The Observational Shadow Features of a Renormalization Group Improved Black Hole Considering Spherical Accretions

Abstract

The study of black hole shadows by considering the surrounding kinds of matter has attracted interest in recent years. In this paper, we use the ray-tracing method to study shadows and photon spheres of renormalization group improved (RGI) black holes, taking into account the different thin spherical accretion models. We find that an increase in the parameters Ω and γ, which are excited by renormalization group theory, can decrease the event horizon and the radius of the photon sphere while increasing the effective potential. For static and infalling accretions, these results indicate that black hole shadows are related to the geometry of spacetime, and are nearly unaffected by spherical accretions. However, due to the Doppler effect, the shadow in the infalling case is darker than the static one, and the intensities of the photon sphere decay more slowly from the photon sphere to infinity. In addition, the peak intensities out of the shadow increase with the parameters Ω and γ. Finally, it can be seen that the effect of Ω on the shadow is more distinct by comparing it with that of γ at the same parameter level.

1. Introduction

Black holes are special objects predicated by the theory of general relativity, and have always attracted scientists interested in explore their fascinating properties. Beginning in the 21st century, research on black holes has been carried out in many aspects and numerous important achievements have been presented [1,2,3,4,5,6,7,8]. In particular, the existence of black holes was confirmed when the Laser Interferometer Gravitational-Wave Observatory (LIGO) discovered gravitational waves [9]. Then, the Event Horizon Telescope (EHT) Collaboration first observed an image of a supermassive black hole, specifically the one in the center of the giant elliptical galaxy M87, in 2019 [10,11,12,13,14,15]. In this image, a dark disk can be observed with a bright ring around it, which are called the black hole shadow and photon ring, respectively. Black hole shadows and photon rings are important features of black holes, reflecting the spacetime properties of black holes and revealing the properties of matter around them.

A ray of light from a distant place is deflected or falls into a black hole due to its gravity. As a result, a dark region appears to observers; this area is the black hole shadow. Near the black hole, the critical state of light accumulates photons due to the gravity, and as a result a glowing ring called the photon ring can be observed. To a distant observer, a photon ring is two-dimensional. In fact, this ring should be identified as the photon sphere of the black hole. As early as 1966, Synge analyzed the shadow of a Schwarzschild black hole [16]. For a Kerr black hole, the size and displacement of the shadow are related to the mass of the black hole, and are further affected by its magnitude and inclination of spin. Thus, such a black hole may form a D-shaped shadow or even a cardioid-shaped shadow [17,18]. By considering applications of the curvature radius of a Kerr black hole shadow, Wei proposed three new methods for determining the black hole’s spin and the observer’s inclination [19]. Chen et al. found that a rotating non-Kerr black hole can cast a cusp shadow, and further discovered a pair of cusps in the shadow of a slowly rotating Kerr black hole enveloped by a thin shell of slowly rotating dark matter, which may be a fractured shadow [20,21]. In view of this, Cunha discussed the effect of Proca hair on the cusp shadow in Kerr black holes [22]. Zhang identified the relationship between the shadow and the polarization of photons coupled to the Weyl tensor. Coupled photons with different polarization directions propagate along different paths, forming a double shadow [23]. At the same time, an interesting view that has emerged recently is that in the event of a pressure singularity, the redshift z may statisfy $z\le 0.01$ with respect to the impact on the shadow of supermassive black holes at cosmological distances [24]. To date, the shadows of many black holes have been explored [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. The size and shape of a black hole’s shadow depends on the geometry of spacetime, on the position of the observer, and especially on the matter near the black hole [43,44].

In curved spacetime, Bardeen and Petterson studied the accretion disk around a Kerr black hole. If the accretion disk around the black hole is not at the equatorial plane, the Lense–Thirring effect causes structural changes in the accretion disk [45]. Moncrief studied the stability of spherical accretion around a Schwarzschild black hole [46]. Hui et al. investigated the relevant physics properties of baryon disk accretion and dark matter accretion [47]. By considering the accretions around black holes, Luminet used an IBM 7040 mainframe to obtain the first shadow image of a black hole surrounded by a thin accretion disk in 1979 [48]. In recent years, the study of accretion matter around black holes has always attracted much attention. For optically thick and thin accretion disks, it has been shown that the size of the shadow is closely related to the spacetime geometry [49,50]. In 2020, Zeng et al. studied the shadow of a black hole in a quintessence dark energy model and confirmed this fact [51]. Subsequently, the shadows of many other black holes surrounded by accretion matter have been investigated [52,53,54,55,56,57,58,59,60,61,62]. In addition, Peng discussed the photon trajectory and deflection angle of asymmetric thin-shell wormholes and provided optic observational evidence to distinguish ultra-compact objects (UCOs) from black holes [59]. In the context of interactions between the Maxwell field and Weyl tensor, a polarized image of a black hole with a thin accretion disk has been obtained [61]. Although it was found that the size and shape of the shadow can be characterized by the spacetime geometry, interestingly, it is hardly influenced by the geometrically thin spherical accretion [63]. In 2020, the shadows and photon spheres of a four-dimensional Gauss–Bonnet black hole with different spherical accretion models were studied, with the result implying that the size of the shadow is indeed irrelevant to the spherical accretion [53]. In a ‘double-logarithmic’ nonlinear electrodynamics black hole, the charge term caused by the coupling of gravity and dynamics in the nonlinear electrodynamics model increases the apparent size of the shadow and reduces the intensity of the incident light in infalling spherical accretion environments [64]. In view of this, the study of the shadow when a black hole is enveloped by the thin spherical accretions is very interesting, as this situation can present the main characteristics of spacetime through comparison with the thin disks.

It is well known that while general relativity achieves great results around black holes, it tells us that there must be a singularity or spacetime singularity at the center of a spherically symmetrical black hole. At the singularity, the curvature of spacetime or the degree of bending is infinite, the volume of matter is infinitesimal, and the density is infinite. Unfortunately, both general relativity and quantum mechanics fail in the event of a singularity, and cannot discuss its properties. The renormalization group is a kind of nonlinear transformation group appearing in classical mechanics. The motion equation of classical systems with time-independent potential defines the transformation that forms the group in phase space [65]. Following the first proposal of quantum effects, many interesting studies have emerged, notably the Hawking evaporation process. Through Hawking evaporation, a black hole may completely evaporate, leading to strange physical processes; alternatively, the curvature of the singularity may be exposed. If the evaporation is not complete, it ends when the Schwarzschild radius approaches the Planck length, meaning that semiclassical results are no longer applicable [66]. In general, this problem could presumably be solved by a theory of quantum gravity. In the renormalization group improved method, quantum gravity effects can be incorporated to obtain a new quantum gravity correction line element, where the parameter ${\Omega }$ characterizes the effect of quantum gravitational effects on the metric [66]. Particularly in RGI black holes, evaporation stops when the critical mass is reached, and thus the singularity does not appear. RGI both enriches the properties of spacetime and helps scientists to understand black holes more deeply. By introducing the renormalization group into the study of black holes, the classical singularity at the center of the black hole can be properly removed [67]. In this case, a new black hole solution is obtained, resulting in an RGI black hole [66]. The gravitational lensing features, thermodynamic properties, critical points, and dynamics of the particles around such black holes are carefully explored in [67,68,69]. However, the shadow and photon sphere of such a black hole which is surrounded by spherical accretion remain unknown, which attracts our attention and interest. In this paper, we investigate the black hole shadow and photon sphere when an RGI black hole is surrounded by thin spherical accretions, and further reveal the effects of the renormalization group parameters on the black hole shadow.

The structure of this article is as follows: In Section 2, we briefly introduce the photon orbits of RGI black holes and their effective potentials; in Section 3, we show images of black hole shadows and specific intensities under different accretion scenarios; finally, in Section 4, we provide a brief discussion and conclusion.

2. Photon Sphere and Effective Potential of RGI Black Holes

In this section, we will mainly discuss the photon sphere and effective potential of a renormalization group improved black hole. This improved black hole can be expressed as [68]

$$d{s}^2=-A\left(r\right)d{t}^2+B{\left(r\right)}^{-1}d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\phi }^2,$$

(1)

with

$$A\left(r\right)=B\left(r\right)=1-{\displaystyle \frac{2M}{r}}{\left(1+{\displaystyle \frac{{\Omega }{M}^2}{r^2}}+{\displaystyle \frac{\gamma {\Omega }{M}^3}{r^3}}\right)}^{-1}.$$

(2)

In Equation (2), $\gamma$ and ${\Omega }$ are the free parameters motivated by non-perturbative renormalization group theory, and characterize the quantum corrections [68]. When $A\left(r\right)=0$, we have the cubic equation $r^3-2M{r}^2+{\Omega }{M}^{2}r+\gamma {\Omega }{M}^3=0$. The number of horizons depends on how many roots there are, and may be none, one, or two. The discriminant of this cubic equation is $\Delta =-{\Omega }{M}^6({\Omega }-{{\Omega }}_{+})({\Omega }-{{\Omega }}_{-})$, which ${{\Omega }}_{\pm }=-\frac{9}{2}\gamma -\frac{27}{8}{\gamma }^2\pm \frac{1}{8}\sqrt{(\gamma +2){(9\gamma +2)}^3}+\frac{1}{2}$. For the discriminant, we demand $\Delta \ge 0$, which means that ${\Omega }\le {{\Omega }}_{+}$ and ${{\Omega }}_{-}<0$. Here, we introduce a new dimensionless parameter, X, and define ${\Omega }=X{{\Omega }}_{+}$. In this paper, we only consider the existence of one or two horizons, thus satisfying $0<X\le 1$ [67].

Then, we explore the motion of light around a black hole. Here, we first consider the Euler–Lagrange equation $\frac{d}{d\lambda }\left(\frac{\partial \mathcal{L}}{\partial {\dot{x}}^{\mu }}\right)=\frac{\partial \mathcal{L}}{\partial x^{\mu }}$, where $\lambda$ is the affine parameter, ${\dot{x}}^{\mu }$ is the four-velocity of a photon, and $\mathcal{L}$ is the Lagrangian. The Lagrangian can be written as $\mathcal{L}=\frac{1}{2}{g}_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }$. If the light is fixed at the equatorial plane, i.e., $\dot{\theta }=0$, $\theta =\frac{\pi }{2}$, we have

$$\mathcal{L}={\displaystyle \frac{1}{2}}\left[-A\left(r\right){\dot{t}}^2+{\displaystyle \frac{{\dot{r}}^2}{A\left(r\right)}}+r^2{\dot{\phi }}^2\right].$$

(3)

In Equation (2), there is no time t and azimuth $\phi$ in the metric equation; therefore, we can obtain two constants, i.e., the energy E and angular momentum L, as follows: $E=-\frac{\partial \mathcal{L}}{\partial \dot{t}}=\dot{t}A\left(r\right)$, $L=\frac{\partial \mathcal{L}}{\partial \dot{\phi }}=\dot{\phi }{r}^2$. Combining this with the null geodesic $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=0$, we obtain

$${\dot{r}}^2={\displaystyle \frac{1}{b^2}}-{\displaystyle \frac{A\left(r\right)}{r^2}},\dot{t}={\displaystyle \frac{1}{bA\left(r\right)}},\dot{\phi }=\pm {\displaystyle \frac{1}{r^2}},$$

(4)

where the affine parameter $\lambda$ is redefined as $\lambda /\left|L\right|$ and $b=\left|L\right|/E$ is the impact parameter of the RGI black hole. The + and − signs in Equation (4) indicate the clockwise and counterclockwise direction of light, and we can introduce the effective potential $V_{eff}$ to reexpress the $\dot{r}$, that is,

$${\dot{r}}^2+V_{eff}={\displaystyle \frac{1}{b^2}},$$

(5)

with $V_{eff}=A\left(r\right)/r^2$. At the photon sphere, the conditions $\dot{r}=0$ and $\ddot{r}=0$ should be satisfied. Thus, for a four-dimensional symmetric black hole, the radius of the photon sphere $r_p$ and the impact parameter $b_p$ satisfy

$$r_p^2=b_p^{2}A\left(r\right),2b_p^{2}A{\left(r\right)}^2=r_p^3{A}^{\prime }\left(r\right).$$

(6)

According to Equation (6), the event horizon $r_{eh}$, the radius of the photon sphere $r_p$, and the impact parameter $b_p$ are shown in the following tables. In

Table 1. The event horizon $r_{eh}$, the radius $r_p$, and the impact parameter $b_p$ at different ${\Omega }$ values when $M=1$ and $\gamma =0.01$.

	${\Omega }=0.01$	${\Omega }=0.1$	${\Omega }=0.3$	${\Omega }=0.5$	${\Omega }=0.7$	${\Omega }=0.8$	${\Omega }=0.9$	${\Omega }=0.99$
$r_{eh}$	1.99496	1.94841	1.83568	1.70503	1.54357	1.44096	1.30513	1.0160
$r_p$	2.99441	2.94296	2.82017	2.68223	2.52194	2.42919	2.32357	2.21168
$b_p$	5.19035	5.13710	5.01140	4.87296	4.71684	4.62951	4.53348	4.43673

, it can be seen that the values of $r_{eh}$, $r_p$ and $b_p$ all decrease with increasing ${\Omega }$, as $\gamma =0.01$. However, when the parameter ${\Omega }$ is unchanged, $r_{eh}$, $r_p$, and $b_p$ all decrease as $\gamma$ rises, as listed in

Table 2. The event horizon $r_{eh}$, the radius $r_p$, and the impact parameter $b_p$ at different $\gamma$ values when $M=1$ and ${\Omega }=0.6$.

	$\mathit{\gamma }=0.01$	$\mathit{\gamma }=0.1$	$\mathit{\gamma }=0.2$	$\mathit{\gamma }=0.3$	$\mathit{\gamma }=0.4$	$\mathit{\gamma }=0.5$	$\mathit{\gamma }=0.6$	$\mathit{\gamma }=0.7$
$r_{eh}$	1.62954	1.60212	1.56878	1.53148	1.48865	1.43736	1.37059	1.25628
$r_p$	2.60555	2.58515	2.5615	2.53665	2.51046	2.4827	2.45311	2.42133
$b_p$	4.79755	4.78124	4.76255	4.74319	4.7231	4.70219	4.68036	4.65746

.

In Figure 1a, it can be seen from the $V_{eff}-r$ graph that the effective potential $V_{eff}$ has a maximum value corresponding to the position of the photon sphere. At the event horizon, $V_{eff}=0$. The larger the value of ${\Omega }$, the lager the maximum value of $V_{eff}$. In Figure 1b, it is obvious that when the value of $\gamma$ changes, the curve does not change very much in comparison with the case of ${\Omega }$. Based on the $V_{eff}$, when $b<b_p$ the light falls into the black hole because there is no barrier to prevent the movement of photons. When $b=b_p$, the light rotates around the black hole repeatedly and accumulates photons. When $b>b_p$, the light is reflected off the barrier when it encounters obstacle, and can then be seen by an observer at a distance.

To observe the motion trajectory of light, we can obtain $dr/d\phi =\pm r^2\sqrt{1/b^2-A\left(r\right)/r^2}$ from Equation (4). By introducing a new parameter, $u=1/r$, we can obtain

$$R(u,b)={\displaystyle \frac{du}{d\phi }}=\sqrt{{\displaystyle \frac{1}{b^2}}-u^2\left[1-2Mu{\left(1+{\Omega }{M}^2{u}^2+\gamma {\Omega }{M}^3{u}^3\right)}^{-1}\right]}.$$

(7)

Using Equation (7), we are able to obtain light trajectory graphs with different values of ${\Omega }$ and $\gamma$, as shown in Figure 2. In Figure 2, the black hole is represented as a black disk and the dashed line around the black hole represents the radius of photon ring; which is simply the two-dimensional expression of the photon sphere. When $b<b_p$, corresponding to the black line in the figure, this means that all of the light falls into the black hole. When $b=b_p$, corresponding to the red line, this means that the light rotates around the black hole and is neither captured nor is able to escape, forming a photon ring. When $b>b_p$, corresponding to the green line, this means that the light is deflected as it approaches the black hole. In addition, we can intuitively see from Figure 2 that the size of the black hole changes more obviously as ${\Omega }$ changes, and more or less unchanged with changing $\gamma$.

3. Static Spherical Accretion and Infalling Spherical Accretion

Accretion is a process in which celestial objects accumulate around matter through gravitational attraction, and is one of the universal processes of astrophysics. As a special object, when a black hole is always surrounded by accretion matter its shadow is closely associated with the accretion matter. In view of this, in this section we study the properties of shadows and photon spheres of RGI black holes by considering two spherical accretion models in which the accretion is sufficiently thin geometrically and optically.

3.1. Shadows and Photon Spheres for the Static Spherical Accretion

Here, we study the shadow images and photon spheres of black holes surrounded by spherical accretion. For a distant observer, the observed specific intensity of spherical accretion can be derived by the following integration [70]:

$$I_{obs}={\int }_{\rho }{g}^{3}j\left({\nu }_e\right)d{l}_{prop},$$

(8)

where g is the redshift factor, which is $g={\nu }_{obs}/{\nu }_e$, ${\nu }_{obs}$ is the observed photon frequency, ${\nu }_e$ is the emitted photon frequency, $j\left({\nu }_e\right)$ is the emissivity per unit volume, and $d{l}_{prop}$ is the infinitesimal proper length. For an RGI black hole, the redshift factor g can be expressed as $g=A{\left(r\right)}^{1/2}$. Here, we assume that

$$j\left({\nu }_e\right)\propto {\displaystyle \frac{\delta ({\upsilon }_e-\upsilon )}{r^2}},$$

(9)

with the radiation of light being monochromatic; v is the rest-frame frequency of the radiation. The emission luminosity from r to infinity is $1/r^2$, which, while an approximation, is sufficient for our purposes in this paper, and $\delta$ is the delta function. In this spacetime, $d{l}_{prop}=\sqrt{A{\left(r\right)}^{-1}d{r}^2+r^{2}d{\phi }^2}$ can be changed to

$$d{l}_{prop}=\sqrt{A{\left(r\right)}^{-1}+r^2{\left({\displaystyle \frac{d\phi }{dr}}\right)}^2}dr.$$

(10)

Substituting Equation (10) into Equation (8), the specific intensity can be rewritten as

$$I_{obs}={\int }_{\rho }{r}^{-2}A{\left(r\right)}^{3/2}\sqrt{A{\left(r\right)}^{-1}+r^2{\left({\displaystyle \frac{d\phi }{dr}}\right)}^2}dr.$$

(11)

where the ray-tracing method taken into account here is used to calculate the intensity profile [63]. In this method, our code is very simple, though it is enough for the purposes of this paper. For more complicated black holes, the recently developed explicit symplectic integrator approach should be used to deal with this problem, as it is more reliable than the Runge–Kutta integrator [71,72,73,74,75,76].

Obviously, according to Equation (11), we can plot $I_{obs}$ as a function of b under different values of ${\Omega }$ and $\gamma$, as shown in Figure 3.

In Figure 3, when $b<b_p$, because the light intensity is mostly absorbed by the black hole, the intensity of light captured by the observer is minimal, although the specific intensity does not go to zero because a small amount of radiation escapes from the black hole, thereby presenting a small finite intensity. When $b=b_p$, it means that at this point the light is in a critical state and rotates around the black hole repeatedly to obtain arbitrary intensity values; consequently, the specific intensity has a maximum in this case. When $b>b_p$, the light encounters a barrier and is refracted to the distant observer. The light intensity captured by the observer at this time is mainly the refracted light. In subfigure (a) of Figure 3, it can be seen that larger values of ${\Omega }$ result in higher peak values, meaning that the corresponding b is lower. In Figure 3b, we find that the curve changes less with changes of the $\gamma$ parameter, which is another way of saying that the relationship between the specific intensity $I_{obs}$ and the impact parameter b is almost unaffected by changing $\gamma$. The corresponding observational shadow features of an RGI black hole as seen by an observer at a distance under different parameters are presented in Figure 4.

3.2. Shadows and Photon Spheres for Infalling Spherical Accretion

In real cosmic environments, the common thin accretion around the black hole is dynamic, and the accretion matter moves towards the black hole due to its gravity. Here, we consider the simple case of free-falling matter; in this case, the redshift factor of the infalling accretion can be expressed as $g=\frac{k_{\zeta }{u}_{obs}^{\zeta }}{k_{\eta }{u}_e^{\eta }}$. Moreover, $k^{\mu }\equiv {\dot{x}}_{\mu }$ is the 4-velocity of the photon and $u_{obs}^{\mu }\equiv (1,0,0,0)$ is the 4-velocity for the distant observer. In particular, $u_e^{\mu }$ is 4-velocity of radiation emitted by the accretion matter, which is

$$u_e^t=A{\left(r\right)}^{-1},u_e^r=-{[1-A\left(r\right)]}^{1/2},u_e^{\theta }=u_e^{\phi }=0.$$

(12)

According to Equation (4), $k_t$ is a constant for photons, and $k_r$ can be obtained from $k_{\alpha }{k}^{\alpha }=0$; then,

$${\displaystyle \frac{k_r}{k_t}}=\pm A{\left(r\right)}^{-1}\sqrt{A\left(r\right)[A{\left(r\right)}^{-1}-b^2{r}^{-2}]},$$

(13)

where the + and − signs in Equation (13) indicate the photons approaching and moving away from the black hole. Similar to the static spherical accretion, we use $j\left({\nu }_e\right)$ here to investigate the observed intensity; however, for the infalling case, the redshift factor g should be

$$g={\left(u_e^r{\displaystyle \frac{k_r}{k_e}}+u_e^t\right)}^{-1}.$$

(14)

The infinitesimal proper length $d{l}_{prop}$ can be expressed as

$$d{l}_{prop}=k_{\eta }{u}_e^{\eta }d\lambda ={\displaystyle \frac{k_t}{|k_r|}}{g}^{-1}dr,$$

(15)

where $\lambda$ is the affine parameter along the photon path $\rho$. Hence, the total observed flux is

$$I_{obs}\propto {\int }_{\rho }{\displaystyle \frac{k_t}{|k_r|}}{g}^3{r}^{-2}dr.$$

(16)

Obviously, the total observed flux $I_{obs}$ obtained by a distant observer is a function of the impact parameter b; therefore, the image of observed specific intensities under different RGI black hole parameters is as shown in Figure 5. From Figure 5, as the impact parameter b increases, the intensity increases gradually at first, increases dramatically at the critical state $b=b_p$, then decreases after reaching the peak value. Because the critical state can accumulate many photons, the observed intensity has a maximum. Figure 3 and Figure 5 indicate that the effects of the ${\Omega }$ and $\gamma$ parameters on the black hole shadow are very similar for both static and infalling accretion. More importantly, the size of the black hole shadow and the position of the photon sphere are unchanged for the infalling case. This further confirms that black hole shadows are features of spacetime geometry, and are not affected by spherical accretions. However, by comparison with static spherical accretion, the shadow is darker in the infalling case. This can be explained by the Doppler effect as well as by the intensities of photon sphere decaying more slowly from the photon sphere to the infinity. In Figure 6, we show the corresponding observational shadow features of an RGI black hole and the infalling accretion around it. Briefly, a black hole presents different observed features when including the renormalization group parameters ${\Omega }$ and $\gamma$, meaning that this class of black holes can be distinguished from Schwarzschild black holes.

4. Conclusions and Discussions

The spacetime characteristics of black holes can be represented by the photon spheres and shadows. In this paper, we have explored the observed characteristics of RGI black holes, including the shadow and photon sphere wrapped by accretion matter. In the background of an RGI black hole, the geometry of the black hole shadow is affected by the spacetime parameters ${\Omega }$ and $\gamma$, which are excited in non-perturbative renormalization group theory. We first investigated the effective potential $V_{eff}$ and the photon orbit of RGI black holes using the null geodesic and plotted the light ray trajectories with different parameters. Then, we studied the shadow images of black holes and the specific intensity $I_{obs}$ of two different accretion cases.

Different values of ${\Omega }$ and $\gamma$ change the effective potential $V_{eff}$, horizon $r_{eh}$, photon sphere radius $r_p$, and impact parameter $b_p$ of the black hole, with the influence of the ${\Omega }$ value being obviously more significant. In the study of light ray trajectories, we further showed that the ${\Omega }$ parameter plays a more important role. When the value of ${\Omega }$ changes, it is obvious that as the value of ${\Omega }$ increases, the radius of black hole shadow and photon sphere decreases. This means that in the background of our study, it is mainly that the ${\Omega }$ parameter that affects the spacetime structure of black holes at the same parameter level. Notably, the event horizon $r_{eh}$ value drops rapidly as the ${\Omega }$ parameter increases, as can be seen in Table 1, where the black disk represents the black hole in the light ray trajectories (Figure 2). However, the impact parameter $b_p$ has a small variation range, as shown in Figure 4. In comparison with the trajectory images, the change in the central dark area is much less obvious. Similarly, based on the images of observed intensity and black hole shadow for a distant observer, the intensity increases with the increase of ${\Omega }$, while $\gamma$ has little influence. In addition, we discovered that the radius of the photon sphere and black hole shadow is not affected by thin spherical accretions, in agreement with the findings in [63]. Due to the Doppler effect, the observed light intensities in the infalling case decay more slowly from the photon sphere to infinity compared with static spherical accretion, and the black hole shadow is significantly darker.

Finally, we used the shadow of M87 as detected by the EHT to constrain the ${\Omega }$ parameter. As described in [10,15,77], the diameter of M87’s shadow as detected by the EHT is $d_{M_{87}}\equiv \frac{D\delta }{M}\approx 11.0\pm 1.5$, where D is the distance to M87 and $\delta$ represents the size of the shadow angle of the black hole at the center of the M87 galaxy. With $1\sigma$ uncertainties, the diameter ranges from 9.5 to 12.5. When $\gamma =0.1$ is determined, it is clearly possible to obtain a rough range for ${\Omega }$ from the shadow cast by an RGI black hole. For $1\sigma$ uncertainties, the upper limit of the parameter can be obtained as ${\Omega }\approx 0.6374M$.

In this paper, we investigated the case of RGI black holes surrounded by thin spherical accretion. Interestingly, thick accretions have been studied in [50]. In this case, RGI black holes may exhibit different observational features that may further help us to find black holes. In our forthcoming research, we intend to study in the case of RGI black holes surrounded by thick accretion.