Observational features of deformed Schwarzschild black holes illuminated by an anisotropic accretion disk

Considering a possible deviation from the Schwarzschild geometry, the authors of [8] proposed a parameterized static spherically symmetric black hole based on the work in [50], which can be expressed in Schwarzschild coordinates as [14, 15, 16, 17, 18, 19]

in which $g_{\mu\nu}$ represents the covariant metric tensor with nonzero components

The metric potential and the deformation function are defined in geometric units as

and

respectively, with $\varepsilon_{k}$ denotes the deformation parameter. Although theoretically, the deformation function is of infinite order, the deformation parameters are constrained by the current astrophysical observations. Specifically, parameters $\varepsilon_{0}$ and $\varepsilon_{1}$ must vanish to ensure the asymptotic flatness of the deformed Schwarzschild spacetime and its rotating counterpart [8]. According to the Lunar Laser Ranging experiment, parameter $\varepsilon_{2}$ satisfies $|\varepsilon_{2}|\leq 4.6\times 10^{-4}$ and can safely be approximated as $0$ [8, 51]. Furthermore, it is conceivable that the higher-order deformation terms ($k\geq 4$) have a negligible impact on the black hole spacetime. Therefore, the present research only focuses on the case where $k=3$. Consequently, we rearrange the deformation function as

where $\varepsilon=\varepsilon_{3}$. Clearly, the degree to which the deformed Schwarzschild black hole deviates from the Schwarzschild black hole is determined by the deformation parameter $\varepsilon$. In particular, the black hole is prolate for $h(r)>0$, but becomes oblate for $h(r)<0$. When the deformation function $h(r)=0$, the metric reduces to Schwarzschild case.

It is worth mentioning that the deformation occurs inside the event horizon and is hardly measurable, but the impact of the deformation on the spacetime structure can be observed through the embedding diagram. For the sake of conciseness and coherence in the paper, we present the method for drawing the embedding diagram in the Appendix. Figure 1 depicts the profile of the deformed Schwarzschild spacetime embedding diagram for various $\varepsilon$, with all curves starting from the event horizon at $r_{\textrm{eh}}=2$²²2Several authors have confirmed that the event horizon radius of the deformed Schwarzschild black hole remains consistent with that of the Schwarzschild black hole when the deformation parameter satisfies $\varepsilon>-8$ [16, 19].. It is observed that as the deformation parameter increases, the curves converge towards the polar region, indicating a transition of the black hole morphology from oblate to prolate. Additionally, the slope of the curves correlates positively with the deformation parameter, suggesting an inverse relationship between the strength of the black hole’s gravitational field and the deformation parameter.

The motion of light rays in deformed Schwarzschild spacetime satisfies the Lagrangian formula

where the four-velocity $\dot{x}^{\mu}$ is a derivative of the generalized coordinate $x^{\mu}$ with respect to an affine parameter $\lambda$. According to the Euler-Lagrange equation $p_{\mu}=\partial\mathscr{L}/\partial\dot{x}^{\mu}$, we have photon’s covariant four-momentum

and

Here, $E_{\textrm{n}}$ and $L_{\textrm{n}}$ are two motion constants during photon propagation due to time-translational and rotational invariance, corresponding to the specific energy and specific angular momentum of the photon, respectively. Next, we would like to introduce the Hamiltonian $\mathscr{H}$ that governs the motion of particles, which can be derived via the Legendre transformation and is expressed as

where $g^{\mu\nu}$ denotes the contravariant metric tensor and satisfies $g^{\mu\nu}=1/g_{\mu\nu}$ in spherically symmetric spacetime. Note that $\mathscr{H}=0$ for null-like particles, while $\mathscr{H}=-1/2$ applies to massive particles. Then, the propagation of light rays can be calculated using the canonical equations,

Photons can be trapped in unstable circular orbits (critical photon orbits) around a black hole, where these orbits satisfy the condition of $p_{r}=\dot{p_{r}}=0$. Hence, according to eq. (2.11), we define the effective potential for photon motions, read as³³3Note that due to the spherical symmetry of the deformed Schwarzschild spacetime, the propagation of light rays within any plane can safely be extended to the entire spacetime, allowing us to set $\theta=\pi/2$ and $p_{\theta}=\dot{p_{\theta}}=0$.

The unstable circular orbits of photons correspond to the local maximum of the effective potential, and their radius, labeled as $r_{\textrm{p}}$, can be numerically calculated using the relation $\partial V_{\textrm{eff}}^{\textrm{n}}/\partial r=0$. It is worth emphasizing that the circular orbits of photons are unstable; once perturbed, photons either fall into the event horizon of the deformed Schwarzschild black hole or escape to infinity, thereby potentially forming a closed curve in the observation plane. This profile is termed as “critical curve” in [28], which serves as the boundary of the traditional concept known as the “black hole shadow”. Once the value of $r_{\textrm{p}}$ is determined, the radius of the critical curve, $b_{\textrm{p}}$, can be obtained using eq. (2.13), expressed as

Table $1$ lists the values of $r_{\textrm{p}}$ and $b_{\textrm{p}}$ of deformed Schwarzschild black holes for various $\varepsilon$. It is evident that as the deformation parameter increases, the critical photon orbit moves closer to the central object, accompanied by a shrinking of the critical curve. This trend not only confirms a negative correlation between the deformation parameter and the gravitational field of deformed Schwarzschild black holes but also provides a basis for testing the no-hair theorem of black holes. Furthermore, it is noteworthy that decreasing the mass of the deformed Schwarzschild black hole can also lead to such a trend, indicating a degeneracy between the deformation parameter and the black hole mass.

The dynamics of the accreting material play a crucial role in the simulation of black hole images. In this study, we assume that the light “illuminating” the deformed Schwarzschild black hole is provided by an optically thin, geometrically thin accretion disk located on the equatorial plane. This accretion disk consists of two parts: the Keplerian region where $r_{\textrm{s}}\geq r_{\textrm{ISCO}}$, and the plunging region where $r_{\textrm{eh}}<r_{\textrm{s}}<r_{\textrm{ISCO}}$, in which $r_{\textrm{s}}$ and $r_{\textrm{ISCO}}$ represent the radial coordinate of the accreting material and the radius of the innermost stable circular orbit (ISCO) of timelike particles, respectively. We then discuss the motion behavior of accreting particles in each region.

In the Keplerian region, particles move in a stable circular orbit around the central body, and the specific energy $E_{\textrm{k}}$ and specific angular momentum $L_{\textrm{k}}$ of the particles are read as [52]

Here, “$()^{\prime}$” denotes the first-order partial derivative with respect to $r$, and the value of $g_{tt}$ is evaluated at $r=r_{\textrm{s}}$. Notably, the motion of time-like particles is also governed by the Lagrangian (2.6) or Hamiltonian (2.11). Therefore, we can further derive the four-velocity of the particle in the Keplerian region, expressed as

and

since the particle is moving in a stable circular orbit on the equatorial plane.

In astrophysics, the viscosity of the accretion disk causes the accreting material to transfer angular momentum outward as it orbits the black hole. Consequently, particles in the Keplerian region gradually migrate closer to the ISCO. Once particles cross the ISCO, they accelerate and spiral toward the black hole until they cross the event horizon. The dynamics of particles during this process have attracted considerable scientific attention and have been confirmed to correlate with astrophysical observations [41, 53, 54]. To simplify the calculations, we adopt the strategy outlined in the literature [42, 55], specifically assuming that the specific energy $E_{\textrm{p}}$ and specific angular momentum $L_{\textrm{p}}$ of particles in the plunging region are identical to those at the ISCO. Thus, we have

where the value of $g_{tt}$ is evaluated at $r=r_{\textrm{ISCO}}$. The value of $r_{\textrm{ISCO}}$ can be determined by the iteration relation [38]

in which “$()^{\prime\prime}$” denotes the second-order partial derivative with respect to $r$. Similar to the Keplerian region, we have the four-velocity of the particle in the plunging region:

and

Here, the values of $g_{tt}$ and $g_{rr}$ are evaluated at $r=r_{\textrm{s}}$, eq. (2.26) is derived from the condition $g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$, where “$-$” indicates that the particle is spiralling towards the black hole.

The propagation of light bridges the details of the accreting material and the black hole image. Performing forward ray tracing from the accretion flow to ascertain which rays reach the observation plane is a computationally intensive process. Fortunately, by leveraging the principle of light path reversibility, we can trace light rays backward in time from the observation plane and identify those that intersect with the accretion disk and the event horizon. Rays intersecting the accretion disk contribute to the image brightness, while those intersecting the event horizon delineate the observable shadow of the black hole.

Figure 2 presents a schematic diagram of this backward ray-tracing process. In this diagram, the spacetime of the black hole and the observer’s local frame are represented by the Cartesian coordinate systems $(x^{\prime},y^{\prime},z^{\prime})$ and $(x,y,z)$, respectively, where the $z$-axis of the latter aligns with $\overline{oo^{\prime}}$ and points towards the black hole singularity. The green disk and yellow block denote the anisotropic accretion disk and the observation plane, respectively. Notably, the pixel coordinates within the observation plane satisfy $z=0$. The angle $\omega$ between $\overline{oo^{\prime}}$ and $\overline{o^{\prime}z^{\prime}}$ represents the observation inclination, ranging from $[0,\pi/2]$. The red and blue solid lines illustrate two representative rays emitted perpendicular to the observation plane. It is evident that the intersection points of the photons emitted from different pixels with the accretion disk vary. These photons can also traverse the accretion disk multiple times under the influence of the black hole’s gravitational field, accumulating their “brightness” [28]. By calculating the total specific intensity of the rays corresponding to each pixel, the black hole image can be reconstructed.

It is noteworthy that, in general, the wavevector of a photon does not hit the accretion disk perpendicularly but rather at an angle $\delta$ relative to the normal of the disk plane (see zoomed-in inset in figure 2). For an anisotropic accretion disk, it is evident that $\delta$ influences the disk’s emission and has the potential to modify the characteristics of the black hole’s image. This phenomenon is referred to as the “projection effect” of the anisotropic accretion disk on radiation. Let us denote the radiation along the normal to the disk as $\mathscr{F}$. Then, the radiation at an angle $\delta$ to the normal can be represented as $\kappa\mathscr{F}$, where $\kappa$ is a function of $\delta$. While Bambi proposed $\kappa=1/2+(3/4)\cos\delta$ in [49], we adopt a more straightforward approach in this study, setting $\kappa=\cos\delta$. The influence of $\kappa$ on the black hole image features will be examined in detail in our forthcoming work.

We now have a clear understanding of the principles of ray-tracing and the projection effect. The next task is to use a numerical integrator to trace the fate of light rays emitted from each pixel on the observation plane. Typically, this operation needs to be performed in the local frame of the black hole, which means we need to determine the initial conditions of the rays, $(x^{\mu},p_{\mu})=(t,r,\theta,\varphi,p_{t},p_{r},p_{\theta},p_{\varphi})$, in the local coordinate system $(x^{\prime},y^{\prime},z^{\prime})$. As shown in figure 2, the observer’s local coordinate system can be aligned with the black hole’s local coordinate system through multiple rotations and translations, thus yielding [56, 57, 58]

Here, $r_{\textrm{obs}}$ denotes the distance between the deformed Schwarzschild black hole and the observer. Further, we have the initial position of the photon in Schwarzschild coordinates,

Assuming that the observation distance $r_{\textrm{obs}}$ is sufficiently large (i.e., $r_{\textrm{obs}}=1000$), the observer’s spacetime can be approximated as flat. Consequently, it is reasonable to assume that the light rays are emitted perpendicularly from the observation plane. Therefore, the initial velocity of the photon measured in the $(x,y,z)$ coordinate system can be represented as $(\dot{x},\dot{y},\dot{z})=(0,0,1)$. Utilizing the differential form of eq. (3.1), we have

Substituting this result into the differential form of eq. (3.1), we obtain the initial velocity of the photon in Schwarzschild coordinates,

Recalling eqs. (2.8)-(2.10), solving for the initial conjugate momentum of the photon in the local frame of the black hole is straightforward⁴⁴4The constant of motion $p_{t}$ can be obtained from the Hamiltonian constraint (2.11)..

We now have the initial conditions of the photon corresponding to each pixel $(x,y,0)$ within the observation plane. Using a fifth- and sixth-order Runge-Kutta-Fehlberg integrator with variable step sizes to integrate eq. (2.12), we can simulate the propagation of light rays and obtain the intersection points with the accretion disk as well as the angle $\delta$.

The relative motion between the accreting particles and the observer introduces a Doppler effect, which can significantly impact the characteristics of the black hole image. For a stationary observer, with four-velocity $u^{\mu}_{\textrm{o}}=(1,0,0,0)$, the redshift factor of the light ray is given by

Here, $p_{\mu}$ is the photon’s conjugate momentum, which can be directly read from the ray-tracing; $u^{\nu}_{\textrm{e}}$ is the four-velocity of the emitting source. The four-velocity of particles in the Keplerian region and the plunging region can be calculated using eqs. (2.17)-(2.19) and eqs. (2.23)-(2.26), respectively.

Figure 3 shows the distributions of redshift factors in the direct images of deformed Schwarzschild black holes for $\varepsilon=-7$, $-3$, $0$, $3$, and $7$ from left to right. The first and second rows correspond to observation angles of $17^{\circ}$ and $85^{\circ}$, respectively. Each redshift factor is visualized using a continuous, linear color map, where red represents redshift $(g<1)$ and blue is associated with blueshift $(g>1)$. Each panel’s central black region represents the black hole’s inner shadow, with its boundary defined by the projection of the event horizon [41]. The size of the inner shadow is modulated by the deformation parameter, while its shape is affected by the observation inclination. Surrounding the inner shadow is a prominent “red ring”, composed of light emitted by particles within the plunging region, resulting in a significant redshift. At low observation inclinations, blueshift features are almost indiscernible. This is because, at smaller observation angles, the projection of the accretion disk’s motion along the line of sight is negligible, resulting in gravitational redshift becoming the dominant effect. At high observation inclinations, the blueshift and redshift regions predominantly appear on the west and east sides of the image, respectively, determined by the direction of the accretion disk’s rotation. Additionally, we find that the redshift and blueshift are suppressed with increasing deformation parameter but are enhanced with increasing observation inclination.

The impact of observation inclination on the distribution of the redshift factors is quite intuitive: at larger observation inclinations, the line of sight is closer to the disk plane, and the component of the accreting material’s velocity along the line of sight increases, thereby enhancing the Doppler effect. However, the mechanism by which the deformation parameter affects the redshift factor is not immediately apparent. We plot figure 4 to elucidate the intrinsic nature of the changes in the redshift factor due to the deformation parameter. The left and right panels respectively show the evolution of the angular velocity $(\Omega_{\textrm{k}}=\dot{\varphi}_{\textrm{k}}/\dot{t}_{\textrm{k}})$ in the Keplerian region and the absolute value of radial velocity $|\dot{r}_{\textrm{p}}|$ in the plunging region of the accreting particles as a function of radius for different $\varepsilon$. It can be confirmed that the values of $\Omega_{\textrm{k}}$ and $|\dot{r}_{\textrm{p}}|$ of the oblate deformed Schwarzschild black holes $(\varepsilon<0)$ are greater than those of the prolate deformed Schwarzschild black holes $(\varepsilon>0)$. This implies that with increasing $\varepsilon$, the angular (radial) velocity of particles in the Keplerian (plunging) region decrease, leading to a reduction in the Doppler effect on the light rays.

We also investigate the distribution of the redshift factors in the secondary images, as shown in figure 5. Similar to the results in figure 3, increasing the observation angle introduces blueshift. However, in this scenario, the influence of the deformation parameter on the redshift factor is difficult to discern. Simultaneously, we find that the central dark region in the secondary image is larger than that in the direct image. This is because the deflection angle of the light rays forming the secondary image is larger, making the image closer to the critical curve [28].

To make the simulated images of black holes observationally meaningful, it is crucial to set the radiation of the accretion disk properly. Theoretically, the radiation is related to the pressure, temperature, density, and spatial distribution of the plasma, and the relevant calculations involve general relativistic magnetohydrodynamic (GRMHD) simulations and integrating the radiative transfer equation. Undoubtedly, this is an extremely complex and tedious process. Interestingly, in recent years, some authors have derived analytical expressions for the radiation of accretion disks by fitting time-averaged images from GRMHD simulations [41, 59]. These profiles not only serve as a powerful tool for qualitatively revealing black hole observational features but also significantly simplify the computational process.

In this study, we adopt the analytical accretion disk model established in the literature [41], where the disk emission is a function of the source coordinate $r_{\textrm{s}}$ and is expressed in log-space as

Here, $p_{1}$ and $p_{2}$ are parameters associated with the observation frequency. Specifically, when $p_{1}=0$ and $p_{2}=-3/4$, the model corresponds to $86$ GHz, and when $p_{1}=-2$ and $p_{2}=-1/2$, it matches $230$ GHz. The observed intensity of the light ray corresponding to each pixel within the observation plane is then given by

where $n$ denotes the number of intersections of the light ray with the accretion disk, and $N_{\textrm{max}}$ represents the maximum number of times the light ray can pass through the disk. The coefficient $C_{n}$, referred to as the “fudge factor”, accounts for the varying contributions of high-order rings to the image brightness. Here, we set $C_{1}=1$ and $C_{n}=2/3$ for $n>1$ to ensure that model (3.7) aligns with GRMHD simulations. It is important to note that, compared to equation (16) in the literature [41], our model incorporates an additional projection effect $\kappa$ arising from the anisotropic accretion disk. As previously mentioned, we set $\kappa=\cos\delta$, where $\delta$ is obtained from the ray-tracing. When $\kappa=1$, model (3.7) degenerates into the case of an isotropic accretion disk.

In the subsequent simulations, we fix the observation distance at $r_{\textrm{obs}}=1000$ M, the field of view at $30\times 30$ M, and the image resolution at $800\times 800$ pixels. Figure 6 presents the images of deformed Schwarzschild black holes with an isotropic accretion disk ($\kappa=1$) at an observation frequency of $86$ GHz, across varying deformation parameters and inclination angles. When the observation inclination is $0^{\circ}$ (leftmost column), the black hole image exhibits central symmetry and can be viewed as a series of concentric circles with varying radii. This symmetry arises because, when the line of sight is perpendicular to the disk plane, the motion of the accretion disk has no component along the line of sight, resulting in a redshift factor $g$ that lacks a Doppler component and includes only gravitational redshift. As non-zero inclination angles are introduced, the image develops an east-west asymmetry that becomes increasingly pronounced with larger inclination angles. In particular, at high inclination angles (e.g., $\omega=50^{\circ}$, $85^{\circ}$), crescent-shaped and eyebrow-like bright spots emerge on the left side of the image, with their brightness and size increasing with $\omega$. This is attributed to the Doppler effect becoming more significant as the inclination angle increases. Additionally, It is observed that increasing the deformation parameter slightly enhances the image brightness. Furthermore, the inner shadow of the black hole is influenced by both the deformation parameter and the inclination angle. Specifically, as the inclination angle increases, the shape of the inner shadow transitions from circular to elliptical, eventually appearing arched at $\omega=85^{\circ}$; the size of the inner shadow decreases noticeably with increasing deformation parameter. It is important to emphasize that in figure 6, the critical curve is barely discernible, with its size decreasing as the deformation parameter increases. This implies that in the $86$ GHz simulations, the contributions of higher-order subrings are easily obscured by the image background, presenting certain challenge for using these images to test gravitational theories.

Figure 7 presents the images of deformed Schwarzschild black holes surrounded by an isotropic accretion disk at $230$ GHz, across different deformation parameters and observation inclinations. The effects of the inclination angle and deformation parameter on the image features are consistent with those observed in figure 6: increasing the deformation parameter reduces the sizes of both the inner shadow and the critical curve, while enhancing the image brightness; increasing the inclination angle modifies the shape of the inner shadow and accentuates the east-west asymmetry in the image. However, there are two fundamental distinctions between the images at $86$ GHz and $230$ GHz. Firstly, the brightness of the $230$ GHz images is significantly lower than that of the $86$ GHz images. Secondary, in the $230$ GHz images, a clear, thin, and complete critical curve is easily discernible, aligning with the results listed in Table 1. The reason for this difference is that the accretion disk has a higher optical depth for low-frequency light, which causes the contribution of higher-order rings to the image to be washed out [41]. In other words, simulations at higher frequencies are more likely to preserve the features of higher-order images, thereby enhancing the ability to test gravitational theories.

Next, we turn our attention to the image characteristics of deformed Schwarzschild black holes surrounded by an anisotropic accretion disk, and elucidate the impact of the accretion disk’s projection effect on black hole images. The projection effect primarily arises from the introduction of the angle $\delta$ between the light rays and the normal to the accretion disk, making it essential to examine the evolution of $\delta$ with respect to the observation coordinates $(x,y)$. Figure 8 illustrates the distribution of $\cos\delta$ in the direct images of deformed Schwarzschild black holes. Here, dark red represents lower $\cos\delta$, which is associated with larger $\delta$, while white corresponds to higher $\cos\delta$ and smaller $\delta$, indicating that light rays are less deviated from the normal to the disk plane. It is observed that $\delta$ correlates with the deformation parameter: a smaller deformation parameter corresponds to a larger $\delta$, suggesting that the convergence effect of the deformed Schwarzschild black hole on light rays is more pronounced. This finding further supports the negative correlation between the curvature of the deformed Schwarzschild spacetime and the deformation parameter. Moreover, it is apparent that light rays originating from the upper half of the observation plane typically exhibit smaller $\delta$. This is due to gravitational lensing effects, which cause these light rays to be bent by the black hole and hit the accretion disk behind the black hole at an angle nearly parallel to the disk’s normal (e.g., the red ray in figure 2). Consequently, this portion of the image satisfies $\kappa\approx 1$, and is thus almost unaffected by the projection effect. However, light rays with smaller $\delta$ are concentrated within a limited region, while the remaining portions of the observation plane generally feature rays hitting the accretion disk at larger $\delta$. This phenomenon becomes more pronounced at higher observation inclinations (e.g., the second row of figure 8 or the blue ray in figure 2). These observations indicate that the projection effect of an anisotropic accretion disk indeed has the potential to alter the observed characteristics of black hole images, particularly those of direct images, highlighting the need for further investigation.

Figures 9 and 10 present the results of re-simulations of figures 6 and 7, respectively, incorporating the projection effect from an anisotropic accretion disk. A comparison of figures 6 through 10 reveals that the projection effect significantly suppresses the image brightness, particularly affecting the intensity of bright spots and the eastern and western regions of the image. This suppression effect becomes more pronounced with increasing observation angle. This is anticipated because, as previously mentioned, light rays rarely cross the accretion disk perpendicularly during propagation, and with increasing observation inclination, the rays approach the disk nearly parallel to its plane, resulting in larger $\delta$ when hitting the disk. The projection effect diminishes the background brightness of image, leading to sharper and more discernible critical curves, as demonstrated in figures 6(s) and 9(s). This phenomenon arises because higher-order images are primarily contributed by light rays passing almost perpendicularly through the accretion disk, where the projection effect has minimal impact on their specific intensities. Conversely, light rays forming the direct image typically possess larger $\delta$, resulting in a significant reduction in their specific intensities. In summary, the projection effect exerts a more substantial influence on direct images compared to higher-order images, leading to reduced background brightness and more prominent higher-order subrings. Furthermore, it is important to note that in figures 9 and 10, the impacts of deformation parameter and observation inclination on image features such as the inner shadow, critical curve, and bright spots, as shown in figures 6 and 7, can still be accurately assessed.

To conduct a more detailed analysis of the projection effect on black hole images, we extracted the specific intensity distribution of light rays along the $x$-axis of the observation plane, as depicted in figure 11. The black solid line represents the scenario where $\kappa=1$, while the red dashed line corresponds to the case of $\kappa=\cos\delta$. When the observation inclination is $0^{\circ}$, the curve exhibits symmetry about $x=0$, with the two peaks representing the critical curve. Under these conditions, the reduction in specific intensity due to the projection effect is negligible. As the observation inclination increases, the peak positions remain constant, but the symmetry is disrupted, with the left and right peaks corresponding to Doppler blueshift and redshift, respectively. In this case, the projection effect increasingly attenuates the specific intensity, particularly in the slope and bulge near the Doppler blueshift peak. This attenuation enhances the prominence of the peaks, facilitating the precise extraction of geometric information from the critical curve. It is noteworthy that this suppression effect is more pronounced in the $86$ GHz simulations compared to the $230$ GHz images. Furthermore, it is observed that in all parameter spaces, the black solid line and the red dashed line overlap perfectly in the region where $I_{\textrm{obs}}=0$, indicating that the projection effect does not impact the size of the inner shadow.

We also extracted the specific intensity distribution along the $y$-axis, as shown in figure 12. At an observation inclination of $0^{\circ}$, the curves are identical to those in figure 11, which is expected given the central symmetry of the images at this inclination. As the observation inclination increases, a pronounced suppression of the projection effect on the specific intensity of light rays becomes evident. Additionally, in the case of $\omega=50^{\circ}$, this suppression effect is more significant in the region where $y<0$ compared to where $y>0$. This difference is due to gravitational lensing, where light rays associated with the $y>0$ region have a smaller $\delta$ during their initial intersection with the accretion disk, while those corresponding to the $y<0$ region generally exhibit a larger $\delta$ during their first crossing of the disk. It is also observed that the projection effect does not change the size of the inner shadow along the $y$-axis.

The use of black hole images for constraining parameters has become a popular area of research, with many authors pioneering methods in this field [59, 60, 61, 62, 63, 64, 65, 66, 67]. However, these constraining techniques rely on the precise extraction of the geometric information of the critical curve, a task that remains challenging given the current angular resolution of the EHT. Based on our previous findings, we can confirm that the features of the black hole’s inner shadow, such as its contour and size, are unaffected by the projection effect and are instead determined solely by the intrinsic properties of the black hole and the observation angle. Consequently, the inner shadow naturally emerges as a viable alternative tool for parameter constraints.

We propose that for spherically symmetric black holes, obtaining just two measurements of the black hole’s inner shadow can uniquely determine the observation inclination and the black hole’s parameter. Figure 13 illustrates the inner shadow of a deformed Schwarzschild black hole with an observation inclination of $\omega=80^{\circ}$ and a deformation parameter of $\varepsilon=-5$. Here, $x_{\textrm{max}}$, $x_{\textrm{min}}$, $y_{\textrm{max}}$, and $y_{\textrm{min}}$ correspond to the leftmost, rightmost, uppermost, and lowermost points of the inner shadow, respectively. The maximum horizontal distance of the inner shadow, $\Delta x$, defined mathematically as $x_{\textrm{max}}-x_{\textrm{min}}$, serves as the first observable used for parameter constraints. The second observable is the maximum vertical distance of the inner shadow, $\Delta y=y_{\textrm{max}}-y_{\textrm{min}}$. Clearly, both $\Delta x$ and $\Delta y$ are related to the deformation parameter, with $\Delta y$ being more sensitive to the observation angle than $\Delta x$. We simulated the inner shadow of deformed Schwarzschild black holes for different parameter spaces within an observation field of $6\times 6$ M with a resolution of $1200\times 1200$ pixels. The resulting contour maps of $\Delta x$ and $\Delta y$ with respect to observation angle and deformation parameter are shown in figure 14, where solid and dashed lines represent $\Delta x$ and $\Delta y$, respectively. It is evident that any given solid (dashed) line will intersect with any other given dashed (solid) line at most once. This implies that once $\Delta x$ and $\Delta y$ are measured, a unique pair of deformation parameter and observation inclination can be determined. It is also worth mentioning that we reasonably speculate that this method could be effective for any spherically symmetric black hole scenario characterized by one additional hair parameter beyond its mass.