# Shadows and photon spheres with spherical accretions in the four-dimensional
Gauss-Bonnet black hole

Xiao-Xiong Zeng1,2111E-mail<EMAIL_ADDRESS>Hai-Qing
Zhang3,4222E-mail<EMAIL_ADDRESS>Hongbao Zhang5,6333E-mail:
<EMAIL_ADDRESS>

1 State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing
Jiaotong University, Chongqing 400074, China
2 Department of Mechanics, Chongqing Jiaotong University, Chongqing 400074,
China
3 Center for Gravitational Physics, Department of Space Science, Beihang
University,
Beijing 100191, China
4 International Research Institute for Multidisciplinary Science, Beihang
University,
Beijing 100191, China
5 Department of Physics, Beijing Normal University, Beijing 100875, China
6 Theoretische Natuurkunde, Vrije Universiteit Brussel, and The International
Solvay
Institutes, Pleinlaan 2, B-1050 Brussels, Belgium

###### Abstract

We investigate the shadows and photon spheres of the four-dimensional Gauss-
Bonnet black hole with the static and infalling spherical accretions. We show
that for both cases, there always exit shadows and photon spheres. The radii
of the shadows and photon spheres are independent of the profiles of accretion
for a fixed Gauss-Bonnet constant, implying that the shadow is a signature of
the spacetime geometry and it is hardly influenced by accretion. Because of
the Doppler effect, the shadows of the infalling accretion are found to be
darker than that of the static one. We also investigate the effect of the
Gauss-Bonnet constant on the shadow and photon spheres, and we find that the
larger the Gauss-Bonnet constant is, the smaller the radii of the shadow and
photon spheres will be. In particular, the observed specific intensity
increases as the Gauss-Bonnet constant grows.

##  1 Introduction

The Event Horizon Telescope (EHT) collaboration has recently obtained an ultra
high angular resolution image of the accretion flow around the supermassive
black hole in M87* [1, 2, 3, 4, 5, 6]. The image shows that there is a dark
interior with a bright ring surrounding it. The dark interior is called black
hole shadow while the bright ring is called photon ring, respectively. The
shadow of a black hole is caused by gravitational light deflection [7, 8, 9,
10, 11]. Specifically, when light emitting from the accretion passes through
the vicinity of the black hole toward the observer, its trajectory will be
deflected. The intensity of the light observed by the distant observer differs
accordingly, leading to a dark interior and bright ring. So far, the shadows
of various black holes have been investigated. It is generally known that the
shadows of spherically symmetric black holes are round and those of rotating
black holes are not precisely round but deformed.

Since the release of the image and data by EHT, its various implications have
been explored. For instance, the extra dimensions could be determined from the
shadow of M87* [12, 13], where a rotating braneworld black hole was
considered. The shadows of high-redshift supermassive black holes may serve as
the standard rulers [14], whereby the cosmological parameters can be
constrained. The black hole companion for M87* can also be constrained through
the image released by EHT [15]. Moreover, the information given by EHT can be
used to impose constraints on particle physics via the mechanism of
superradiance [16, 17]. In particular, for the vector boson, it may constrain
some of the fuzzy dark matter parameter space. In addition, dense axion cloud
can also be induced by rapidly rotating black holes through superradiance
[18].

Accretion matters are apparently important for the shadows of black holes,
since many astrophysical black holes are believed to be surrounded by
accretion matters. The first image of a black hole surrounded by an thin disk
accretion was pictured out in [19]. For a geometrically and optically thick
accretion disk [20], it was found that the mass of the disk would affect the
shadow of the black hole, and as the mass grows the shadow becomes more
prolate. In particular, by reanalyzing the trajectory of the light ray, the
shadow of a Schwarzschild black hole with both thin and thick accretion disks
have been clarified and detailed recently [21]. It was found that there
existed not only the photon ring444Note that in this paper, the photon ring is
defined by the light ray that intersects the plane of the disk three or more
times, which is different from other references such as Ref.[22]. To
distinguish them, we call the photon ring in Ref.[22] as the photon sphere in
this paper.  but also the lensing ring. The lensing ring makes a significant
contribution to the observed flux while the photon ring makes little. In
addition, the observed size of the central dark area was found to be
determined not only by the gravitational redshift but also by the emission
profile. When the accretion matter is spherically symmetric, there is also a
shadow for the black hole [23]. The location of the shadow edge is found to be
independent of the inner radius at which the accreting gas stops radiating
[22]. The size of the observed shadow can serve as a signature of the
spacetime geometry, since it is hardly influenced by the details of the
accretion. This result is different from the case in which the accretion is a
disk [21].

In this paper, we intend to investigate the shadow of a four-dimensional
Gauss-Bonnet black hole with spherical accretions [24]. The Gauss-Bonnet term
in the Lagrangian is topologically invariant in four dimensional spacetime.
Thus in order to consider the dynamical effect of Gauss-Bonnet gravity, one is
generically required to work in higher dimensions [25, 26]. Very recently,
Glavan and Lin has proposed a Gauss-Bonnet modified gravity in four dimension
by simply rescaling the Gauss-Bonnet coupling constant $\alpha\to\
\alpha/(D-4)$ and taking the limit $D\rightarrow 4$ [24]. However, as many
authors have pointed out, this theory is not well-defined with the initial
regularization scheme [27, 28, 29, 30]. Recently, the authors in [31] have
proposed a consistent theory of four dimensional Gauss-Bonnet gravity using
ADM decomposition of the spacetime. They successfully found a four dimensional
Gauss-Bonnet theory of two dynamical degrees of freedom by breaking the
temporal diffeomorphism invariance. Thus, the cosmological and black hole
solutions naively given in [24] can be accounted as exact solutions in the
theory of [31]. Our background of the black hole solution indeed also
satisfies the well-defined theory of [31]. Many other characteristics of the
four-dimensional Gauss-Bonnet black hole have been investigated, see for
instance [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].

In particular, gravitational lensing by black holes in ordinary medium and
homogeneous plasma in four-dimensional Gauss-Bonnet gravity have been studied
in [48, 49]. The shadows cast by the spherically symmetric [32, 50] and
rotating [51] four-dimensional Gauss-Bonnet black hole have also been studied.
It will be more interesting to investigate the corresponding light intensity
of the shadow, which comprises the main issue of this paper. To be more
precise, in this paper, we are interested in the spherical accretions, which
can be classified into the static and infalling one. On the one hand, we want
to explore how the Gauss-Bonnet constant affects the radii of the shadow and
photon sphere as well as the light intensity observed by a distant observer.
On the other hand, we want to explore how the dynamics of the accretion
affects the shadow of the black hole. As a result, we find that the larger the
Gauss-Bonnet constant is, the smaller the radii of the shadow and photon
sphere will be, and the larger the intensity will be. In addition, the shadow
of the infalling accretion are found to be darker than that of the static case
because of the Doppler effect.

The remainder of this paper is organized as follows. In Section 2, we
investigate the motion of the light ray near the four-dimensional Gauss-Bonnet
black hole and figure out how it is deflected. In Section 3, we investigate
the shadows and photon spheres with the static spherical accretion. To explore
whether the dynamics of the accretion will affect the shadow and photon
sphere, the accretion is supposed to be infalling in Section 4. Section 5 is
devoted to the conclusions and discussions. Throughout this paper, we set
$G=\hbar=c=k_{B}=1$.

##  2 Light deflection in the four-dimensional Gauss-Bonnet black hole

Starting from the following Einstein-Hilbert action with an additional Gauss-
Bonnet term

$I=\frac{1}{16\pi
G}\int\sqrt{-g}d^{4}x\left[R+\alpha(R_{\mu\nu\lambda\delta}R^{\mu\nu\lambda\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2})\right],$
(1)

by rescaling the Gauss-Bonne coupling constant $\alpha\to\ \alpha/(D-4)$ and
taking the limit $D\to 4$, one can obtain the four-dimensional spherically
symmetric Gauss-Bonnet black hole as

$ds^{2}=-F(r)dt^{2}+\frac{dr^{2}}{F(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (2)

with

$F(r)=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha M}{r^{3}}}\right),$
(3)

where $M$ is the mass of the black hole. Note that the same solution was
already found previously in [52] by considering the Einstein gravity with Weyl
anomaly. Solving the equation $F(r)=0$, one can obtain two solutions,

$r_{\pm}=M\pm\sqrt{M^{2}-\alpha},$ (4)

in which $r_{+}$ and $r_{-}$ correspond to the outer horizon (event horizon)
and inner horizon, respectively. In order to assure the existence of a
horizon, the Gauss-Bonnet coupling constant should be restricted in the range
$-8\leq\alpha/M^{2}\leq 1$. For the case $\alpha>0$ there are two horizons,
while for the case $\alpha<0$ there is only one single horizon.

In order to investigate the light deflection caused by the four-dimensional
Gauss-Bonnet black hole, we need to find how the light ray moves around the
black hole. As we know, the light ray satisfies the geodesic equation, which
can be encapsulated in the following Euler-Lagrange equation

$\frac{d}{d\lambda}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)=\frac{\partial\mathcal{L}}{\partial
x^{\mu}},$ (5)

with $\lambda$ the affine parameter, $\dot{x}^{\mu}$ the four-velocity of the
light ray and $\mathcal{L}$ the Lagrangian, taking the form as

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\frac{1}{2}\left(-F(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{F(r)}+r^{2}\left(\dot{\theta}^{2}+\sin^{2}{\theta}~{}\dot{\phi}^{2}\right)\right).$
(6)

As in [7, 8, 9], we focus on the light ray that moves on the equatorial plane,
i.e., $\theta=\frac{\pi}{2}$ and $\dot{\theta}=0$. In addition, since none of
the metric coefficients depends explicitly on time $t$ and azimuthal angle
$\phi$, there are two corresponding conserved quantities, $E$ and $L$.
Combining Eqs.(3), (5) and (6) together, the time, azimuthal and radial
component of the four-velocity can be expressed as

$\displaystyle\dot{t}=\frac{1}{b\left[1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha
M}{r^{3}}}\right)\right]},$ (7) $\displaystyle\dot{\phi}=\pm\frac{1}{r^{2}},$
(8)
$\displaystyle\dot{r}^{2}+\frac{1}{r^{2}}\left[1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha
M}{r^{3}}}\right)\right]=\frac{1}{b^{2}},$ (9)

where we have redefined the affine parameter $\lambda\rightarrow\lambda/|L|$,
and $b=\frac{|L|}{E}$, which is called the impact parameter. The $+$ and $-$
in Eq.(8) correspond to the light ray traveling in the counterclockwise and
clockwise along azimuthal direction, respectively. Eq.(9) can also be
rewritten as

$\dot{r}^{2}+V(r)=\frac{1}{b^{2}},$ (10)

where

$V(r)=\frac{1}{r^{2}}\left[1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha
M}{r^{3}}}\right)\right],$ (11)

is an effective potential. The conditions for the photon sphere orbit are
$\dot{r}=0$ and $\ddot{r}=0$, which can be translated to

$\displaystyle V(r)=\frac{1}{b^{2}},~{}~{}~{}V^{{}^{\prime}}(r)=0,$ (12)

where the prime ′ denotes the first derivative with respect to the radial
coordinate $r$. Based on this equation, we can obtain the radius $r_{ph}$ and
impact parameter $b_{ph}$ for the photon sphere, which are shown together with
the size of the event horizon $r_{+}$ in Table 1 for different $\alpha$. From
this table, we can see that the three parameters, i.e., $r_{ph}$, $b_{ph}$ and
$r_{+}$ all decrease as $\alpha$ increases.

Table 1. The radius $r_{ph}$, impact parameter $b_{ph}$ of the photon sphere
and the event horizon $r_{+}$ for different $\alpha$ with $M=1$.
$\alpha=-7.7$ $\alpha=-5.5$ $\alpha=-3.3$ $\alpha=-1.1$ $\alpha=0.111$
$\alpha=0.333$ $\alpha=0.555$ $\alpha=0.777$ $r_{ph}$ 4.70134 4.36744 3.95844
3.40373 2.94939 2.83932 2.71357 2.56483 $b_{ph}$ 6.7815 6.46004 6.07084
5.55557 5.15252 5.05903 4.95501 4.83671 $r_{+}$ 3.94958 3.54951 3.07364
2.44914 1.94287 1.8167 1.66708 1.47223

(a) $\alpha=-5.5$ (b) $\alpha=0.555$

Figure 1: The profiles of the effective potential for $\alpha=-5.5$ (left
panel) and $\alpha=0.555$ (right panel) with $M=1$. For both panels, Regions 2
correspond to the red lines where $V(r)=1/b_{ph}^{2}$, while Regions 1 and
Regions 3 correspond to $V(r)<1/b_{ph}^{2}$ and $V(r)>1/b_{ph}^{2}$,
respectively.

Here we would like to take $\alpha=-5.5,0.555$ as two examples with the
corresponding effective potential depicted in Figure 1. We can see that at the
event horizon, the effective potential vanishes. It increases and reaches a
maximum at the photon sphere, and then decreases as the light ray moves
outwards. As a light ray moves in the radially inward direction, the effective
potential will affect its trajectory. In Region 1, the light will encounter
the potential barrier and then be reflected back in the outward direction. In
Region 2, namely $b=b_{ph}$, the light will asymptotically approach the photon
sphere. Since the angular velocity is non-zero, it will revolve around the
black hole infinitely many times. In Region 3, the light will continue moving
in the inward direction since it does not encounter the potential barrier.
Eventually, it will enter the inside of the black hole through the event
horizon.

(a) $\alpha=-5.5$ (b) $\alpha=0.555$

Figure 2: The trajectory of the light ray for different $\alpha$ with $M=1$ in
the polar coordinates $(r,\phi)$. The red line corresponds to $b=b_{ph}$, the
black line corresponds to $b<b_{ph}$, and the green line corresponds to
$b>b_{ph}$. The spacing in impact parameter is $1/5$ for all light rays. The
black hole is shown as a solid disk and the photon orbit as a dashed red line.

Furthermore, the trajectory of the light ray can be depicted according to the
equation of motion. Combining Eqs.(8) and (9), we have

$\frac{dr}{d\phi}=\pm
r^{2}\sqrt{\frac{1}{b^{2}}-\frac{1}{r^{2}}\left[1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha
M}{r^{3}}}\right)\right]}.$ (13)

By setting $u=1/r$, we can transform (13) into

$\frac{du}{d\phi}=\sqrt{\frac{1}{b^{2}}-u^{2}\left(\frac{1-\sqrt{8\alpha
Mu^{3}+1}}{2\alpha u^{2}}+1\right)}\equiv G(u).$ (14)

From Eq.(14) we can solve $\phi$ with respect to $u$. Employing the
ParametricPlot555In many references such as in Ref.[22], the ray-tracing code
is employed to plot the trajectory of the light ray. , we can plot the
trajectory of the light ray, which is shown in Figure 2. The black, red and
green line correspond to $b<b_{ph}$, $b=b_{ph}$ and $b>b_{ph}$, respectively.
As one can see, for the case of $b<b_{ph}$, the light ray drops all the way
into the black hole, which corresponds to Region 3 in Figure 1. For the case
of $b>b_{ph}$, the light ray near the black hole is reflected back, which
corresponds to Region 1 in Figure 1. And for the case of $b<b_{ph}$, the light
ray revolves around the black hole, which corresponds to Region 2 in Figure 1.
Note that for $b>b_{ph}$, in order to plot the geodesic, we should find a
turning point, where the light ray changes its radial direction. The turning
point is determined by the equation $G(u)=0$, where $G(u)$ has been defined in
Eq.(14).

##  3 Shadows and photon spheres with rest spherical accretion

In this section, we will investigate the shadow and photon sphere of the four-
dimensional Gauss-Bonnet black hole with static spherical accretion, which is
assumed to be optically thin. To this end, we should find the specific
intensity observed by the observer $\rm(ergs^{-1}cm^{-2}str^{-1}Hz^{-1})$. The
observed specific intensity $I$ at the observed photon frequency $\nu_{\rm o}$
can be found by integrating the specific emissivity along the photon path [53,
54]

$I(\nu_{\rm o})=\int_{\gamma}g^{3}j(\nu_{\rm e})dl_{\rm prop},$ (15)

where $g=\nu_{\rm o}/\nu_{\rm e}$ is the redshift factor, $\nu_{\rm e}$ is the
photon frequency of the emitter, $dl_{\rm prop}$ is the infinitesimal proper
length, and $j(\nu_{\rm e})$ is the emissivity per unit volume measured in the
rest frame of the emitter.

(a) $\alpha=-5.5$ (b) $\alpha=0.555$

Figure 3: Profiles of the specific intensity I(b) seen by a distant observer
for a static spherical accretion. We set $M=1$ and take $\alpha=-5.5$ (left
panel), and $\alpha=0.555$ (right panel) as two examples.

In the four-dimensional Gauss-Bonnet black hole, $g=F(r)^{1/2}$. Concerning
the specific emissivity, we also assume that it is monochromatic with rest-
frame frequency $\nu_{r}$, that is

$j(\nu_{\rm e})\propto\frac{\delta(\nu_{e}-\nu_{r})}{r^{2}}.$ (16)

According to Eq.(2), the proper length measured in the rest frame of the
emitter is

$\displaystyle dl_{\rm prop}$ $\displaystyle=$
$\displaystyle\sqrt{F(r)^{-1}dr^{2}+r^{2}d\phi^{2}}$ (17) $\displaystyle=$
$\displaystyle\sqrt{F(r)^{-1}+r^{2}(\frac{d\phi}{dr})^{2}}dr,$

in which $d\phi/dr$ is given by Eq.(13). In this case, the specific intensity
observed by the infinite observer is

$I(\nu_{\rm
o})=\int_{\gamma}\frac{F(r)^{3/2}}{r^{2}}\sqrt{F(r)^{-1}+r^{2}(\frac{d\phi}{dr})^{2}}dr.$
(18)

The intensity is circularly symmetric, with the impact parameter $b$ of the
radius, which satisfies $b^{2}=x^{2}+y^{2}$.

Next we will employ Eq.(18) to investigate the shadow of the four-dimensional
Gauss-Bonnet black hole with the static spherical accretion. Note that the
intensity depends on the trajectory of the light ray, which is determined by
the impact parameter $b$. So we will investigate how the intensity varies with
respect to the impact parameter. For different $\alpha$, the numerical results
of $I(b)$ are shown in Figure 3. From this figure, we see that the intensity
increases rapidly and reaches a peak at $b_{ph}$, and then drops to lower
values with increasing $b$. This result is consistent with Figure 1 and Figure
2. Since for $b<b_{ph}$, the intensity originating from the accretion is
absorbed mostly by the black hole. And for $b=b_{ph}$, the light ray revolves
around the black hole many times, so the observed intensity is maximal. While
for $b>b_{ph}$, only the refracted light contributes to the intensity of the
observer. As $b$ becomes larger, the refracted light becomes less. The
observed intensity thus vanishes for large enough $b$. In principle, the peak
intensity at $b=b_{ph}$ should be infinite because the light ray revolves
around the black hole infinite times and collect an arbitrarily large
intensity. However, because of the numerical limitations and the logarithmic
form of the intensity, the real computed intensity never goes to infinity,
which has also been well addressed in [21, 22]. From Figure 3, we can also
observe how the Gauss-Bonnet coupling constant affects the observed intensity.
For all the $b$, the larger the coupling constant is, the stronger the
intensity will be.

(a) $\alpha=-5.5$ (b) $\alpha=0.555$

Figure 4: The black hole shadows cast by the static accretion for different
$\alpha$ with $M=1$ in the $(x,y)$ plane. The bright ring is the photon
sphere.

The shadow cast by the four-dimensional Gauss-Bonnet black hole in the $(x,y)$
plane is shown in Figure 4. We can see that outside the black hole shadow,
there is a bright ring, which is the photon sphere. The radii of the photon
spheres for different $\alpha$ have been listed in Table 1. Obviously, the
results in Figure 4 are consistent with those in Table 1. That is, the larger
the Gauss-Bonnet constant is, the smaller the radius of the photon sphere will
be.

Moreover, we can see that inside the shadow, the intensity does not go to zero
but having a small finite value. The reason is that part of the radiation has
escaped to infinity. For $r>r_{ph}$, the solid angle of the escaping rays is
$2\pi(1+\cos\theta)$, while for $r<r_{ph}$, the solid angle of the escaping
rays is $2\pi(1-\cos\theta)$, where $\theta$ is given by

$\sin\theta=\frac{r_{ph}^{3/2}}{r}\left[1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\alpha
M}{r^{3}}}\right)\right]^{1/2}.$ (19)

By only counting the escaping light rays, we have the net luminosity observed
at infinity as

$L_{\infty}=\int_{r_{+}}^{r_{ph}}4\pi r^{2}j(\nu_{\rm
e})2\pi(1-\cos\theta)dr+\int_{r_{ph}}^{\infty}4\pi r^{2}j(\nu_{\rm
e})2\pi(1+\cos\theta)dr.$ (20)

Table 2. The net luminosity of the escaping rays for different $\alpha$ with
$M=1$.
$\alpha=-7.7$ $\alpha=-5.5$ $\alpha=-3.3$ $\alpha=-1.1$ $\alpha=0.111$
$\alpha=0.333$ $\alpha=0.555$ $\alpha=0.777$ $L_{\infty}$ 0.169698 0.18729
0.215428 0.265181 0.323863 0.34006 0.359979 0.386447

For different $\alpha$, the numerical results are listed in Table 2. We can
see that the net luminosity increases with increasing $\alpha$. For the
Schwarzschild black hole, the net luminosity is found to be $L_{\infty}=0.32$
[22]. Obviously, for the positive $\alpha$, the net luminosity in the four-
dimensional Gauss-Bonnet black hole is larger than that in Schwarzschild black
hole, while for the negative $\alpha$, the net luminosity in this spacetime is
smaller than that in Schwarzschild black hole.

##  4 Shadows and photon spheres with infalling spherical accretion

In this section, we allow the optically thin accretion to move towards the
black hole. This model is thought to be more realistic than the static
accretion model since most of the accretions are mobile in the universe. For
simplicity, we assume that the accretion free falls on to the black hole from
infinity. We still employ Eq.(18) to investigate the shadow of the four-
dimensional Gauss-Bonnet black hole.

Different from the static accretion, the redshift factor for the infalling
accretion should be evaluated by

$g=\frac{k_{\beta}u^{\beta}_{\rm o}}{k_{\gamma}u^{\gamma}_{\rm e}},$ (21)

in which $k^{\mu}=\dot{x_{\mu}}$ is the four-velocity of the photon,
$u^{\mu}_{\rm o}=(1,0,0,0)$ is the 4-velocity of the distant observer, and
$u^{\mu}_{\rm e}$ is the 4-velocity of the accretion under consideration,
given by

$\displaystyle u^{t}_{\rm e}=\frac{1}{F(r)},~{}~{}~{}u^{r}_{\rm
e}=-\sqrt{1-F(r)},~{}~{}~{}u^{\theta}_{\rm e}=u^{\phi}_{\rm e}=0.$ (22)

The four-velocity of the photon has been obtained previously in Eq.(7)-
Eq.(9). We know that $k_{t}=1/b$ is a constant, and $k_{r}$ can be inferred
from $k_{\gamma}k^{\gamma}=0$. Therefore,

$\frac{k_{r}}{k_{t}}=\pm\frac{1}{F(r)}\sqrt{1-\frac{b^{2}F(r)}{r^{2}}},$ (23)

where the sign $+$($-$) correspond to the case that the photon gets close to
(away from) the black hole. With this equation, the redshift factor in Eq.(21)
can be simplified as

$g=\frac{1}{u^{t}_{e}+k_{r}/k_{e}u^{r}_{e}}.$ (24)

In addition, the proper distance can be defined as

$dl_{\rm prop}=k_{\gamma}u^{\gamma}_{\rm e}d\lambda=\frac{k_{t}}{g|k_{r}|}dr,$
(25)

where $\lambda$ is the affine parameter along the photon path $\gamma$. We
also assume that the specific emissivity is monochromatic, therefore, Eq.(16)
can be used. The intensity in Eq.(15) thus can be expressed as

$\displaystyle
I(\nu_{o})\propto\int_{\gamma}\frac{g^{3}k_{t}dr}{r^{2}|k_{r}|}.$ (26)

Now we will use Eq.(26) to investigate the shadow of the black hole
numerically. For different $\alpha$, the intensities with respect to $b$
observed by the distant observer are shown in Figure 5. Similar to the static
accretion, we find that as $b$ increases, the intensity increases first, then
reaches a maximum intensity at $b=b_{ph}$, and then drops away. We can also
observe the effect of $\alpha$ on the intensity from Figure 5. That is, the
larger the value of $b$ is, the larger the observed intensity will be.

(a) $\alpha=-5.5$ (b) $\alpha=0.555$

Figure 5: The profiles of the specific intensity I(b) seen by a distant
observer for an infalling accretion. For both cases, we set $M=1$.

(a) $\alpha=-5.5$ (b) $\alpha=0.555$

Figure 6: The shadow of the black hole cast by the infalling accretion for
different $\alpha$ with $M=1$ in the $(x,y)$ plane. The brightest ring outside
the black hole is the photon sphere.

The 2-dimensional image of the shadow and photon sphere seen by a distant
observer are shown in Figure 6. We can see that the radius of the shadow and
the location of the photon sphere are the same as those with the static
accretion. A major new feature is that in the central region, the shadow with
infalling accretion is darker than that with the static accretion, which is
well accounted for by the Doppler effect. Nearer the event horizon of the
black hole, this effect is more obvious.

It has been argued that in the universe, the accretion flows do have inward
radial velocity, and the velocity tends to be large precisely at the radii of
interest for the shadow formation. Therefore the model with radially infalling
gas is most appropriate for comparison with the image of M87*.

In order to explore how the profile of the specific emissivity affects the
shadow of the black hole, we will choose different profiles of $j(\nu_{\rm
e})$. The corresponding intensities are shown in Figure 7. From this figure,
we see clearly that the intensity in these cases has the behavior similar to
the case $j(\nu_{\rm e})=1/r^{2}$. That is, the peak is always located at
$b=b_{ph}$. The difference is that the intensity decays faster for the higher
power of $1/r$, which makes the peak more prominent. The corresponding
2-dimensional image of shadow and photon sphere are shown in Figure 8.

(a) $j(\nu_{\rm e})=1/r^{4}$ (b) $j(\nu_{\rm e})=1/r^{5}$

Figure 7: Profile of the specific intensity I(b) seen by a distant observer
with different profiles of specific emissivity. For both cases, we set $M=1$,
$\alpha=-5.5$.

(a) $j(\nu_{\rm e})=1/r^{4}$ (b) $j(\nu_{\rm e})=1/r^{5}$

Figure 8: The shadow of the black hole cast by the infalling accretion with
different profiles of specific emissivity in the $(x,y)$ plane. We set
$\alpha=-5.5$, $M=1$.

Our results in Figure 7 and Figure 8 show that although the profile of the
spherical accretion affects the intensity of the shadow, it does not affect
the characteristic geometry such as the radius of the shadow, which is
determined only by the geometry of the spacetime.

##  5 Discussions and conclusions

In this paper, we have investigated the shadows and photon spheres cast by the
four-dimensional Gauss-Bonnet black hole with spherical accretions. We first
obtain the radius of the photon sphere and critical impact parameter for
different Gauss-Bonnet constants, and find that the larger the Gauss-Bonnet
constant is, the smaller the radius of the photon sphere and critical impact
parameter will be, which is consistent with the previous results [50, 51]. It
should be noted that, a simple approximation of the radius of the shadow was
derived in Section V of [32], in which the authors mainly studied the quasi-
normal modes and stability of the four dimensional spherical Gauss-Bonnet
black hole. In the concise Section V of [32] the authors analytically obtained
a linear relation between the Gauss-Bonnet constant $\alpha$ with respect to
the radius of the shadow, in the units of event horizon. This linear relation
indeed was only satisfied in the small $\alpha$ regime. In fact, from the
numerics in Table 1 in our paper we can check that in the units of event
horizon, the radius of the shadow also increase as $\alpha$ grows. However,
for larger $\alpha$’s this linear relation will be destroyed. Therefore, our
numerical evaluations actually go beyond the simple derivations in [32].

More importantly, we obtain the specific intensity $I(\nu_{o})$ observed by a
distant observer, in which the accretion was supposed to be either static or
infalling. For both cases, we find that the specific intensity increases with
the increasing Gauss-Bonnet constant. We plot the image of the shadows in the
$(x,y)$ plane, and find that there is a bright sphere ring outside the dark
region. The interior region of the shadow with the infalling accretion turns
out to be darker than that with the static accretion, due to the Doppler
effect. We also investigate the effect of the profile of the accretion on the
shadow. As a result, it is found that although the profile will affect the
intensity of shadow, it does not affect the characteristic of the geometry
such as the radius of the photon sphere. In Ref.[22], the emission originating
from the accretion was cut-off at different locations, the size of the shadow
was found to be independent of the locations. Obviously, our result is
consistent with the observation in Ref.[22].

The EHT Collaboration has molded M87* with the Kerr black hole, and claimed
that the observation supports the General Relativity. In this paper, we did
not consider the Kerr-like black hole in the four-dimensional Gauss-Bonnet
gravity since the spherically symmetric black hole, in some case, may produce
qualitatively similar results [53]. For example, the simplified spherical
model captures the key features that also appear in state of the art general-
relativistic magnetohydrodynamics models [5], whether they are spinning or
not.

In addition, the real accretion flows are generically not spherically
symmetric. The hot accretion flow in M87* and most other galactic nuclei
consist of a geometrically thick and quasi-spherical disk. It will be more
interesting to investigate the shadow with a thick disk accretion. Recently,
Ref.[21] has investigated the shadow with a thin and thick accretion. They
reanalyzed the orbit of photon and redefined the photon ring and lensing ring,
in which the lensing ring is the light ray that intersects the plane of the
disk twice and the photon ring is that intersects the plane three or more
times. They defined a total number of orbits as $n\equiv\phi/2\pi$. In this
case, $n>3/4$ corresponds to the light ray crossing the equatorial plane at
least twice, $n>5/4$ corresponds to the light ray crossing the equatorial
plane at least three times, and $n<3/4$ corresponds to the light ray crossing
the equatorial plane only once. For the case of $\alpha=-5.5$, the trajectory
of the light ray is shown in Figure 9. Compared it with Figure 2, we see that
the photon ring is around the photon sphere, and the lensing ring is around
the photon ring. It will be interesting to investigate the shadow, photon
ring, and lensing ring with a thin or thick disk in the four-dimensional
Gauss-Bonnet black hole. We leave it as future work.

Figure 9: The behavior of light rays as a function of impact parameter $b$. We
treat $(r,\phi)$ as the Euclidean polar coordinates. The red lines, blue lines
and green lines correspond to the direct, lensed, and photon ring
trajectories, respectively. The spacing in impact parameter is
$1/5,1/100,1/1000$ in the direct, lensed, and photon ring bands. The black
hole is shown as a solid disk and the photon orbit as a dashed line. We set
$\alpha=-5.5$, $M=1$.

## Acknowledgements

We are grateful to Xiaoyi Liu for her invaluable discussions throughout this
project. This work is supported by the National Natural Science Foundation of
China (Grant Nos. 11875095, 11675015, 11675140, 11705005). In addition, H.Z.
is supported in part by FWO-Vlaanderen through the project G006918N, and by
the Vrije Universiteit Brussel through the Strategic Research Program “High-
Energy Physics”. He is also an individual FWO fellow supported by 12G3518N.

## References

  * [1] K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. 875, no. 1, L1 (2019)
  * [2] K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. II. Array and Instrumentation,” Astrophys. J. 875, no. 1, L2 (2019)
  * [3] K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. III. Data Processing and Calibration,” Astrophys. J. 875, no. 1, L3 (2019)
  * [4] K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole,” Astrophys. J. 875, no. 1, L4 (2019)
  * [5] K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring,” Astrophys. J. 875, no. 1, L5 (2019)
  * [6] K. Akiyama et al. [Event Horizon Telescope Collaboration], “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” Astrophys. J. 875, no. 1, L6 (2019)
  * [7] J. L. Synge, “The Escape of Photons from Gravitationally Intense Stars,” Mon. Not. Roy. Astron. Soc. 131, no. 3, 463 (1966). doi:10.1093/mnras/131.3.463
  * [8] J. M. Bardeen, W. H. Press and S. A. Teukolsky, “Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation,” Astrophys. J. 178, 347 (1972).
  * [9] S. E. Gralla and A. Lupsasca, “Lensing by Kerr Black Holes,” Phys. Rev. D 101, no. 4, 044031 (2020)
  * [10] A. Allahyari, M. Khodadi, S. Vagnozzi and D. F. Mota, “Magnetically charged black holes from non-linear electrodynamics and the Event Horizon Telescope,” JCAP 2002, 003 (2020)
  * [11] P. C. Li, M. Guo and B. Chen, “Shadow of a Spinning Black Hole in an Expanding Universe,” Phys. Rev. D 101, no. 8, 084041 (2020)
  * [12] I. Banerjee, S. Chakraborty and S. SenGupta, “Silhouette of M87*: A New Window to Peek into the World of Hidden Dimensions,” Phys. Rev. D 101, no. 4, 041301 (2020)
  * [13] S. Vagnozzi and L. Visinelli, “Hunting for extra dimensions in the shadow of M87*,” Phys. Rev. D 100, no. 2, 024020 (2019)
  * [14] S. Vagnozzi, C. Bambi and L. Visinelli, “Concerns regarding the use of black hole shadows as standard rulers,” Class. Quant. Grav. 37, no. 8, 087001 (2020)
  * [15] M. Safarzadeh, A. Loeb and M. Reid, “Constraining a black hole companion for M87* through imaging by the Event Horizon Telescope,” Mon. Not. Roy. Astron. Soc. 488, no. 1, L90 (2019)
  * [16] H. Davoudiasl and P. B. Denton, “Ultralight Boson Dark Matter and Event Horizon Telescope Observations of M87*,” Phys. Rev. Lett. 123, no. 2, 021102 (2019)
  * [17] R. Roy and U. A. Yajnik, “Evolution of black hole shadow in the presence of ultralight bosons,” Phys. Lett. B 803, 135284 (2020)
  * [18] Y. Chen, J. Shu, X. Xue, Q. Yuan and Y. Zhao, “Probing Axions with Event Horizon Telescope Polarimetric Measurements,” Phys. Rev. Lett. 124, no. 6, 061102 (2020)
  * [19] J.-P. Luminet, “Image of a spherical black hole with thin accretion disk,” Astron. Astrophys. 75, 228 (1979).
  * [20] P. V. P. Cunha, N. A. EirÅ¡Â®, C. A. R. Herdeiro and J. P. S. Lemos, “Lensing and shadow of a black hole surrounded by a heavy accretion disk,” JCAP 2003, no. 03, 035 (2020)
  * [21] S. E. Gralla, D. E. Holz and R. M. Wald, “Black Hole Shadows, Photon Rings, and Lensing Rings,” Phys. Rev. D 100, no. 2, 024018 (2019)
  * [22] R. Narayan, M. D. Johnson and C. F. Gammie, “The Shadow of a Spherically Accreting Black Hole,” Astrophys. J. 885, no. 2, L33 (2019)
  * [23] H. Falcke, F. Melia and E. Agol, “Viewing the shadow of the black hole at the galactic center,” Astrophys. J. 528, L13 (2000)
  * [24] D. Glavan and C. Lin, “Einstein-Gauss-Bonnet gravity in 4-dimensional space-time,” Phys. Rev. Lett. 124, no. 8, 081301 (2020)
  * [25] R. G. Cai, “Gauss-Bonnet black holes in AdS spaces,” Phys. Rev. D 65, 084014 (2002)
  * [26] D. G. Boulware and S. Deser, “String Generated Gravity Models,” Phys. Rev. Lett. 55, 2656 (1985).
  * [27] R. A. Hennigar, D. Kubiznak, R. B. Mann and C. Pollack, “On Taking the $D\to 4$ limit of Gauss-Bonnet Gravity: Theory and Solutions,” arXiv:2004.09472 [gr-qc].
  * [28] S. X. Tian and Z. H. Zhu, “Comment on ”Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime”,” arXiv:2004.09954 [gr-qc].
  * [29] F. W. Shu, “Vacua in novel 4D Einstein-Gauss-Bonnet Gravity: pathology and instability?,” arXiv:2004.09339 [gr-qc].
  * [30] S. Mahapatra, “A note on the total action of $4D$ Gauss-Bonnet theory,” arXiv:2004.09214 [gr-qc].
  * [31] K. Aoki, M. A. Gorji and S. Mukohyama, “A consistent theory of $D\rightarrow 4$ Einstein-Gauss-Bonnet gravity,” [arXiv:2005.03859 [gr-qc]].
  * [32] R. A. Konoplya and A. F. Zinhailo, “Quasinormal modes, stability and shadows of a black hole in the novel 4D Einstein-Gauss-Bonnet gravity,” arXiv:2003.01188 [gr-qc].
  * [33] R. A. Konoplya and A. F. Zinhailo, “Grey-body factors and Hawking radiation of black holes in $4D$ Einstein-Gauss-Bonnet gravity,” arXiv:2004.02248 [gr-qc].
  * [34] S. L. Li, P. Wu and H. Yu, “Stability of the Einstein Static Universe in $4D$ Gauss-Bonnet Gravity,” arXiv:2004.02080 [gr-qc].
  * [35] A. K. Mishra, “Quasinormal modes and Strong Cosmic Censorship in the novel 4D Einstein-Gauss-Bonnet gravity,” arXiv:2004.01243 [gr-qc].
  * [36] C. Y. Zhang, P. C. Li and M. Guo, “Greybody factor and power spectra of the Hawking radiation in the novel $4D$ Einstein-Gauss-Bonnet de-Sitter gravity,” arXiv:2003.13068 [hep-th].
  * [37] H. Lu and Y. Pang, “Horndeski Gravity as $D\rightarrow 4$ Limit of Gauss-Bonnet,” arXiv:2003.11552 [gr-qc].
  * [38] Y. P. Zhang, S. W. Wei and Y. X. Liu, “Spinning test particle in four-dimensional Einstein-Gauss-Bonnet Black Hole,” arXiv:2003.10960 [gr-qc].
  * [39] R. A. Konoplya and A. Zhidenko, “Black holes in the four-dimensional Einstein-Lovelock gravity,” Phys. Rev. D 101, 084038 (2020)
  * [40] P. G. S. Fernandes, “Charged Black Holes in AdS Spaces in $4D$ Einstein Gauss-Bonnet Gravity,” arXiv:2003.05491 [gr-qc].
  * [41] P. Liu, C. Niu and C. Y. Zhang, “Instability of the novel 4D charged Einstein-Gauss-Bonnet de-Sitter black hole,” arXiv:2004.10620 [gr-qc].
  * [42] S. A. Hosseini Mansoori, “Thermodynamic geometry of novel 4-D Gauss Bonnet AdS Black Hole,” [arXiv:2003.13382 [gr-qc]].
  * [43] R. Roy and S. Chakrabarti, “A study on black hole shadows in asymptotically de Sitter spacetimes,” arXiv:2003.14107 [gr-qc].
  * [44] D. V. Singh and S. Siwach, “Thermodynamics and P-v criticality of Bardeen-AdS Black Hole in 4-D Einstein-Gauss-Bonnet Gravity,” arXiv:2003.11754 [gr-qc].
  * [45] A. Aragš®n, R. BšŠcar, P. A. Gonzš¢lez and Y. Vš¢squez, “Perturbative and nonperturbative quasinormal modes of 4D Einstein-Gauss-Bonnet black holes,” arXiv:2004.05632 [gr-qc].
  * [46] R. Kumar and S. G. Ghosh, “Rotating black holes in the novel $4D$ Einstein-Gauss-Bonnet gravity,” arXiv:2003.08927 [gr-qc].
  * [47] X. X. Zeng and H. Q. Zhang, “Influence of quintessence dark energy on the shadow of black hole,” [arXiv:2007.06333 [gr-qc]].
  * [48] S. U. Islam, R. Kumar and S. G. Ghosh, “Gravitational lensing by black holes in $4D$ Einstein-Gauss-Bonnet gravity,” arXiv:2004.01038 [gr-qc].
  * [49] X. H. Jin, Y. X. Gao and D. J. Liu, “Strong gravitational lensing of a 4-dimensional Einstein-Gauss-Bonnet black hole in homogeneous plasma,” arXiv:2004.02261 [gr-qc].
  * [50] M. Guo and P. C. Li, “The innermost stable circular orbit and shadow in the novel $4D$ Einstein-Gauss-Bonnet gravity,” arXiv:2003.02523 [gr-qc].
  * [51] S. W. Wei and Y. X. Liu, “Testing the nature of Gauss-Bonnet gravity by four-dimensional rotating black hole shadow,” arXiv:2003.07769 [gr-qc].
  * [52] R. G. Cai, L. M. Cao and N. Ohta, “Black Holes in Gravity with Conformal Anomaly and Logarithmic Term in Black Hole Entropy,” JHEP 1004, 082 (2010)
  * [53] M. Jaroszynski and A. Kurpiewski, “Optics near kerr black holes: spectra of advection dominated accretion flows,” Astron. Astrophys. 326, 419 (1997)
  * [54] C. Bambi, “Can the supermassive objects at the centers of galaxies be traversable wormholes? The first test of strong gravity for mm/sub-mm very long baseline interferometry facilities,” Phys. Rev. D 87, 107501 (2013)