#  The shadows and observational appearance of a noncommutative black hole
surrounded by various profiles of accretions

###### Abstract

The accretion around the black hole plays a pivotal role in the theoretical
analysis of black hole shadow, and of the black hole observation in
particular. We mainly study the shadow and observation characteristics of
noncommutative Schwarzschild black holes wrapped by three accretion models,
and then explore the influence of noncommutative parameters on the observation
appearance and spacetime geometry of black holes. When the black hole is
surrounded by an optically and geometrically thin accretion disk, it shows
that the direct emissions always dominate the total observed intensity, while
the lensing ring superimposed upon the direct emission produces a thin ring,
which improves the observation intensity of the black hole image. However, the
photon rings makes a negligible contributions to the total observed brightness
due to its exponential narrowness, although the photon ring intersects the
thin plane more than three times to pick up larger intensity. More
importantly, when the noncommutative parameters changed, the corresponding
regions and observation intensities of photon ring, lensing ring and direct
emission all change correspondingly. For optical thin spherically symmetric
accretion, we consider the static and infalling matters, respectively. We find
that the observation intensity of the two spherical accretion models increase
with the increase of noncommutative parameters. In addition, due to the
Doppler effect of the infalling movement, the shadow image of infalling
accretion is darker than that of static accretion. In a word, the different
accretion models and noncommutative parameters will lead to different shadow
images and optical appearances of noncommutative Schwarzschild black holes.

Xiao-Xiong Zeng,a,b,111E-mail<EMAIL_ADDRESS>Guo-Ping Li,c,222E-mail:
<EMAIL_ADDRESS>Ke-Jian He∗,d,333*Corresponding<EMAIL_ADDRESS>
aState Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing
Jiaotong University, Chongqing 400074, China
b Department of Mechanics, Chongqing Jiaotong University, Chongqing 400074,
China
cPhysics and Space College, China West Normal University, Nanchong 637000,
China
dCollege Of Physics, Chongqing University, Chongqing 401331, China

##  1 Introduction

The black hole, as one of the important predictions of general relativity, has
been widely concerned and discussed intensively by physicists. In recent
years, the observation of gravitational waves provides a new platform for
investigating the properties of black holes [1]. In 2015, in the first and
second observation runs, the Laser Interferometer Gravitational wave
Observatory (LIGO) detector has identified ten gravitational wave events,
which are caused by the merger of binary black holes [2]. In addition, as a
major event to observe astrophysical black holes, the Event Horizon Telescope
(EHT) collaboration released the first ultra-high angular resolution image of
the supermassive black hole M$87^{\ast}$ in 2019, which opened a new window
for the black hole observation[3, 4, 5, 6, 7, 8, 9, 10]. A striking feature of
this image is that a dark interior is surrounded by a bright ring, in which
the dark area and the bright ring are dubbed shadow and photon ring of the
black hole, respectively. The strong gravitational field near the black hole
makes the light deflect, which is widely considered to be an important
determinant of the formation of black hole shadow[11, 12, 13]. For the distant
observer, the image of black hole shadow is projected onto the local sky of
the observer due to the gravitationally lensed, which is realized by the
extremely strong gravitational field around the black hole. Hence, the study
of shadow and gravitational lensing of black hole can be used as a feasible
method to explore the gravitational field[14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30].

The study of the shadow of black hole will excavate valuable geometric
information near the black hole, and the observation of black hole shadow can
have a deeper understanding of the related properties of black hole. The
theoretical analysis of the shadow formation of Schwarzschild black hole began
with the work in[31], and then the shadow of Kerr black hole was also
discussed in[11]. Additionally in the universe, the real astrophysical black
holes are always surrounded by various accretion materials, which will play an
important role for the observation and imaging of black holes. In 1979,
Luminet simulated the first image of a black hole with an emitting accretion
thin disk, which shows that the strong gravity of the black hole makes the
light around the black hole bend [32]. Although the actual accretion flow is
generally not spherically symmetric, the simplified spherical model can give
rise to some thoughtful understandings on fundamental properties of accretion
appeared in the usual general-relativistic magnetohydrodynamics models[7, 33].
In this sense, by considering the spherically symmetric accretion, the shadow
of Schwarzschild black hole has been investigated [34]. It is shown that the
size of the observed shadow and photon sphere has nothing to do with the
distribution of accretion, which means that shadow is a feature of spacetime
geometry and is unrelated to the details of accretion process. When
considering the case that the Schwarzschild black hole is surrounded by an
optically and geometrically thin disk accretion, Wald et al. found that the
detail and emission form of accretion will affect the appearance and
observation intensity of black hole shadow[35]. Meanwhile, they found that the
black hole image was composed of the dark region and bright region with
inclusion of direct emission, lensing rings and photon rings, which provide
different degrees of help for the observation intensity. The study of black
hole shadow has ushered in a new climax, the shadow and observed appearance of
the black hole have also been widely studied in general relativity, as well as
modified gravity theory, e.g., the size and shape of shadow, the spherical
accretion flows, and the thin or thick accretion disks [36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,
79, 80, 81].

In recent years, noncommutative spacetime in gravity theories has been an
important research object [82], and it is considered to be an alternative way
to the quantum gravity[83]. In particular, considering the influence of
noncommutative on black hole is a far-reaching subject, and several methods
are proposed to implement a noncommutative spacetime in theories of gravity
[84, 85, 86, 87, 88, 89]. And then, it shows that noncommutativity can be
implemented in General Relativity by modifying the source of matter, so that
the Dirac delta function is replaced by a Gaussian distribution [90] or
alternatively by a Lorentzian distribution [91]. Several works have been
proposed to investigate in the noncommutative spacetime, e.g., the
thermodynamic property, the quasinormal modes, and the characteristics of
geometry[92, 93, 94, 95, 97, 98, 99, 96]. However, the observed intensity and
appearance characteristics of black holes wrapped by accreted matter have
rarely been investigated in the noncommutative spacetime. Therefore, our aim
in this paper is to explore the shadow and optical appearance of
noncommutative Schwarzschild black hole, the influence of noncommutative
parameters on the shadows and photon rings of black holes. Here, we mainly
consider the noncommutative Schwarzschild black holes surrounded by different
accretion models, namely the geometrically and optically thin disk accretion
and the spherically symmetric accretion. More than that, in the case of thin
disk accretion, we will explore the black hole shadow, observation appearance,
and corresponding luminosity intensity under different emission functions. In
addition, we also explore whether the change of relevant state parameters in
noncommutative background will affect the shadows, lensing rings and photon
rings of black holes.

This paper is arranged as follows: Section 2 is devoted to introduce the
noncommutative Schwarzschild black hole via Lorentzian distribution and
discuss the effective potential and photon orbits of it. In section 3, when an
optically thin disk accretion surrounded the black hole, we give the shadow
image of the black hole and its observation intensity. In section 4, we show
the images of the black hole with spherically symmetric accretions; Finally,
section 5 ends up with conclusions and discussions.

##  2 The effective potential and photon orbit of the noncommutative
Schwarzschild black hole

In this section, we aim to deduce the photon motion near the noncommutative
Schwarzschild black hole, so as to obtain the characteristics of the ray
trajectories near the black hole. We consider that the mass density of a
static, spherically symmetric, particle-like gravitational source is no longer
a function distribution, but given by a Lorentzian distribution as follows
[91],

$\rho_{\theta}=\frac{\sqrt{\theta}M}{\pi^{3/2}\left(\pi\theta+r^{2}\right)^{2}}.$
(1)

Here, $\theta$ is the strength of noncommutativity of spacetime and $M$ is the
total mass diffused throughout a region with linear size $\sqrt{\theta}$. For
the smeared matter distribution, we can further obtain [92],

$\mathcal{M}_{\theta}=\int_{0}^{r}\rho_{\theta}(r)4\pi
r^{2}dr=\frac{2M}{\pi}\left(\tan^{-1}(\frac{r}{\sqrt{\pi\theta}})-\frac{\sqrt{\pi\theta}r}{\pi\theta+r^{2}}\right)=-\frac{4\sqrt{\theta}M}{\sqrt{\pi}r}+M+\mathcal{O}(\theta^{3/2}).$
(2)

In this way, the noncommutative Schwarzschild black hole metric is given by

$ds^{2}=-A(r)dt^{2}+\frac{1}{A(r)}dr^{2}+r^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right),$ (3)

and

$A(r)=1-\frac{2\mathcal{M}_{\theta}}{r}=1-\frac{2M}{r}+\frac{8\sqrt{\theta}M}{\sqrt{\pi}r^{2}}+\mathcal{O}(\theta^{3/2}).$
(4)

The event horizon of this line element can be got at $A(r)=0$, and the larger
root corresponds to the event horizon of the black hole and the smaller root
is so called Cauchy (or inner) horizon, which is

$r_{e}=M+\frac{\sqrt{M^{2}\pi-8\sqrt{\pi}\sqrt{\theta}M}}{\sqrt{\pi}},$ (5)

and

$r_{c}=M-\frac{\sqrt{M^{2}\pi-8\sqrt{\pi}\sqrt{\theta}M}}{\sqrt{\pi}}.$ (6)

In the next step, we study the motion of photons near the black hole. In this
case, the motion of photon is described by the Euler-Lagrange equation in the
background (3), given by

$\frac{d}{d\tau}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)=\frac{\partial\mathcal{L}}{\partial
x^{\mu}}.$ (7)

Here, $\tau$ is the affine parameter and $\dot{x}^{\mu}$ represents the four-
velocity of light ray. In the context of static spacetime, we can get the
Lagrangian $\mathcal{L}$ of photons, that is,

$\mathcal{L}=\frac{1}{2}\mathit{g}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\frac{1}{2}\left(-A(r)\dot{t}^{2}+A(r)^{-1}\dot{r}^{2}+r^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}\right)\right).$ (8)

The spherical symmetry allows us to choose a particular plane in spacetime,
which for convenience is chosen to be the equatorial plane with
$\vartheta=\frac{\pi}{2}$ [100, 31, 11]. Obviously, the Lagrangian is
independent of both coordinates $t$ and $\varphi$, and we have two conserved
quantities

$\displaystyle
E=-\frac{\partial\mathcal{L}}{\partial\dot{t}}=A(r)\dot{t},\quad
J=\frac{\partial\mathcal{L}}{\partial\dot{\varphi}}=r^{2}\dot{\varphi}.$ (9)

Here, $E$ and $J$ are the energy and angular momentum of photons,
respectively. The constraint on
$\mathit{g}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$ (null geodesics) for the
line element therefore reads as

$\displaystyle-A(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{A(r)}+r^{2}\dot{\varphi}^{2}=0.$
(10)

Further, using the equations (9) and (10) we get

$\dot{r}^{2}=E^{2}-\frac{J^{2}}{r^{2}}A(r).$ (11)

At this point, the effective potential $V_{eff}(r)$ is used to represent the
radial geodesic write as follows

$\dot{r}^{2}+V_{eff}(r)=\frac{1}{b},$ (12)

and

$V_{eff}(r)=\frac{1}{r^{2}}(1-\frac{2M}{r}+\frac{8\sqrt{\theta}M}{\sqrt{\pi}r^{2}}).$
(13)

In which, $b$ is the impact parameter $b=L/E$, it is defined as the vertical
distance between the geodesic line and the parallel line passing through the
origin. In addition, the affine parameter $\tau$ has been replaced by
$\frac{\tau}{\left|J\right|}$. In this spacetime, there exists one null
geodesic on the equatorial plane, which is circular. This is essentially the
photon sphere, projected on the equatorial plane, yielding a circle, known as
photon circular orbit. For the spherically symmetric black hole with a photon
sphere, the position of the maximum of the potential corresponds to the
stability threshold for the circular null geodesic around the structure
$b_{c}$. When $b>b_{c}$, the existence of $dr/d\varphi=0$, that is, there is a
turning point for the null geodesic, and if there is no inflection point, the
light will be captured by the black hole. For the critical case $b=b_{c}$, the
photon is in the critical state of being captured or escaping from the black
hole. At this time, the photon will move to an unstable circular orbit near
the black hole, and the corresponding surface of the circular orbit is the
surface of the photon sphere. At the same time, the motions of the light ray
on the sphere plane should have a definite value

$\dot{r}=0,\quad\ddot{r}=0.$ (14)

Therefore, we have

$V_{eff}(r)=\frac{1}{b^{2}},~{}~{}~{}V_{eff}^{{}^{\prime}}(r)=0.$ (15)

Based on this equation, the effective potential of the black holes exhibits a
maximum for the photon sphere radius $r_{p}$ corresponding to the real and the
positive. Therefore, the radius $r_{p}$ and impact parameter $b_{c}$ of photon
sphere satisfy

$\displaystyle r_{p}{}^{2}-b_{c}^{2}A(r)=0,\quad
2b_{c}^{2}A(r)^{2}-r_{p}{}^{3}A^{\prime}(r)=0.$ (16)

Because of the change of correlation parameters, the spacetime structure will
be different, which means that the motion behavior of photon is different.
Therefore, we give a table of various important physical quantities under the
different strength of noncommutativity $\theta$ in this spacetime, which is
presented in Table 1.

Table 1: The values of the relevant physical quantity for different state parameter $\theta$ which $M=1$. Parameter | $\theta=0.001$ | $\theta=0.005$ | $\theta=0.01$ | $\theta=0.02$ | $\theta=0.03$ | $\theta=0.04$ | $\theta=0.049$
---|---|---|---|---|---|---|---
$r_{e}$ | 1.92589 | 1.82513 | 1.74071 | 1.60141 | 1.46716 | 1.31192 | 1.02984
$r_{p}$ | 2.90162 | 2.76952 | 2.66073 | 2.48667 | 2.32854 | 2.16678 | 2.00178
$b_{c}$ | 5.06897 | 4.89998 | 4.76285 | 4.54807 | 4.35966 | 4.17583 | 4.00178

When the value of noncommutative parameter $\theta$ increases, it is easy to
see that the radius of the event horizon $r_{e}$ and the radius of the photon
sphere $r_{p}$ decrease, and the impact parameter $b_{c}$ of photon sphere
also decreases. As a cross validation of this result, we can compare the
results in Table 1 with those in Schwarzschild spacetime. We know that in
Schwarzschild spacetime, the radius of the photon sphere is $r_{p}\sim 3M$
(the Schwarzschild spacetime ). From Table 1, it goes to show that the radius
of the photon sphere is smaller than that in the Schwarzschild spacetime, and
the difference is more obvious with the increase of the strength of
noncommutativity $\theta$. Concurrently, taking $\theta=0.01$ and
$\theta=0.049$ as examples, we show the effective potential $V_{eff}$ (or
impact parameter $b$) versus the radius $r$ in Figure 1.

Figure 1:  The profiles of the effective potential $V_{eff}$ and $r$ for
$\theta=0.01$ (left panel) and $\theta=0.049$ (right panel) with $M=1$.

When $V_{eff}=0$, it corresponds to the position of event horizon. With the
increase of the radius $r$, the effective potential increases, and the maximum
value is reached in the position of the photon sphere, i.e $r\sim b_{c}$.
After that, the effective potential begins to decrease with the increase of
the radius $r$, i.e $r>b_{c}$. In general, when a light ray moves in the
radially inward direction, the light ray will show different motion behavior.
The reflection occurs when light rays hit the potential barrier generated by
the effective potential, which corresponds to Region 1 ($b>b_{c}$). And, when
the impact parameter of the light ray is close to the radius of the photon
sphere, the light ray will rotate around the black hole, which corresponds to
Region 2 ($b=b_{c}$). If there are no obstacles to affect the behavior of
light rays, it will fall into the black hole, which corresponds to Region 3
($b<b_{c}$). After that, we are going to introduce a parameter $u_{0}=1/r$ so
that we can rewrite the equation (12) as

$\displaystyle\Phi(u_{0})=\frac{du_{0}}{d\varphi}=\sqrt{\frac{1}{b^{2}}-u_{0}^{2}\left(1-2Mu_{0}+\frac{8\sqrt{\theta}M}{\sqrt{\pi}u_{0}^{-2}}\right)}.$
(17)

By solving the above equation numerically and with the help of the ray-tracing
model, we can obtain the behavior of the geodesic lines for different values
of the impact parameter $b$. In Figure 2, taking $\theta=0.01$ (left panel)
and $\theta=0.049$ (right panel) as two examples, we show the ray trajectories
near the black hole.

Figure 2:  The trajectory of the light ray in the polar coordinates
$(r,\varphi)$ with different noncommutative parameter values, where
$\theta=0.01$ (left panel) and $\theta=0.049$ (right panel).

In Figure 2, we have a black disk representing the limit of the event horizon,
and the red dotted circle represents the location of the photon sphere. We
show the change of trajectory of light ray under different impact parameter
$b$, and here we set the distance between different impact parameter $b$ to
$1/5$. When $b<b_{c}$ (the black line in Figure 2), light will eventually be
sucked into the black hole, and can not be captured by the observer at
infinity. When $b>b_{c}$ (green line in Figure 2), the light deflects so that
the light can reach the observer at an infinite distance. Especially in the
critical case of $b=b_{c}$ (the red line in Figure 2), the light will revolve
around the black hole. Moreover, the green line, red line and black line in
Figure 2 correspond to region 1, region 2 and region 3 in Figure 1,
respectively. In addition, due to the change of noncommutative parameter, the
size of black hole shadow and the binding ability to photons will change
accordingly, which also leads to the different deflection degree of each ray
path in Figure 2.

##  3 Shadows, photon rings and lensing rings with thin disk accretions

It is well known that there are always many accretion materials around black
holes in the real universe. Nonetheless, considering many factors involved in
the analysis of black hole shadow in the noncommutative spacetime, such as
potential geometric background, geometry, optics and emission of accretion
disk model, it is useful to consider some simplified assumptions in order to
study the observational characteristics of black hole shadow. In this section,
we consider an optically and geometrically thin disk around the equatorial
plane of the black hole in noncommutative spacetime, and the observed light
intensity is mainly provided by the thin disk.

###  3.1 Direct emission, Lensed ring and photon ring

The light ray trajectories near the black hole are an important basis for
studying the observational appearance of a black hole. Meanwhile, in order to
obtain the optical appearance of the noncommutative black hole through the
ray-tracing procedure, we define the total number of orbits generated by a
beam of light from the light source to the observer as the change of azimuthal
angle, that is,

$n(b)=\frac{\varphi}{2\pi},$ (18)

which is a function of impact parameter $b$. The number of orbits obviously
depends on how close the impact parameters $b$ are to the critical parameters
$b_{c}$. In addition, it also depends on the geometry of noncommutative
spacetime. In the previous study of Wald et al.[35], they found that the
number of photon orbits can be divided into three regions, namely

  * •

$n<3/4$ : Direct emission, which means that the trajectories of light rays
will intersect with the equatorial plane only once.

  * •

$3/4<n<5/4$: Lensing ring, which means that the trajectories of light rays
will intersect with the equatorial plane at least twice.

  * •

$n>5/4$: Photon ring, which means the trajectories of light rays will
intersect with the equatorial plane at least three times.

It is worth mentioning that we use $M=1$ as the unit, and the numerical
regions of the direct emission, lensing ring and photon ring related to impact
parameter $b$ are given under different noncommutative parameter values
$\theta$, as shown in Table 2.

Table 2: Under different noncommutative parameters, the region of direct emission ring, lensing ring and photon ring is related to the impact parameters $b$ for the case $M=1$. Parameter | $\theta=0.01$ | $\theta=0.03$ | $\theta=0.049$
---|---|---|---
Direct emission | $b<4.55053$ | $b<4.08502$ | $b<3.45242$
$n<3/4$ | $b>5.80841$ | $b>5.50959$ | $b>5.28931$
Lensing ring | $4.55053<b<4.75069$ | $4.08502<b<4.33838$ | $3.45242<b<3.92753$
$3/4<n<5/4$ | $4.80392<b<5.80841$ | $4.41739<b<5.50959$ | $4.09183<b<5.28931$
Photon ring | $4.75069<b<4.80392$ | $4.33838<b<4.41739$ | $3.92753<b<4.09183$
$n>5/4$ |  |  |

Further, we want to show the direct emission, lensing ring and photon ring
with different noncommutative parameters more intuitively. Therefore, we take
different values of $\theta$ and show their comparative results in Figure 3.
Meanwhile, the corresponding photon trajectories of the noncommutative
spacetime in the polar coordinates ($b$, $\varphi$), are shown in Figure 4.

Figure 3:  The number of orbits $n$ versus the impact parameter $b$ for
different noncommutative parameter values $\theta$,.

Figure 4:  The relationship between the photon trajectories and impact
parameter $b$ near the black hole for different values of noncommutative
parameters $\theta$. In which, $\theta=0.01$ (left panel), $\theta=0.03$ (mid
panel) and $\theta=0.049$ (right panel). The spacings of $b$ are $1/5$,
$1/100$, $1/1000$ for the direct (redline), lensed (blue line) and photon
(green line) ring trajectories, respectively. And, the red dotted line
represents the photon orbit, and the black disk represents the event horizon
of the black hole.

Combined with Table 2, Figure 3 and Figure 4, we can notice that with the
increase of noncommutative parameter $\theta$, the range of lensing ring and
photon ring are enlarged. In other words, with the increase of noncommutative
parameters, there will be wider lensing rings and photon rings in spacetime,
and the corresponding brightness of these regions will also be enhanced. In
addition, when the impact parameter $b$ is close to the critical case $b\sim
b_{c}$, the total number of photon orbits $n$ will reach a peak in a narrow
region. After that, if the impact parameter $b$ increases to a large enough
value, the light ray trajectories are the direct emissions no matter what the
value of noncommutative parameter $\theta$ is.

###  3.2 Transfer functions

Then, we are in position to study the image of a black hole with an optically
and geometrically thin disk accretion in the noncommutative spacetime, and the
disk is located on the equatorial plane of the black hole. For convenience,
the thin disk accretion is located in the rest frame of static worldlines, and
the photons emitted from it should follow the basic property of isotropy. In
addition, we consider that the static observer locates at the north pole, and
the observed intensity is mainly provided by the thin disk, that is, the
influence of other light sources in spacetiime is ignored.

In this way, the specific intensity and frequency of the emission are
expressed as $I_{\nu}^{{em}}(r)$ and $\nu_{em}$, and the observed specific
intensity and frequency are defined as $I_{\nu^{\prime}}^{{obs}}(r)$ and
$\nu_{obs}$. The observed specific intensity can be expressed as

$I_{\nu^{\prime}}^{{obs}}=[1-\frac{2M}{r}+\frac{8\sqrt{\theta}M}{\sqrt{\pi}r^{2}}]^{3/2}I_{\nu}^{{em}}(r).$
(19)

Then, by integrating the full frequency, we can get the expression of total
observation and emission intensity in the whole path, namely

$I_{{OBS}}=\int
I_{\nu^{\prime}}^{{obs}}(r)\,d\nu_{obs}^{\prime}=\int[1-\frac{2M}{r}+\frac{8\sqrt{\theta}M}{\sqrt{\pi}r^{2}}]^{2}I_{\nu}^{{em}}(r)\,d\nu_{em}=[1-\frac{2M}{r}+\frac{8\sqrt{\theta}M}{\sqrt{\pi}r^{2}}]^{2}I_{{EM}}(r),$
(20)

where $I_{{EM}}(r)=\int I_{\nu}^{{em}}(r)d\nu_{em}$. In addition, $I_{OBS}$
and $I_{EM}$ represent the total observed and emission intensity,
respectively. Based on the previous consideration, the thin disk provides the
main contribution of the observation intensity, that is, when the light ray
trajectories pass through the thin disk once, it will get brightness from the
thin disk once. Finally, the observation intensity that the observer at
infinity can obtain should be the total brightness that the light can obtain
from the thin disk. Hence, the total received luminosity will be the sum of
all the intensities from all these crossings with the accretion disk, which is

$I_{{OBS}}(r)=\sum_{n}(A(r))^{2}I_{{EM}}(r)|_{r=r_{n}(b)}.$ (21)

Here, $r_{n}(b)$ is defined as transfer function, which contains the
information about the radial position of the $n_{th}$ intersection with the
thin emission disk. In addition, the slope of the transfer function is
$dr/db$, which is called the demagnification factor, and it represents the
demagnified scale of the transfer function[35, 60].

Figure 5:  The first three transfer functions of a black hole under different
noncommutative parameters. Here, $\theta=0.01$ (left panel), $\theta=0.03$(mid
panel) and $\theta=0.049$ (right panel).

In Figure 5, the black line (first transfer function $n=1$ ), which
corresponds to the transfer function for the direct emission. Although the
value of noncommutative parameter changes, the slope of the black line is
$\frac{dr}{db}|_{1}\sim Constant$. In fact, the direct image profile is the
redshifted source profile since its slope is approximatively equal to $1$. The
orange line (second transfer function $n=2$ ), which corresponds to the
transfer function for the lensing ring. It can be seen that when the impact
parameter $b$ is close to the critical case $b\sim b_{c}$, there is a
relatively gentle area, that is, the slope of the transfer function will have
a relatively small value. However, as the value of impact parameter $b$
continues to increase, the slope of the second transfer function begins to
increase sharply to a very large value. In other words, the image of the back
side of the disk will be demagnified, because its slope is much greater than
$1$. The red line ( third transfer function $n=3$), which corresponds to the
transfer function for the photon ring. It is obvious that the slope
$\frac{dr}{db}|_{3}$ is close to infinity $\frac{dr}{db}|_{3}\sim\infty$.
Namely, the image of the front side of the disk will be extremely demagnified.
Hence, the total observed flux is mainly provided by the direct emission, and
the luminosity provided by photon ring and lensing ring only accounts for a
very small part of the total observed flux. In particular, the image provided
by the later transfer function is more demagnetized and negligible, so we only
consider the first three transfer functions.

###  3.3 The observational features of noncommutative Schwarzschild black hole

With the previous preparation, we can further study the observed specific
intensity. For the observer, the observed light intensity is mainly provided
by the thin disk, and the specific brightness only depends on the radial
coordinate $r$. Therefore, we consider three toy models about the emissivity
of thin disks.

  * •

Model I: we consider that the emission starts from the position of the radius
of the innermost stable circular orbit $r_{isco}$, and the emitted function
$I_{EM}(r)$ is a decay function suppressed by the second power, which is

$\displaystyle
I_{EM}^{1}(r)=\begin{cases}\left(\frac{1}{r-(r_{isco}-1)}\right)^{2},&r>r_{{isco}}\\\
0,&r\leq r_{{isco}}\end{cases}$ (22)

  * •

Model II: we consider that the emission has an obvious peak on the photon
sphere layer, which has the similar center and asymptotic characteristics as
model I. But the attenuation is greater, so that $I_{EM}(r)$ is a decay
function suppressed by the third power function, which is described by

$\displaystyle
I_{EM}^{2}(r)=\begin{cases}\left(\frac{1}{r-(r_{p}-1)}\right)^{3},&r>r_{p}\\\
0,&r\leq r_{p}\end{cases}$ (23)

  * •

Model III: we consider that the emission starts right off the event horizon,
but its decay rate is slower than the first two models, as follows

$\displaystyle
I_{EM}^{3}(r)=\begin{cases}\frac{1-\tan^{-1}(r-(r_{isco-1)}))}{1-\tan^{-1}(r_{p})},&r>r_{e}\\\
0,&r\leq r_{e}\end{cases}$ (24)

In this case, by using those emitted functions and with the help of equation
(20), we take $\theta=0.01$ and $\theta=0.049$ as two examples to show the
results of intensity of the emission and observation in Figure 6 and Figure 7.

In Figure 6 ($\theta=0.01$) and Figure 7 ($\theta=0.049$), the left column is
the emission profile of the three modes, which is the relationship between the
emission specific intensity and the radial coordinate $r$, and from top to
bottom are model I, model II and model III, respectively. The middle column
shows the observed specific intensity $I_{OBS}$, which is related to the
impact parameter $b$. In addition, the optical appearance of observed
intensities is given in the column on the right.

Figure 6:  Observational appearance of the thin disk with different emission
profiles near the black hole ($\theta=0.01$), viewed from a face-on
orientation.

Figure 7:  Observational appearance of the thin disk with different emission
profiles near the black hole ($\theta=0.049$), viewed from a face-on
orientation.

From the left column in the first row of Figure 6 and 7, we find that the peak
region of emission is near the critical case of the impact parameter $r\sim
b_{c}$ , and then decreases sharply to zero with the increase of radial
coordinate $r$. In this case, the photon sphere is in the interior of the
emission region. And meanwhile, due to the influence of gravitational lensing
effect in the observation process, we clearly see that there are peaks of
observed intensity in the region of photon and lensing ring, and the two peaks
are separated independently. However, the peak value of observation intensity
of the photon and lensing ring are not only smaller than those of the direct
emission, but also their observation area is very narrow. As a result, the
observed intensities is dominated by the direct emission, with the lensing
ring emission providing only a small contribution to the total flux and the
photon ring providing a negligible contribution. In the column on the right,
it shows that the main optical appearance is provided by direct emission, and
the photon ring is very difficult to detect.

For the second row, the emission peak in the left column is near the photon
sphere $r\sim r_{p}$. The results of the middle column show that the direct
emission will make the observation intensity reach a peak at first, and then
show a gentle attenuation trend with the increase of $r$. More importantly,
the region of photon and the lensing ring almost coincide, which makes the
total observation intensity of this area significantly improved. Hence, the
observation intensity will reach a new peak due to the contribution of photon
ring, lensing ring and direct emission. However, the photon and lensing ring
are highly demagnetized and confined to a narrow region. In other words, the
direct emission is still dominant in observations. It can be seen from the
right column that the lensing ring has some contribution to the total
observation intensity, but the mainly contribution still is the direct
emission.

When the emitted region has been extended to the position near the event
horizon $r_{e}$, as shown in the left column of the third row. Moreover, the
lensing ring, photon ring and direct emission are overlapped in a larger
range. The observation intensity increases gently from the position slightly
larger than the event horizon, but suddenly increases sharply and reaches a
peak in the emission region of the photon ring. Then, due to the contribution
of the lensing ring emission, the observation intensity will reach a larger
peak. After that, the observation intensity began to decline gently. In this
case, the contribution of the lensing ring to the total observation intensity
is more obvious than that of the first two models. In addition, the optical
appearance shows a bright ring, which is composed of the contribution by the
lensing ring, photon ring and direct emission.

Comparing Figure 6 and 7, we can find that the change of noncommutative
parameters will have a direct impact on observation in this spacetime.
Firstly, with the increase of noncommutative parameters, the radius of photon
and lensing ring decreases, but the area occupy becomes wider. It shows that
the optical appearance of model III, the bright ring when $\theta=0.049$ is
wider than that when $\theta=0.01$. However, for all three emitted functions,
when the noncommutative parameter increases, the maximum observation intensity
decreases. In particular, when $\theta=0$ (the Schwarzschild spacetime [35]),
the observation intensity is significantly higher than that case of
$\theta=0.049$ (the noncommutative spacetime). Therefore, these results can
provide an effective help to distinguish the noncommutative Schwarzschild
black hole from the Schwarzschild black hole in spacetime.

##  4 Shadows of the noncommutative Schwarzschild black hole with spherical
accretion

In this section, our main goal is to study the image of the noncommutative
Schwarzschild black hole with a spherical accretion, which can be regarded to
be optically thin. Here, we consider two different models, that is, the static
spherical accretion and infalling spherical accretion. More importantly, we
want to explore the influence of noncommutative parameters on the spacetime
structure and observational characteristics.

We first study the shadow image and photon sphere of a black hole wrapped by
static spherical accretion in the noncommutative spacetime. That is, we assume
that the accretion is static around the black hole. For the observer at
infinity, the specific intensity of observation (measured usually in $\rm
ergs^{-1}cm^{-2}str^{-1}Hz^{-1}$) is determined by [101, 102]

$\displaystyle{I}({\nu_{obs}})=\int_{\gamma}g^{3}j(\nu_{e})dl_{{prop}},$ (25)

and

$\displaystyle g=\frac{\nu_{obs}}{\nu_{e}},$ (26)

among them, $g$, $\nu_{e}$ and $\nu_{obs}$ are redshift factor, the photon
frequency and observed photon frequency, respectively. In the rest frame of
the emitter, $j(\nu_{e})$ is the emissivity per unit volume, and $dl_{{prop}}$
is the infinitesimal proper length. In the noncommutative Schwarzschild
spacetime, the redshift factor has a form

$\displaystyle g=\frac{\nu_{obs}}{\nu_{e}}=A(r)^{1/2},$ (27)

Then, we can consider that the radiation of light is not only monochromatic,
but also monochromatic $\nu_{f}$ is the fixed frequency,. When the emission
has a $1/r^{2}$ radial profile [102], we can get the following equation

$\displaystyle
j\left(\nu_{e}\right)\propto\frac{\delta\left(\nu_{e}-\nu_{f}\right)}{r^{2}},$
(28)

where $\delta$ stands for the delta function. In this spacetime, we can also
obtain

$\displaystyle{dl}_{{prop}}=\sqrt{\frac{1}{A(r)}{dr}^{2}+r^{2}{d\varphi}^{2}}.$
(29)

Specially, we can take another form of the above equation, which is

$\displaystyle{dl}_{{prop}}=\sqrt{\frac{1}{A(r)}+r^{2}\left(\frac{{d\varphi}}{{dr}}\right)^{2}}{dr}.$
(30)

Under the help of the equations (27), (28) and (29), we can get the specific
intensity observed by a static observer at infinity is

$\displaystyle
I(\nu_{obs})=\int_{\gamma}\frac{A(r)^{3/2}}{r^{2}}\sqrt{\frac{1}{A(r)}+r^{2}\left(\frac{{d\varphi}}{{dr}}\right)^{2}}{dr}.$
(31)

Based on the above equation, we can explore the shadow image and corresponding
intensity of black hole surrounded by static accretion model in noncommutative
spacetime. It is worth mentioning that we want to study the influence of the
change of noncommutative parameters on the observation intensity and make a
comparison with Schwarzschild spacetime. In Figure 8, we depict the observed
specific intensity at spatial infinity when the noncommutative parameter
$\theta$ takes different values. At the same time, in Figure 9, we also show
these properties of the black hole shadow image and its brightness in
spacetime with the different noncommutative parameters $\theta$.

Figure 8:  For different noncommutative parameter values $\theta$, the
observed specific intensity at spatial infinity with $M=1$.

As illustrated in Figure 1, with the increase of impact parameter $b$, the
observed intensity $I(\nu_{obs})$ began to increase gently until it reached
the peak at $b_{c}$, and then showed a trend of attenuation.

Figure 9:  Images of the black hole shadow with static spherical accretion for
the noncommutative parameters $\theta$, in witch $\theta=0$ (left panel),
$\theta=0.01$ (middle panel) and $\theta=0.049$ (right panel).

From figure 8, we can clearly see that the specific intensity $I(\nu_{obs})$
will increase sharply with the increase of the impact parameter $b$, and reach
the peak value in the critical case $b\sim b_{c}$, which is applicable to
different values of the noncommutative parameter $\theta$. When the impact
parameter $b$ is larger than the critical condition and continues to increase
$b>b_{c}$, the specific intensity $I(\nu_{obs})$ shows a decreasing trend.
And, when $b$ tends to infinity $b\sim\infty$, the observation intensity will
be infinitely close to zero $I(\nu_{obs})\sim 0$. Moreover, we find that the
change of noncommutative parameter will affect the observation intensity,
which is shown in that the increase of noncommutative parameter will increase
the peak value of observation intensity. In other words, the larger the
noncommutative parameter is, the brighter the optical appearance image will be
seen by the observer at infinity. Obviously, these features are also reflected
in Figure 9, the observed optical appearance in noncommutative spacetime
($\theta=0.049$ or $\theta=0.01$) is much brighter in Schwarzschild spacetime
($\theta=0$). However, the radius of the photon sphere of the noncommutative
Schwarzschild black hole is obviously smaller than that of the Schwarzschild
black hole. In addition, the inner region of the photon sphere in Figure 9 is
not completely black, and there is little brightness near the photon sphere,
which is caused by a small part of radiation escaping from the black hole.

Now, we will study the spherical accretion around the noncommutative black
hole, and the radiative gas moves radially toward the black hole. It is a
dynamic model, the equation (25) is still applicable, but the corresponding
redshift factor becomes

$g=\frac{\mathcal{K}_{\rho}u_{0}^{\rho}}{\mathcal{K}_{\sigma}u_{e}^{\sigma}},\quad\mathcal{K}^{\mu}=\dot{x}_{\mu},$
(32)

where $\mathcal{K}^{\mu}$ is the four-velocity of the photon and
$u_{0}^{\mu}=(1,0,0,0)$ is the four-velocity of the static observer. In
addition, the quantity $u_{e}^{\mu}$ is the four-velocity of the infalling
accretion, which has a form

$u_{e}^{t}={A(r)}^{-1},\quad u_{e}^{r}=-\sqrt{{1-A(r)}},\quad
u_{e}^{\theta}=u_{e}^{\varphi}=0.$ (33)

From the null geodesic, we can obtain the four-velocity of photon, that is

$\mathcal{K}_{t}=\frac{1}{b},\quad\frac{\mathcal{K}_{r}}{\mathcal{K}_{t}}=\pm\frac{1}{A(r)}\sqrt{1-A(r)\frac{b^{2}}{r^{2}}},$
(34)

Considering that photon may approach or escape from the black hole, the above
equation is preceded by a sign $\pm$. Then, for the case of infalling
spherical accretion, the redshift factor can be written as follows

${g}_{i}=[u_{e}^{t}+(\frac{\mathcal{K}_{r}}{\mathcal{K}_{e}})u_{e}^{r}]^{-1}.$
(35)

And, the proper distance can be defined by

$dl_{p}=\mathcal{K}_{\mu}u_{e}^{\mu}d\lambda=\frac{\mathcal{K}_{t}}{{g}_{i}|\mathcal{K}_{r}|}dr.$
(36)

Similarly, we also assume that the specific emissivity is monochromatic.
Therefore, the observation intensity in the case of infalling spherical
accretion can be expressed as

${I}^{\ast}({\nu_{obs}})\propto\int_{\gamma}\frac{{g}_{i}^{3}\mathcal{K}_{t}dr}{r^{2}|\mathcal{K}_{r}|}.$
(37)

Using the equation (37), we can study the influence of noncommutative
parameters on black hole shadow image and its brightness distribution, as
shown in Figure 10 and Figure 11.

Figure 10:  The observed specific intensity at spatial infinity for different
noncommutative parameter values $\theta$ with $M=1$.

Figure 11:  Images of the black hole shadow with infalling spherical accretion
for the noncommutative parameters $\theta$, in witch $\theta=0$ (left panel),
$\theta=0.01$ (middle panel) and $\theta=0.049$ (right panel).

For different state parameters $\theta$, the maximum observed intensity is
still at the critical position of the impact parameter $(b\sim b_{c})$. When
$b<b_{c}$, the observation intensity increases with the increase of $b$ until
it reaches the peak. However, when $b>b_{c}$, the observation intensity tends
to decrease with the increase of $b$. In addition, the radius of black hole
shadow image decreases with the increase of noncommutative parameter, but the
observed brightness increases. Both Figure 10 and Figure 11 show that the
observed luminosity at $\theta=0.049$ is significantly higher than that at
$\theta=0.01$ and $\theta=0$ (the Schwarzschild spacetime). It is worth noting
that the central region of the infalling spherical accretion is much darker
than that of the static accretion case, which is due to the Doppler effect.

##  5 Conclusions and discussions

In this paper, we mainly study the shadow and observation characteristics of
noncommutative Schwarzschild black holes wrapped by three accretion models,
and then explore the influence of noncommutative parameters on the observation
appearance and spacetime geometry of black holes. Based on the null geodesic
of the noncommutative black hole, we study the effective potential and photon
orbits in this spacetime. Due to the change of noncommutative parameters, the
spacetime structure changes, which also leads to the change of corresponding
physical parameters. Specifically, the event horizon $r_{e}$, the radius
$r_{p}$ and impact parameter $b_{c}$ of photon sphere are all decreased with
the increase of the noncommutative parameter $\theta$, but the effective
potential increased. In order to study the observational appearance and
intensity around a noncommutative black hole surrounded by accretion matter,
we mainly take three optical and geometric accretion models as examples to
present the observational appearance of the noncommutative black hole.

When the noncommutative Schwarzschild black hole is surrounded by an optically
and geometrically thin accretion disk in the frame of the static worldlines,
we study the ray trajectories near the black hole according to the total
number of orbits $n=\frac{\varphi}{2\pi}$. Those ray trajectories can be
divided into the direct emission, lensing ring and photon ring. In this sense,
it turns out that there is not only a dark central shadow area, but also the
photon rings and lensing ring in the outer area of the shadow. Intriguingly,
by studying the first three transfer functions, we find that the lensing ring
is highly demagnetized, while the photon ring is extremely demagnetized. In
addition, we employ three typical toy model functions of the emission profile
to further study the observational appearance of the noncommutative black
hole, and then compare the observed specific intensities of direct emission,
lensing ring and photon ring. It shows that the photon ring is a highly curved
light, and it intersects with the disk plane at least three times, but its
contribution to the total brightness can be ignored because its area is so
narrow. Then, the area occupied by the lensing ring is wider than photon ring,
and its demagnetization is not too high in a certain range of the impact
parameter, thus making a bigger contribution to the flux of the observed
image. But, this contribution is also very small by comparing with the direct
emission. Hence, our results show that the direct emissions always dominate
the total observed intensity for all three toy-model functions. When the
noncommutative parameters increased, the range of lensing ring and photon ring
will increase. Also, the observation intensity influenced naturally, i.e., one
can see in Figure 7 that there appears a wide and bright annulus in the
optical observation appearance of model III. Moreover, in order to get some
insight into the effects of these rings on a realistic observation, we blur
the images of the accretion disk, which shown in Figure 12. Here, we mimic the
EHT resolution by using a Gaussian filter with standard derivation equal to
$1/12$ the field of view[35]. Obviously, the characteristics of the lensing
ring in the simulation observation results are washed out, and the photon ring
can not be observed directly. Indeed, the blurred rings and direct emission
are indistinguishable for all three emitted functions, even if they are
distinguishable before blurring. Although the characteristics of photon ring
and lensing ring are weakened, the third emission mode has a brighter and
wider blurred ring compared with the first two emission modes. In this case,
the shadow radius is quite different, which means that the location of the
direct emission determines the size of the shadow.

(a) model I (b) model II

(c) model III

Figure 12:  The blurred images from Figure 7, and The high resolution images
are blurred to correspond roughly to the EHT resolution.

In our universe, there may still be spherically symmetric accretion around
black holes. It is of great significance to further discuss the observational
appearance of black holes surrounded by spherical accretion. For convenience,
we here consider only two simple relativistic spherical accretion models,
i.e., the static spherical accretion and infalling spherical accretion. In
both spherical accretion models, the maximum observed brightness always
appears at the impact parameter $b\sim b_{c}$, and the brightness of the inner
area of the shadow is very low, but the observation brightness outside the
shadow increases obviously. Moreover, with the increase of the noncommutative
parameter $\theta$, the observation intensity of the optical appearance in
both two models also increase. However, it is obvious from Figures 8 and 10
that the infalling accretion model increase significantly higher than the
static accretion. In addition, due to the Doppler effect, the inner area of
the shadow of the infalling accretion is darker than that of the static. This
model is considered to be more realistic than the static spherical accretion
model since most of the accretion in the universe is dynamic.

Acknowledgments

The authors would like to thank the anonymous reviewer for their helpful
comments and suggestions, which helped to improve the quality of this paper.
This work is supported by the National Natural Science Foundation of China
(Grant Nos. 11675140, 11705005, 11903025 and 11875095).

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