of photons in the background spacetime, the invariant phase space structures should also play an important role in the formation mechanism of the shadow. For a dynamical system of photons travelling in a curved spacetime, its fixed points can be determined by the conditions

where $q^{\mu}=(t,r,\theta,\varphi)$ and $p_{\nu}=(p_{t},p_{r},p_{\theta},p_{\varphi})$, $H$ is the Hamiltonian of the system. Actually, the light rings on the equatorial plane are the fixed points for the photon motion Grover and Wittig (2017); Cunha et al. (2017a). In the vicinity of the fixed points, one can linearize the equations (3) and obtain a matrix equation

where $\mathbf{X}=(q^{\mu},p_{\nu})$ and $J$ is the Jacobian matrix. The eigenvalues $\mu_{j}$ of the Jacobian matrix $J$ determine the local dynamical properties of the system near the fixed points (see also in Fig.LABEL:lx0). The stable and unstable invariant manifolds correspond to the cases of ${\rm Re}(\mu_{j})<0$ and ${\rm Re}(\mu_{j})>0$, respectively; while the center manifold corresponds to the case with ${\rm Re}(\mu_{j})=0$ where the eigenvalues $\mu_{j}$ are pure imaginary numbers. Due to the special properties of the invariant manifolds, there is no trajectory crossing the invariant manifolds. Points in the unstable (stable) invariant manifold move to the fixed points exponentially in backward (forward) time. In terms of Lyapunov’s central limit theorem, the eigenvalue with ${\rm Re}(\mu_{j})=0$ leads to the so-called Lyapunov orbits, which is a one-parameter family $\gamma_{\epsilon}$ of periodic orbits Mingotti et al. (2009); Koon et al. (2001); Henon (1997); Richardson (1980); Grover and Wittig (2017). These orbits $\gamma_{\epsilon}$ in the center manifold collapse into a fixed point as $\epsilon\rightarrow 0$. Similarly, the periodic orbits also have their own stable and unstable manifolds. These invariant phase space structures are also shown in Fig. 1 in Grover and Wittig (2017). Obviously, the photon spheres and other periodic orbits can be generalized to the Lyapunov orbits related to the fixed points Grover and Wittig (2017).

These concepts in dynamical systems provide a powerful theoretical foundation for understanding the formation of shadows cast by black holes. The unstable invariant manifold builds a bridge between the photon sphere and the observer because such manifold can approach the fixed points exponentially in backward time Grover and Wittig (2017); Wang et al. (2018a). Only the Lyapunov family of the spherical orbits near the unstable fixed points are responsible for generating the black hole shadow. For a Kerr black hole with scalar hair, there are three unstable light rings $\mathfrak{L}_{1}$, $\mathfrak{L}_{2}$ and $\mathfrak{L}_{3}$. However, the Lyapunov orbits associated with $\mathfrak{L}_{1}$ is non-spherical and only the orbits emanating from $\mathfrak{L}_{2}$ and $\mathfrak{L}_{3}$ are spherical Grover and Wittig (2017). The numerical simulated shadow in Fig.5

shows that the complicated and disconnected boundaries of the shadow are completely determined by the Lyapunov spherical orbits. Thus, the invariant manifolds of certain Lyapunov orbits are directly related to black hole shadows even in the case of complicated non-convex, disconnected shadows. Moreover, the curved streamlines in the unstable invariant manifolds could lead to that the shape of the black hole shadow detected by the observer at spatial infinity differs from that of the photon sphere surface Grover and Wittig (2017); Wang et al. (2018a, 2020).

To probe the matter distribution around black holes, one must simulate images of black hole models by considering different choices and select a model that could accurately represent the main features of the observed images. For the black hole M$87^{*}$, it is well known that it belongs to the class of low luminosity active galactic nuclei, and its spectral energy distribution presents features associated with emission from an optically thin and geometrically thick accretion disk ascribed to the synchrotron radiation with an observed brightness temperature in radio wavelengths in the range of $10^{9}-10^{10}K$ Event Horizon Telescope Collaboration et al. (2019a). Recently, the most salient features appearing in the EHT Collaboration images of M$87^{*}$ were reproduced with impressive fidelity and the corresponding configuration model revealed that there may exist an asymmetric bar-like structure attached to a two-temperature thin disk in the equatorial plane of the black hole Boero and Moreschi (2021). Moreover, the asymmetry in brightness is a robust indicator of the orientation of the spin axis. The simulations using different orientations of the black hole spin show that the spin direction opposite to the observed jet is favored by the asymmetric shape of the observed crescent sector.

As mentioned in the previous part, the comparison between the polarization patterns of the M87^∗ image and the viable GRMHD models reveals the existence of magnetic field near the black hole. Actually, the magnetic field can generate some features of black hole images Wang et al. (2021); Junior et al. (2021). For a rotating black hole immersed in a Melvin magnetic field Wang et al. (2021), the shadow becomes oblate for the weak magnetic field. However, in the case with the strong magnetic field, the multiple disconnected shadows emerge, including a middle oblate shadow and many striped shadows. Moreover, the novel feature in the Melvin-Kerr black hole shadow is the gray regions on both sides of the middle main shadow Wang et al. (2021), which are caused by the stable photon orbits around the stable light rings. In fact, the photons moving along the stable photon orbits are trapped and they can’t enter the black hole. Strictly, the gray regions don’t belong to the black hole shadow, but if there are no light sources in the stable photon orbit regions, the observer also see dark shadows in the gray regions Wang et al. (2021); Junior et al. (2021). The chaotic lensing arising from the magnetic field gives rise to the self-similar fractal structures in the black hole shadows. The chaotic image also occurs for the case illuminated by an accretion disk in the Kerr-Melvin black hole spacetime with a strong enough magnetic field Hou et al. (2022). These new effects in shadows could provide a new way to probe the magnetic field near black holes.

The images of black holes indicate that the supermassive black holes in the centers of galaxies are actually surrounded by plasma. Besides as a light source to illuminate black holes, plasma is a dispersive medium where the index of refraction depends on the spacetime point, the plasma frequency and the photon frequency, so the plasma changes the path of the light traveling through it and further affects the geometrical features of black hole shadows Perlick et al. (2015a); Atamurotov and Ahmedov (2015); Abdujabbarov et al. (2016); Crisnejo and Gallo (2018); Huang et al. (2018); Yan (2019); Babar et al. (2020); Matsuno (2021); Tsupko (2021); Li et al. (2022); Atamurotov et al. (2021a); Wang and Wei (2022); Badía and Eiroa (2021); Zhang et al. (2022b); Atamurotov et al. (2021b); Perlick and Tsupko (2017); Perlick et al. (2015b); Bezdekova et al. (2022). The influence of plasma on the shadows depends mainly on the ratio between the plasma frequency and the photon frequency. If the plasma frequency is smaller than the photon frequency, the shadow is not very much different from the vacuum case. However, if the plasma frequency tends to the photon frequency, the significant changes in the photon regions will lead to a drastic modification of the properties of the shadow. In the realistic case where the plasma frequency is much smaller than the photon frequency, the plasma has a decreasing effect on the size of the shadows if the plasma density is higher at the photon sphere than at the observer position. The above analyses are based on an assumption of plasma with radial power-law density. Recent study of angular Gaussian distributed plasma Zhang et al. (2022b), where the plasma is non-spherically symmetric, shows that the effect of plasma can be qualitatively explained by taking the plasma as a convex lens with the refractive index being less than 1. For the supermassive black holes at the centers of the Milky Way and the galaxy M87, which are the main targets of the current observations by the EHT, it is shown that the plasma effects start to become relevant at radio wavelengths of a few centimeters or more. However, the present and planned instruments focus on the submillimeter range, where the scattering and self-absorption have no significant effect on the emitted radiation around the black holes and the plasma effects are very small Perlick and Tsupko (2017); Perlick et al. (2015b), so a realistic observation of the plasma influence on the shadows seems unfeasible at present.

It is natural to expect to constrain black hole parameters by the using of shadows because the shape and size of shadows depend on the black hole parameters themselves. In general, since black hole shadows have complex shapes in the observer’s sky, the precise description of the shadow boundaries is crucial for measuring black hole parameters.

To fit astronomical observations, several observables were constructed by using special points on the shadow boundaries in the celestial coordinates. For the Kerr black hole, the two observables $R_{s}$ and $\delta_{s}=D_{cs}/R_{s}$ ( as shown in Fig.6) are introduced to measure the approximate size of the shadow and its deformation with respect to the reference circle Hioki and Maeda (2009), respectively. If the inclination angle is given, the values of the mass and spin of the black hole can be obtained by the precise enough measurements of $R_{s}$ and $\delta_{s}$. Recently, the length of the shadow boundary and the local curvature radius are introduced to describe the shadow boundary Wei et al. (2019a). The black hole spin and the observer inclination can be constrained by simply measuring the maximum and minimum of the curvature radius. Moreover, a topological covariant quantity is analyzed to measure and distinguish different topological structures of the shadows Wei et al. (2019b); Cunha and Herdeiro (2020). To further describe the general characterization of the shadow boundaries, a coordinate-independent formalism Abdujabbarov et al. (2015) is proposed where the shadow curves $R_{\psi}(\psi)$ are expressed in terms of Legendre polynomials $R_{\psi}=\sum\limits_{l=0}^{\infty}c_{l}P_{l}(\cos\psi)$ with the expansion coefficients $c_{l}$. The dimensionless deformation parameters $\delta_{n}$ are defined to measure the relative difference between the shadow at $\psi=0$ and at other angles $\psi=\pi/n$, $n=1,2,...k$, and $k$ is an arbitrarily positive integer. These distortions are both accurate and robust so they can also be implemented to analyse the noisy data.

Above analyses are based on an assumption that the black hole shadows are cast by a bundle of photons in parallel trajectories that originating at infinity. For a realistic black hole surrounded by an accretion disk, the shadow is imprinted on the image of the accretion flow. In principle, comparing a detailed model of the accretion disk around the black hole with astronomical observations will yield a measurement of the size and shape of the shadow. However, it is not feasible to predict the details of the brightness profile of the accretion flow image. The first reason is the incompleteness of accretion disk models, and all theoretical models are simplified by introducing some assumptions so they are impossible to be completely consistent with the real disks. The other reason is the observed variability of the emission in the disk, since the inner accretion flow is highly turbulent and variable in the real astronomical environment. Thus, it is necessary to build a procedure to analyze the observation data that focuses on directly measuring the properties of the shadow in a manner that is not seriously affected by our inability to predict the brightness profile of the rest of the image Psaltis et al. (2015). The gradient method Marr and Hildreth (1980); Canny (1983) is such kind of model-independent algorithms in image processing, which has already been applied successfully to interferometric images to quantify the properties of the turbulent structure of the interstellar magnetic field. The basic concept in this algorithm is that the magnitude of the gradient of the accretion flow image has local maxima at the locations of the steepest gradients, such as, in the case of the expected EHT images, which coincide with the edge of the back hole shadow Psaltis et al. (2015). With the obtained gradient image where the rim of the black hole shadow appears as the most discernible feature, a shadow pattern algorithm matching with the Hough/Radon transform is employed to determine the shape and size of the shadow. This algorithm not only measures the properties of the black hole shadow, but also assesses the statistical significance of the results.

The distinct features of black holes originating from deviation parameters in the alternative theory can help test the general relativity. It is shown that the shadow becomes prolate for the negative deviation parameter and becomes oblate for the positive one Broderick et al. (2014). The large deformation parameter in the Konoplya-Zhidenko rotating non-Kerr black hole yields the special cusp-shaped shadow for the equatorial observer Wang et al. (2017). The large deviation arising from the quadrupole mass moment leads to chaotic shadow and the eyeball-like shadows with the self-similar fractal structures Wang et al. (2018b). The similar features of shadows also appear in other non-Einstein theories of gravity including the quadratic degenerate higher-order scalar-tensor theories Long et al. (2020). Moreover, using the priori known estimates for the mass and distance of M87^∗ based on stellar dynamics Event Horizon Telescope Collaboration et al. (2019a, b, c, d, e, f); Gebhardt et al. (2011); Walsh et al. (2013); Kormendy and Ho (2013), the inferred size of the shadow from the horizon-scale images of the object M87^∗ Event Horizon Telescope Collaboration et al. (2019a) is found to be consistent with that predicted from general relativity for a Schwarzschild black hole within $17\%$ for a $68\%$ confidence interval. However, this measurement still admits other possibilities. The size of the black hole shadow M87^∗ can be used as a proxy to measure the deviations from Kerr metric satisfied weak-field tests Psaltis et al. (2020). For the parameterized Johannsen-Psaltis black hole, it has four lowest-order parameters and the shadow depends primarily on the parameter $\alpha_{13}$ and only weakly on spin Johannsen (2013). The 2017 EHT measurement for M87^∗ places a bound on the deviation parameter $-3.6<\alpha_{13}<5.9$ Psaltis et al. (2020). For the modified gravity bumpy Kerr metric Gair and Yunes (2011), the size of the shadow depends primarily on the parameter $\gamma_{1,2}$ and the requirement that the shadow size is consistent with the measurement of M87^∗ within $17\%$ gives a constraint on the deviation parameter $-5.0<\gamma_{1,2}<4.9$ Psaltis et al. (2020). For the Konoplya-Rezzolla-Zhidenko metric Konoplya et al. (2016), the EHT measurements results in the constraint $-1.2<\alpha_{1}<1.3$ Psaltis et al. (2020). For these parametric deviation metrics, the measurements of the shadow size lead to significant constraints on the deviation parameters that control the second post-Newtonian orders. This means that the EHT measurement of the size of a black hole leads to metric tests that are inaccessible in the weak-field tests. In general, such parametric tests cannot be connected directly to an underlying property of the alternative theory. Recently, the EHT measurements have been applied to set bounds on the physical parameters, such as, the electric charge Kocherlakota et al. (2021) and the MOG parameter in the Scalar-Tensor-Vector-Gravity Theory Kuang et al. (2022). The quality of the measurements Kocherlakota et al. (2021) is already sufficient to rule out that M87^∗ is a highly charged dilaton black hole, a Reissner-Nordström naked singularity or a Janis-Newman-Winicour naked singularity with large scalar charge. Similarly, it also excludes considerable regions of the space of parameters for the doubly-charged dilaton and the Sen black holes. Such tests are very instructive Amarilla et al. (2010); Bambi and Freese (2009); Mizuno et al. (2018); Cunha et al. (2019a) because they can shed light on which underlying theories are promising candidates and which must be discarded or modified. The constraints and tests from shadows are complementary to those imposed by observations of gravitational waves from stellar-mass sources.

Black hole shadow may also provide a way to test binary black hole. Nowadays, the gravitational-wave events detected by the LIGO-Virgo-KAGRA Collaborations Abbott et al. (2016a, b, 2017a, 2017b, 2017c) confirm the existence of binary black hole system in the universe, and the systems of binary black hole are expected to be common astrophysical systems. The shadows of the colliding between two black holes were simulated by adopting the Kastor-Traschen cosmological multiblack hole solution, which describes the collision of maximally charged black holes with a positive cosmological constant Nitta et al. (2011); Yumoto et al. (2012). Fig.7 shows the change of the shadows with time $t$ during the collision of the two black holes with equal mass. At $t=0$, the two black holes are mutually away enough and their shadows are separated. However, each shadow is a little bit elongated in the $\alpha$ direction because of the interaction between the two black holes. At $t=1.6$, the eyebrowlike shadows appear around the main shadows. The eyebrowlike shadows can be explained by a fact that light rays bypass one black hole of binary system and enter the other one. With the further increase of time, the eyebrowlike structures grow and the main shadows approach each other. Although not discernible in the figure, in fact there appear the fractal structures of the eyebrows, i.e., infinitely many thinner eyebrows at the outer region of these eyebrows as well as at the inner region of the main shadows Yumoto et al. (2012).

As time elapses, the interval between two black hole shadows becomes indefinitely narrower, and it is expected that the black hole shadows eventually merge with each other Yumoto et al. (2012). However, due to the special properties of Kastor-Traschen metric, the recent investigation also implies that there is no observer who will see the merge of black hole shadows even if the black holes coalesce into one Okabayashi et al. (2020). Another important solution of binary black hole with analytical metric form is Majumdar-Papapetrou solution, which describes the geometry of two extremally charged black holes in static equilibrium where gravitational attraction is in balance with electrostatic repulsion. The similar eyebrowlike shadows are found in the Majumdar-Papapetrou binary black hole system Shipley and Dolan (2016).

Actually, these eyebrowlike shadows with fractal structures also appear in other binary black hole systems, such as, in the double-Schwarzschild and double-Kerr black hole systems Cunha et al. (2018) in which two black holes are separated by a conical singularity. These common key features imprinted in the shadows of binary systems, such as disconnected shadows with characteristic eyebrows, open up a new analytic avenue for exploring four dimensional black hole binaries Hertog et al. (2019).

Dark matter The nature of dark matter is one of the most important open fundamental questions of physics. Dark matter is assumed to be an invisible matter, which constitutes the dominant form of matter in the universe and has feeble couplings with the common visible matter at most. Despite extensive observational data supporting its presence on a large scale, dark matter has not been directly detected by any scientific instrument. Dark matter should influence black hole shadow due to its gravitational effects. A simple spherical model consisting of a Schwarzschild black hole with mass $M$ and a homocentric spherical shell of dark matter halo with mass $\Delta M$ is applied to tentatively study the effects of dark matter on the black hole shadow Konoplya (2019). It is found that the mass of dark matter and its distance over mass distribution lead to larger radius of shadows. However, it must be pointed out that in this simple model the dark matter is unlikely to manifest itself in the shadows of galactic black holes, unless its concentration near black holes is abnormally high Konoplya (2019).

The effect of dark matter halo on black hole shadows has been studied in the spacetimes of a spherically symmetric black hole and of a rotating black hole Hou et al. (2018); Xu et al. (2018); Jusufi et al. (2020, 2019); Cunha et al. (2019b). It is shown that the structures of the black hole shadows in the cold dark matter (CDM) and scalar field dark matter (SFDM) halos are very similar to the cases of the Schwarzschild and Kerr black holes, respectively. Both dark matter models influence the shadows in a similar way and the sizes of the shadows increase with the dark matter parameter $k\equiv\rho_{c}R^{3}$, where the characteristic density $\rho_{c}$ and the radius $R$ are related to the distribution of dark matter halo in two models. In general, the influence of the dark matter on the black hole shadows is minor and only becomes significant when $k$ increases to order of magnitude of $10^{7}$ for both CDM and SFDM models Hou et al. (2018). The calculation of the angular radii of the shadows shows that the dark matter halo could influence the shadow of Sgr A^∗ at a level of order of magnitude of $10^{-3}\mu as$ and $10^{-5}\mu as$, for CDM and SFDM, respectively. However, it is out of the reach of the current astronomical instruments Hou et al. (2018). The current EHT resolution is $\sim 60\mu as$ at 230 GHz and will achieve $15\mu as$ by observing at a higher frequency of 345 GHz and adding more very long baseline interferometry (VLBI) telescopes. The space-based VLBI RadioAstron Kardashev and Khartov (2013) will be able to obtain a resolution of $1-10\mu as$. This is still at least three orders of magnitude lower than the resolution required by the CDM model. The black hole shadow has been studied for a rotating black hole solution surrounded by superfluid dark matter and baryonic matter. Using the current values for the parameters of the superfluid dark matter and baryonic density profiles for the Sgr A^∗ black hole, it is shown that the effects of the superfluid dark matter and baryonic matter on the sizes of shadows are almost negligible compared to the Kerr vacuum black hole Jusufi et al. (2020). Moreover, comparing with the dark matter, the shadow size increases considerably with the baryonic mass. This can be understood by the fact that the baryonic matter is mostly located in the galactic center. Similarly, the baryonic matter in this model yields an increase of the angular diameter of the shadow of the magnitude $10^{-5}\mu as$ for the Sgr A^∗ black hole Jusufi et al. (2020).

The axion is a hypothetical particle beyond the standard model, which is initially proposed to solve the strong CP (charge-conjugation and parity) problem Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978); Di Luzio et al. (2020). Nowadays, axionlike particles are also introduced in fundamental theories and served as an excellent dark matter candidate so there are many search experiments designed to prob axions Marsh (2016); Sikivie (2021); Raffelt (2008, 1990); Anastassopoulos et al. (2017); Payez et al. (2015). Axion cloud around a rotating black hole may be formed through the superradiance mechanism if the Compton wavelength of axion particle is at the same order of the black hole size Strafuss and Khanna (2005); Brito et al. (2015). Due to the existence of the axion cloud, the axion-electromagnetic-field coupling gives rise to that the position angles of linearly polarized photons emitted near the horizon oscillate periodically Carroll et al. (1990); Plascencia and Urbano (2018); Fujita et al. (2019); Fedderke et al. (2019). Along this line, a novel strategy of detecting axion clouds around supermassive black holes is recently proposed by using the high spatial resolution and polarimetric measurements of the EHT Chen et al. (2020).

Fig.8 presents the axion parameter space which is potentially probed by M87^∗ and Sgr A^∗ for different position angle precisions Chen et al. (2020). This method is complementary to the constraints from the black hole spin measurements through gravitational wave detections Arvanitaki et al. (2015). Since the position angle oscillation induced by the axion background does not depend on photon frequency, it is expected that polarimetric measurements at different frequencies in the future can be used to distinguish astrophysical background and to improve the sensitivity of tests of the axion superradiance scenario. Moreover, the possibility of probing ultralight axions by the circular polarization light is also studied in Shakeri and Hajkarim (2022).

Extra dimension The possible existence of extra dimensions is one of the most remarkable predictions of the string theory. The extra spatial dimension could play an important role in fundamental theories within the context of the unification of the physical forces and also in black hole physics. For the high-dimensional black holes, it is shown that the extra dimension influences the shape and size of the shadows Hertog et al. (2019); Singh and Ghosh (2018); Papnoi et al. (2014); Amir et al. (2018). Using the size and deviation from circularity of the shadow of the black hole M87^∗ observed by the EHT collaboration, the curvature radius of AdS₅ in the Randall-Sundrum brane-world scenario is bounded by an upper limit $l\lesssim 170$AU Vagnozzi and Visinelli (2019). This upper limit is far from being competitive with current $\mathcal{O}$ (mm) scale constraints from precision tests of gravity, but greatly improves the limit $l\lesssim 0.535$ Mpc obtained from GW170817 Visinelli et al. (2018). More importantly, it is an independent limit from imaging the dark shadow of M87^∗. Using a rotating black hole solution with a cosmological in the vacuum brane, the black hole shadow together with the observed data of M87^∗ also provides a upper bound for the normalized tidal charge $q<0.004$ Neves (2020a), which is the second best result for the tidal charge to date and is a little higher than the best one $q<0.003$ from a solar system test Boehmer et al. (2008). Moreover, the negative values of the tidal charge are reported to be favored with the M87^∗ and Sgr A^∗ data in the brane contexts by the using of Reissner-Nordström-type geometry Bin-Nun (2010); Zakharov (2018); Horvath and Gergely (2013) and a rotating black hole without a cosmological constant Banerjee et al. (2020).

For the case of the compactified extra dimension, the shadow of a rotating uniform black string has been studied where the extra spatial dimension is treated as a compacted circle with the circumference $l$ Tang et al. (2022). The momentum of photon arising from the fifth dimension enlarges the photon regions and the shadow of the rotating 5D black string while it has slight impact on the distortion. The angular diameter in the EHT observations of M87^∗ leads to the constraint on the length of the compact extra dimension 2.03125 mm $\lesssim l\lesssim$ 2.6 mm Tang et al. (2022). Similarly, from the observations of Sgr A^∗, the constraints 2.28070 mm $\lesssim l\lesssim$ 2.6 mm and 2.13115 mm $\lesssim l\lesssim$ 2.6 mm can be given by the upper bounds of the emission ring and the angular shadow diameter respectively Tang et al. (2022). In particular, within these bounds, the rotating 5D black string spacetime is free from the Gregory-Laflamme instability Tang et al. (2022).

Effects of the specific angular momentum $\xi_{\psi}$ of photon from the fifth dimension on black hole shadow have also been studied for a rotating squashed Kaluza-Klein black hole Long et al. (2019), which is a kind of interesting Kaluza-Klein type metrics with the special topology and asymptotical structure Wang (2006). It has squashed $S^{3}$ horizons so the black hole has a structure similar to a five-dimensional black hole in the vicinity of horizon, but behaves as the four-dimensional black holes with a constant twisted $S^{1}$ fiber in the far region. For this special black hole, the radius $R_{s}$ of the black hole image in the observer’s sky has different values for the photons with different angular momentum $\xi_{\psi}$. The real radius of the black shadow is equal to the minimum value of $R_{s_{\rm min}}$. Especially, as the black hole parameters lie in a certain special range, it is found that there is no shadow for a black hole since the minimum value $R_{s_{\rm min}}=0$ in these special cases Long et al. (2019), which is novel since it does not appear in the usual black hole spacetimes. It must be pointed out that the emergence of black hole without shadow does not mean that light rays can penetrate through the black hole. Actually, it is just because the photons near the black hole with certain range of $\xi_{\psi}$ change their propagation directions and then become far away from the black hole. The phenomenon of black hole without black shadow will vanish if there exists the further constraint on the specific angular momentum $\xi_{\psi}$ of photon from the fifth dimension. In the case where black hole shadow exists, the radius of the black hole shadow increases monotonically with the increase of extra dimension parameter in the non-rotating case. With the increasing of rotation parameter, the radius of the black hole shadow gradually becomes a monotonously decreasing function of the extra dimension parameter. With the latest observation data, the angular radii of the shadows for the supermassive black hole Sgr A^∗ at the centre of the Milky Way Galaxy and the supermassive black hole in M87 are estimated Long et al. (2019), which implies that there is a room for the theoretical model of such a rotating squashed Kaluza-Klein black hole.

Coupling between the photon and background field Analogous to the motion of charged particles in an electromagnetic field, the propagation of light rays in a spacetime is also influenced by the coupling between the photon and background field, which could leave observable effects on the black hole shadow. In the standard Einstein-Maxwell theory, there is only a quadratic term of Maxwell tensor directly related to electromagnetic field, which can be seen as an interaction between Maxwell field and metric tensor. Actually, the interactions between electromagnetic field and curvature tensor could appear naturally in quantum electrodynamics with the photon effective action originating from one-loop vacuum polarization Drummond and Hathrell (1980). Although these curvature tensor corrections appear firstly as an effective description of quantum effects, the extended theoretical models without the small coupling constant limit have been investigated for some physical motivations Hehl and Obukhov (1999); Balakin et al. (2008); Clarke et al. (2001); Bamba and Odintsov (2008).

The coupling between the photon and Weyl tensor leads to birefringence phenomenon so that the paths of light ray propagations are different for the coupled photons with different polarizations. Thus, it is natural to give rise to double shadows for a single black hole because the natural lights near the black hole can be separated into two kinds of linearly polarized light beams with mutually perpendicular polarizations Huang et al. (2016). With the increase of the coupling strength, the umbra of the black hole decreases and the penumbra increases. In the case of an equatorial thin accretion disk around the Schwarzschild black hole, the black hole image and its polarization distribution are also affected by the coupling strength Zhang et al. (2021). The observed polarized intensity in the bright region is stronger than that in the darker region. It is also noted that the effect of the coupling on the observed polarized vector is weak in general and the stronger effect appears in the bright region close to the black hole in the image plane. Moreover, for the different coupling strengths, the observed polarized patterns have a counterclockwise vortex-like distribution with a rotational symmetry as the observed inclination angle $\theta_{0}=0^{\circ}$. The rotational symmetry in polarized patterns gradually vanishes with the increase of the inclination angle. Quantum electrodynamic effects from the Euler-Heisenberg effective Lagrangian on the shadow have been studied in the black hole background Hu et al. (2021). Similarly, in this case, the birefringence effect also yields that observer sees different shadow sizes of a single black hole for different polarization lights.

The coupling between a photon and a generic vector field is also introduced to study black hole shadow Yan et al. (2020). The generic vector field is assumed to obey the symmetries possessed by the black hole and the boundary condition that the vector field vanishes at infinity. It is found that the black hole shadow in edge-on view also has different appearances for different frequencies of the observed light. This is because the coupling form alters the way that the system depends on the initial conditions. These new phenomena about the black hole shadow originating from the coupling between the photon and background vector field are not simply caused by modifications of the metric, which could help give insight into new physics Yan et al. (2020). In particular, such a kind of coupling can affect the motion of photons and phenomenologically depict a violation of equivalence principle Yan et al. (2020). Thus, it is proposed as a mechanism to test the equivalence principle by analyzing black hole shadows. Although the current observation conditions might not allow us to directly detect these novel phenomena, it is expected that the future project of the next generation EHT with other future multi-band observations Blackburn et al. (2019) as well as the related data-processing techniques could allow for tests of these new physics imprinted in the black hole shadows. Moreover, the shadow images of M87^∗ and Sgr A^∗ are recently used to constrain the parameters in the generalized uncertainty principle (GUP) Neves (2020b) and the Lorentz symmetry violation Khodadi and Lambiase (2022), respectively. Although these best upper limits are weaker than those obtained in most other physical frameworks, they are valuable for further understanding black hole images and fundamental problems in physics Vagnozzi et al. (2022); Ozel et al. (2021); Psaltis (2019).