††thanks<EMAIL_ADDRESS>huaifan999@163.com,zhouxn10@let.edu.cn,guo970527<EMAIL_ADDRESS>
# Shadow thermodynamics of non-linear charged Anti-de Sitter black holes

Yun-Zhi Du1, Huai-Fan Li1111the corresponding author, Xiang-Nan Zhou2, Wei-Qi
Guo2, Ren Zhao1 1Institute of Theoretical Physics, Shanxi Datong University,
Datong, China
2 College of Physics and Information Engineering, Shanxi Normal University,
TaiYuan, China

###### Abstract

Non-linear interaction between the electromagnetic fields (EMF) are occurred
when vacuum polarization in quantum electrodynamics (QED) happens. The field
of non-linear electrodynamics which may be resulting from this interaction
could have important effects on black hole physics. In this work, we
investigate the relationship between the observable quantity, the shadow
radius and the first-order phase transition for the non-linear charged AdS
black hole in the frame of the Einstein-power-Yang-Mills (EPYM) gravity.
Through the analysis, we find with a certain condition there exist the non-
monotonic behaviors between the shadow radius, the horizon radius, and
temperature (or pressure). And from the viewpoint of the shadow radius, the
phase transition temperature is higher than that from the viewpoint of the
horizon radius with the same condition. These indicate that the shadow radius
can be regarded a probe to reveal the thermodynamic phase transition
information of black holes. When the system is undergoing the phase transition
in two cases of the different non-linear YM charge parameter values:
$\gamma=1,~{}1.5$, the thermal profiles of the coexistent big and small black
hole phases with the temperature are presented. Furthermore, the effects of
non-linear YM charge parameter on the shadow radius and the thermal profile
are also investigated.

##  I Introduction

In universe there exists a most interesting and bizarre object, black hole.
After a lot of researches in the past years, people believe in the existence
of supermassive black holes in heart of the most galaxies. At least we can
address to the two of them as the Milky Way and neighboring elliptical
$M87^{*}$ galaxies Gillessen2017 ; Avery2015 . The breakthroughs on the
existence of black hole are the Event Horizon Telescope (EHT) Collaboration
who announced the first and the second images of the supermassive black holes
$M87^{*}$ Akiyama2019 ; Akiyama2019a ; Akiyama2019b  and the supermassive
black holes $SgrA^{*}$ Akiyama2022 ; Akiyama2022a . For this purpose, they
used a technique called Very Long Baseline Interferometry (VLBI). As well as
we know, the gravitational attraction around a black hole is so intense that
the nearby objects will fall into it when they reach at a critical radius.
These astounding phenomena are known as the gravitational lensing. It is
expected that we can understand the properties of black holes more better from
the lensing effect. That leads to the more considerable attention focused on
the gravitational lensing in the strong gravity regime of black holes. Some
lensing observable of Kerr black hole have been evaluated in Bozza2002 ;
Vazquez2004 ; Bozza2004  and more discussions can be found in Beckwith2005 ;
Wei2012 ; Chen2010 ; Gralla2020 ; Hsiao2020 ; Islam2021  and therein.

In addition, when the nearby objects are photons emitted by an illuminated
source located behind the black hole, then it brings a shadow which can be
seen by an observer sitting at infinity. The shadow concept has been studied
firstly by Bardeen Bardeen1973 . Specifically, the photons that escape from
the spherical orbits form the boundary of the dark silhouette of the black
hole. This dark silhouette is known as black hole shadow from the outside
observer. Note that the shadows of spherically symmetric black holes are
circular Synge1966 ; Chandrasekhar1998 ; Eiroa2019 ; Belhaj2020 ; Belhaj2022 ,
but for the spinning black holes their shadows are actually deformed
Chandrasekhar1992 ; Falcke2000 ; Belhaj2021 ; Belhaj2021a ; Belhaj2021b ;
Belhaj2021c . Moreover, the photon region of gravitational lensing also
provides key properties in the black hole shadow. And the unstable photon
regions outside the black hole event horizon can realize the possibility to
observe the black hole directly. The first black hole image of $M87^{*}$
published by EHT collaborations in 2019 had given the certain constraints on
some shadow observable. Later the angular shadow radius of $M87^{*}$ was
addressed in Kocherlakota2021 . Recently the angular shadow radius of
$SgrA^{*}$ in the second black hole image has been presented Akiyama2022 ;
Akiyama2022a . It is noted that these constraints on the shadow radius are
obtained with the Kerr geometry in GR as a premise, but they cannot exclude
other black holes in GR or some exotic black holes in the modified gravity
theory. The EHT observation of shadow can be applied as a tool to constrain
the parameters of various black holes in various theories of gravity, and in
the future more precise observational results can even be useful to
distinguish different black holes. Therefore, we should show the theoretically
possible results of various black holes to be tested by the future
observational data of EHT.

On the other hand, the black hole thermodynamics has attracted more attention
because of the remarkable and non-trivial results corresponding to the
criticality and the stability behaviors. A crucial issue has been devoted to
the investigation of the black holes in AdS/dS geometry Kubiznak2012 ;
Altamirano2013 ; Kubiznak2016 ; Marks2021 ; Cai2013 ; Wei2013 ; Banerjee2011 ;
Hendi2017 ; Bhattacharya2017 ; Wei2021 . The needed quantities have been
nicely computed for various gravity theories. Considering the cosmological as
a pressure, certain black holes show similarities with van der Waals fluid
systems where some universalities have been obtained. Furthermore, Hawking and
Page had shown the theoretical evidence of the certain transition in the phase
space for the Schwarzschild-AdS black hole Hawking1983 . The first-order phase
transition of the charged Reissner Nordstrom-AdS (RN-AdS) black hole has been
studied in different backgrounds including Dark Energy (DE) and Dark Matter
(DM) Belhaj2022 ; Belhaj2020a ; Belhaj2019 . In this way, the shadows and the
deflection angle of the light rays by black holes in arbitrary dimensions have
been discussed in Belhaj2021 ; Belhaj2022 ; Belhaj2020a ; He2022 . The effect
of DE and the space-time dimension on the involved optical quantities has been
inspected. In Refs. Belhaj2021a ; Belhaj2021c , for instance, the influence of
DM on the shadows and the photon rings of a stringy black hole illuminated by
certain accretions has been also studied. Motivated by the investigations on
these spacetimes, interplays between the black hole thermodynamics and the
optical properties have been established. It has been shown that the shadow
size can provide information on the HP phase transitions, the critical
behaviors and the microstructure states of the black holes living in AdS
geometries Zhang2020 ; Belhaj2020b ; Cai2021 . And the relation between the
shadow and the thermodynamics of the black hole has been also developed for
regular space-times Guo2022 . Using the elliptic function analysis, it has
been explored further a fundamental connection between the AdS black hole
thermodynamics and the deflection angle of the light rays. Concretely, various
thermodynamics behaviors of such black holes have been approached in terms the
deflection angle variations BelhajPLB . However, there is little work about
the concrete relation between two coexistent black hole phases and their
shadows for a black hole undergoing the first-order phase transition. In this
work, we will investigate this issue for the non-linear charged AdS black
hole.

The linear charged black holes in AdS spacetime Chamblin1999  within a second-
order phase transition show a scaling symmetry: at the critical point the
state parameters scale respects to charge q, i.e., $S\sim q^{2},~{}P\sim
q^{-2},~{}T\sim q^{-1}$ Johnson2018 . It is naturally to gauss whether there
exists the scaling symmetry in the non-linear charged AdS black holes. As a
generalization of the charged AdS Einstein-Maxwell black holes, it is
interesting to explore new non-linear charged systems. Due to infinite self-
energy of point like charges in Maxwell’s theory Born1934 ; Kats2007 ;
Anninos2009 ; Cai2008 ; Seiberg1999 , Born and Infeld proposed a
generalization when the field is strong, bringing in non-linearities Dirac2013
; Birula1970 . The power-law Maxwell electrodynamics is a famous of nonlinear
electrodynamics models, which involves a Yang-Mill field exponentially coupled
to Einstein gravity. As well as we know, the electromagnetic part of its
Lagrangian density is $Tr[F_{\mu\nu}^{(a)}F^{\mu\nu(a)}]^{\gamma}$. There
possesses the conformal invariance and is easy to construct the analogues of
the four-dimensional Reissner-Nordström black hole solutions in higher
dimensions. Additionally several features of the Einstein-power-Yang-Mills
(EPYM) gravity in extended thermodynamics have recently been studied Zhang2015
; Moumni2018 ; Du2021 .

Inspired by these, we mainly investigate the relation between shadow radius
and phase transitions of the charged EPYM AdS black hole. This work is
organized as follows. In Sec. II, we would like to briefly review the
thermodynamic quantities and present the relationship between the observable
quantity (i.e., shadow radius) and the horizon radius of the charged EPYM AdS
black hole. In Sec. III, the behaviour of the shadow radius with the horizon
radius and temperature nearby the phase transition point are investigated. And
the effect of non-linear YM charge parameter on these behaviors is also
presented. In order to exhibit more intuitively the relationship between the
EPYM AdS black hole phase structure and its shadow, we show the thermal
profiles of two coexistent big and small black hole phases in the two-
dimensional plane, see in Sec. IV. A brief summary is given in Sec. V.

##  II Shadow of Non-Linear charged AdS black holes

The solution of the four-dimensional Einstein-power-Yang-Mills (EPYM) AdS
black hole with non-linear Yang-Mills (YM) charge was presented in Ref.
Yerra2018 :

$\displaystyle ds^{2}=-f(r)dt^{2}+f^{-1}dr^{2}+r^{2}d\Omega_{2}^{2},$ (1)
$\displaystyle
f(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{2}+\frac{\left(2q^{2}\right)^{\gamma}}{2(4\gamma-3)r^{4\gamma-2}},$
(2)

where $d\Omega_{2}^{2}$ is the metric on unit $2$-sphere with volume $4\pi$
and $q$ is the YM charge. Note that this solution is valid for the condition
of the non-linear YM charge parameter $\gamma\neq 0.75$, and the power YM term
holds the weak energy condition (WEC) for $\gamma>0$ Corda2011 . In the
extended phase space, $\Lambda$ was interpreted as the thermodynamic pressure
$P=-\frac{\Lambda}{8\pi}$. The black hole event horizon locates at
$f(r_{+})=0$.

The thermodynamical quantities of this system are

$\displaystyle M$ $\displaystyle=$ $\displaystyle\frac{1}{6}\left[8\pi
Pr_{+}^{3/2}+3r_{+}^{\frac{3-4\gamma}{2}}\frac{\left(2q^{2}\right)^{\gamma}}{8\gamma-6}+3r_{+}\right].$
(3) $\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r_{+}}\left(1+8\pi
Pr_{+}^{2}-\frac{\left(2q^{2}\right)^{\gamma}}{2r_{+}^{(4\gamma-2)}}\right),~{}~{}~{}~{}~{}S=\pi
r_{+}^{2},~{}~{}~{}~{}V=\frac{4\pi r_{+}^{3}}{3}.$ (4) $\displaystyle P$
$\displaystyle=$ $\displaystyle
r_{+}\left[\frac{T}{2}-\frac{1}{8\pi}r_{+}+\frac{\left(2q^{2}\right)^{\gamma}}{16\pi}r_{+}^{1-4\gamma}\right].$
(5)

It is known that this non-linear AdS black hole exhibits a vdW’s-like phase
transition, with the critical point given in Refs. Du2021 ; Du2022

$\displaystyle r_{c}^{4\gamma-2}$ $\displaystyle=$
$\displaystyle\left(2q^{2}\right)^{\gamma}f(1,\gamma),\quad
f(1,\gamma)=\gamma(4\gamma-1),~{}S_{c}=\pi\left(2q^{2}\right)^{\frac{\gamma}{2\gamma-1}}f^{1/(2\gamma-1)}(1,\gamma),$
(6) $\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi\left(2q^{2}\right)^{\gamma/(4\gamma-2)}f^{1/(4\gamma-2)}(1,\gamma)}\frac{2\gamma-1}{4\gamma-1},$
(7) $\displaystyle P_{c}$ $\displaystyle=$
$\displaystyle\frac{2\gamma-1}{16\pi\gamma\left(2q^{2}\right)^{\gamma/(2\gamma-1)}f^{1/(2\gamma-1)}(1,\gamma)}.$
(8)

where the YM charge parameter satisfies the condition $\frac{1}{2}<\gamma$. It
is clear that this critical point closely depends on the YM charge and the
non-linear YM charge parameter. From the above quantities we can obtain a
interesting relation, which is only related with the non-linear YM charge
parameter, as

$\displaystyle
S_{c}^{2}T_{c}^{2}P_{c}=\frac{(2\gamma-1)^{3}}{16\pi\gamma(4\gamma-1)^{2}}.$
(9)

Now we will consider a free photon orbiting around a black hole one the
equatorial hyperplane defined by $\theta=\pi/2$ and $p_{\theta}=0$. The
Lagrangian takes the form as

$\displaystyle
H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=\frac{1}{2}\left(-f^{-1}p_{t}^{2}+fp_{r}^{2}+r^{-2}p_{\phi}^{2}\right),$
(10)

where $p_{\mu}$ are the generalized momentums. For this black hole background,
there are two Killing fields $\partial_{t}$ and $\partial_{\phi}$, which lead
two constants, the particle energy $E$ and orbital angular momentum $L$ along
each geodesics

$\displaystyle-E=p_{t}=-f(r)\dot{t},~{}~{}~{}~{}~{}~{}L=p_{\phi}=r^{2}\dot{\phi},$
(11)

where the dot represents the derivative to the affine parameter. The light
rays are the solutions to Hamilton’s equations

$\displaystyle\dot{p}_{\mu}=-\frac{\partial H}{\partial
x^{\mu}},~{}~{}~{}~{}\dot{x}^{\mu}=\frac{\partial H}{\partial p_{\mu}},$ (12)

which read

$\displaystyle\dot{p}_{t}$ $\displaystyle=$ $\displaystyle
0,~{}~{}\dot{p}_{\phi}=0,~{}~{}\dot{p}_{\theta}=0,$ $\displaystyle\dot{p}_{r}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{f^{\prime}(r)p_{t}^{2}}{f^{2}(r)}+f^{\prime}(r)p_{r}^{2}+2f(r)p_{r}p_{r}^{\prime}-\frac{2p_{\phi}^{2}}{r^{3}}\right),$
$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle-\frac{p_{t}}{f(r)},~{}~{}\dot{r}=f(r)p_{r},~{}~{}\dot{\phi}=\frac{p_{\phi}}{r^{2}}.$
(13)

With $H=0$, we have

$\displaystyle\dot{r}^{2}+V_{eff}=0,~{}~{}~{}V_{eff}=\frac{L^{2}f(r)}{r^{2}}-E^{2}.$
(14)

As an example, we exhibit the effective potential in Fig. 1 for the EPYM AdS
black hole with the fixed parameters $\gamma=1.01,~{}P=0.003,~{}q=1.9,~{}M=2$,
and different angular momentums $L/E$. Since the positive of $\dot{r}^{2}$,
there exists the condition: $V_{eff}<0$. Thus the photon can only survive in
the range of negative effective potential. When the conserved quantum $L$ is
small, the photon will fall into the black hole from somewhere with a larger
value of $r$. While, for the larger value of $L$, the peak of effective
potential will increase, that leads to the reflection of photon before it
falls into black hole. Between these two cases, there exists a critical case
which is described by the thickness red line, whose peak approaches zero at
$r=4.3122$ and at the same time the radial velocity of photon vanishes. This
point just corresponds to the photon sphere radius because of the spherically
symmetric static black hole. In the following, we will present the relation
between a spatial static observer and the photon sphere radius.

Figure 1: The effective potential for the EPYM AdS black h ole with the
parameters $\gamma=1.01,~{}P=0.003,~{}q=1.9,~{}M=2$. The angular momentums
$L/E$ of the photon varies from $0.5$ to $36$ from bottom to top.

From the Fig. 1, we knows that the unstable circular photon sphere satisfies

$\displaystyle
V_{eff}=0,~{}~{}\frac{dV_{eff}}{dr}=0,~{}~{}\frac{d^{2}V_{eff}}{dr^{2}}<0.$
(15)

Here we denote the photon sphere radius as $r_{ph}$. From the above second
equation, we find that the radius $r_{ph}$ satisfies

$\displaystyle 2f(r)\mid_{r_{ph}}=rf^{\prime}(r)\mid_{r_{ph}},$ (16)

where the prime represents the derivative to the radial coordinate $r$. For
given the metric function, by solving the equation $V_{eff}=0$, the impact
parameter or the angular momentum of the photon reads

$\displaystyle\mu_{ph}\equiv\frac{L}{E}=\frac{r}{\sqrt{f(r)}}\bigg{|}_{r_{ph}}.$
(17)

Considering light rays sent from an observer at radius coordinate $r_{0}$ into
the past. These light rays can be divided into two classes: Light rays of the
first class go to infinity after being deflected by the black hole. Light rays
of the second one go towards the horizon of the black hole. If there are no
light sources between the observer and the black hole, initial directions of
the second class correspond to darkness on the observer’s sky. This dark
circular disk on the observer’s sky is called the shadow of the black hole.
The boundary of the shadow is determined by the initial directions of light
rays that asymptotically spiral towards the outermost photon sphere. The light
ray send from a static observer at position $r_{0}$ transmits into the past
with an angle $\alpha$ relative to the radial direction, which reads

$\displaystyle\cot{\alpha}=\sqrt{\frac{1}{r^{2}f(r)}}\frac{dr}{d\phi}\bigg{|}_{r_{0}}.$
(18)

If the light ray goes out again after reaching $r_{ph}$, from eq. (13) the
orbit equation can be written as

$\displaystyle\frac{dr}{d\phi}=\pm r\sqrt{\frac{r^{2}}{\mu_{ph}^{2}}-f(r)}.$
(19)

Thus, the angular radius of the shadow becomes

$\displaystyle\cot{\alpha}^{2}=\frac{r_{0}^{2}}{\mu_{ph}^{2}f(r_{0})}-1,$ (20)

and by using a trigonometric identity,
$1+\cot{\alpha}^{2}=\frac{1}{\sin^{2}{\alpha}}$, it can be rewritten as

$\displaystyle\sin{\alpha}=\frac{\mu_{ph}\sqrt{f(r_{0})}}{r_{0}}$ (21)

The shadow radius of the black hole observed by a static observer at $r_{0}$
can be written as

$\displaystyle r_{s}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}\sqrt{f(r_{0})}.$ (22)

##  III Phase transition of EPYM AdS black hole from the viewpoint of shadow
radius

When this system is undergoing the first-order phase transition (HPBH/LPBH
phase transition) with temperature $T$ ($T=\chi T_{c},~{}\chi<1$), we denote
the horizon radius of two coexistence black hole phases as $r_{1}$ and
$r_{2}$, respectively. With the Maxwell’s equal area law, the condition of the
first-order phase transition for this system with the given temperature $T$,
which had been shown in our previous works Du2021 ; Du2022 , reads

$\displaystyle\chi\frac{2\gamma-1}{\gamma^{1/(4\gamma-2)}(4\gamma-1)^{(4\gamma-1)/(4\gamma-2)}}=\frac{1}{f^{1/(4\gamma-2)}(x,\gamma)}\left(1+x-\frac{1-x^{4\gamma}}{2f(x,\gamma)(1-x)x^{4\gamma-2}}\right),~{}~{}\frac{(2q^{2})^{\gamma}}{r_{2}^{4\gamma-2}}=\frac{1}{f(x,\gamma)}$
(23)

with

$\displaystyle f(x,\gamma)=\frac{(3-4\gamma)(1+x)(1-x^{4\gamma})+8\gamma
x^{2}(1-x^{4\gamma-3})}{2x^{4\gamma-2}(3-4\gamma)(1-x)^{3}},~{}~{}x=\frac{r_{1}}{r_{2}}.$
(24)

(a) $\gamma=1$ (b) $\gamma=1$ (c) $\gamma=1.5$

Figure 2: The pictures of the horizon radii for two coexistent black hole
phases with temperature. The YM charge set to $q=1.9$.

As $q=1.9$, by numerically fitting the data the horizon radii of the
coexistent big and small black hole phases nearby the critical point as the
functions of temperature are given as the following

$\displaystyle r_{1}$ $\displaystyle\approx$
$\displaystyle\begin{cases}2.65\times 10^{7}T^{4}-8.16\times
10^{5}T^{3}+8.77\times
10^{3}T^{2}+1.9,&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\gamma=1\\\ 2.39\times
10^{6}T^{4}-1.2\times 10^{5}T^{3}+2.16\times
10^{3}T^{2}+1.77,&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\gamma=1.5\end{cases}$ (25)
$\displaystyle r_{2}$ $\displaystyle\approx$
$\displaystyle\begin{cases}-7.11+3.81\times 10^{-10}/T^{3}-2.52\times
10^{-6}/T^{2}+0.32/T,&~{}~{}~{}~{}~{}~{}\gamma=1\\\ -4.12+8.1\times
10^{-10}/T^{3}-3.27\times
10^{-6}/T^{2}+0.32/T.&~{}~{}~{}~{}~{}~{}\gamma=1.5\end{cases}$ (26)

And the coexistent curves read

$\displaystyle P\approx\begin{cases}-6.42\times 10^{5}T^{8}+6.33\times
10^{3}T^{7}-1.49\times 10^{3}T^{6}+7.27T^{5}+1.13\times
10^{3}T^{4}-5.58T^{3}+1.3T^{2},&\gamma=1\\\
250T^{8}+17.5T^{7}-116.9T^{6}-16.4T^{5}-0.57T^{4}+15.32T^{3}+1.01T^{2}.&\gamma=1.5\end{cases}$
(27)

Nearby the critical point, the radii of two coexistent black hole phases as
the function of temperature are exhibited in Fig. 2. It is obviously that the
horizon radius for the small coexistent phase increases with the increasing of
temperature, while for the large coexistent phase it increases with
temperature.

Now, we will investigate the phase transition information of this system from
the viewpoint of the shadow radius. In order to obtain the analyzed result,
firstly we set the non-liner YM charge parameter equal to one: $\gamma=1$. The
photon sphere radius becomes

$\displaystyle r_{ph}=\frac{1}{2}\left(3M+\sqrt{9M^{2}-8q^{2}}\right).$ (28)

It is clear that this result is exactly equal to that of the asymptotically
flat charged black hole, in which the photon sphere radius depends on the
pressure. Note that the mass of eq. (3) is related with the pressure.
Substituting it into above equation, we have the photon sphere radius

$\displaystyle r_{ph}=\frac{3q^{2}+r_{+}^{2}(3+8\pi
Pr_{+}^{2})+\sqrt{[3q^{2}+r_{+}^{2}(3+8\pi
Pr_{+}^{2})]^{2}-32q^{2}r_{+}^{2}}}{4r_{+}}.$ (29)

At the critical point ($r_{+}=r_{c}$ and $P=P_{c}$), the critical photon
sphere radius reads $r_{phc}=(2+\sqrt{6})q$ for $\gamma=1$. Therefore the
shadow radius as a function of the black hole horizon radius becomes

$\displaystyle
r_{s}=\frac{r_{ph}(r_{+})}{\sqrt{f(r_{ph}(r_{+}))}}\sqrt{f(r_{0})}.$ (30)

Substituting the critical photon sphere radius and pressure into eq. (30), we
can get the critical shadow radius $r_{sc}$, which is related to the YM charge
$q$.

Based on the constraint, a static observer at spatial infinity have
$f(r_{0})=1$ Zhang2020 . With eqs. (29) and (30), the behaviors of the shadow
radius as the function of the black hole horizon radius for different values
of pressure can be obtained, which are exhibited in Fig. 3(a). It is obviously
that there exists a minimum value of the shadow radius for this system with
the given pressure (no matter it is large than or less than the critical
pressure). The shadow radius is decreasing with the increasing of the horizon
radius for small black holes, while for the big ones it is increasing with the
horizon radius until to a constant $\sqrt{3/(8\pi P)}$. It indicates that at
the minimum point the system will undergo a phase transition which is
different the HPBH/LPBH one. Furthermore, for the small EPYM black hole its
shadow radius decreases with the increasing of pressure.

(a) $~{}q=1.9$

(b) $~{}P=0.0006994<P_{c}$

(c) $~{}P=1.5P_{c}$

Figure 3: The non-linear YM charge parameter set to $\gamma=1$. In the left
picture: the pressure set to $P_{c}-0.00022,~{}Pc,$ and $1.5P_{c}$ from bottom
to top. In the middle and right pictures: the YM charge varies from $1.9$ to
$0.8$ from top to bottom. The horizontal thin dashed lines stand for the
limited value of the shadow radius.

(a) $~{}P=P_{c}-0.00022,~{}P_{c},~{}P_{c}+0.0002$

(b) $~{}P=P_{c}-0.00022,~{}P_{c},~{}P_{c}+0.0002$

(c) $~{}P=P_{c}-0.00022$

Figure 4: The parameters set to $\gamma=1,~{}q=1.9$. A static observer at
$r_{0}=100$.

There is a interesting phenomena: the values of the shadow radius under the
limitations of $r_{+}\rightarrow 0$ and $r_{+}\rightarrow\infty$ are the same,
which is related to the pressure $P$, but not to the YM charge $q$ (shown in
Figs. 3(b) and 3(c)). Here we present the limited forms of the shadow radius
with different $\gamma$:

$\displaystyle r_{s}|_{r_{+}\rightarrow
0,\infty}=\begin{cases}\sqrt{-\frac{3}{\Lambda}}=\sqrt{\frac{3}{8\pi
P}},&\gamma=1\\\ \frac{4.3416}{4\pi\sqrt{P}},&\gamma=1.5\end{cases}$ (31)

Note that for two cases of different non-linear YM charge parameter values,
the inverse square coefficient between $r_{s}|_{r_{+}\rightarrow 0,\infty}$
and pressure is the same: $0.345494$. That means for the EPYM AdS black hole
the limited value of the shadow radius is independent with the charge
information ($q$ and $\gamma$) carried by the system, only is related with the
pressure.

For different values of pressure, combining the first-order phase transition
condition (23) and eqs. (2), (4), (8), (29), and (30), we show the $T-r_{+,s}$
phase diagrams in Figs. 4(a) and 4(b). It is obviously that for $P<P_{c}$,
there exists the non-monotonic behavior both in the $T-r_{+}$ and $T-r_{s}$
planes. However, the temperature becomes only a monotone increasing function
of the shadow radius and the horizon radius when the pressure is bigger than
the critical one. At the critical pressure, there exist the deflection points
at $r_{+}=r_{c}$ and $r_{s}=r_{sc}$, respectively. These behaviors are very
similar to the isobar process of a VdW’s system in the $T-S$ plane, which
means there exist the first-order phase transition both from the viewpoint of
the horizon radius and the shadow radius. Thought constructing the Maxwell’s
equal-area law both in $T-r_{+}$ and $T-r_{s}$ planes, we also get the phase
transition points. The corresponding shadow radii are remarked by $r_{s_{1}}$
and $r_{s_{2}}$.

It should be noted that there is a rather peculiar phenomenon: for the given
pressure $P<P_{c}$ the phase transition temperature from the viewpoint of the
horizon radius is slightly lower than that from the viewpoint of the shadow
radius as $\gamma=1$ shown in Fig. 4(c). The reason of this situation may be
caused by the gravitational effect and the non-linear YM charge term. So far,
we have converted the phase transition information of the EPYM AdS black hole
phase transition into the measurable physical quantity, the shadow radius.
That means we can detect whether there is a phase transition by measuring the
shadow radius of the black hole, so as to obtain the microscopic structure of
the black hole.

(a) $~{}q=1.9$

(b) $~{}P=0.0012<P_{c}$

(c) $~{}P=3.5P_{c}$

Figure 5: The non-linear YM charge parameter set to $\gamma=1.5$. In the left
picture: the pressure set to $P_{c}-0.0001,~{}Pc,$ and $1.5P_{c}$ from bottom
to top. In the middle and right pictures: the YM charge varies from $1.9$ to
$1$ from bottom to top. The horizontal thin dashed lines stand for the limited
value of the shadow radius.

In addition, the non-monotonic behavior of $T-r_{s}$ only just indicates the
existence of the first-order phase transition, but not the first-order one
between high potential black hole and low potential black hole. Since the
black hole horizon radius cannot be directly obtained from experiments, while
its shadow can be observed. Therefore, the shadow radius may serve as a probe
for the phase structure of the EPYM AdS black hole. For the case of
$\gamma=1.5$ we also present the numerical results of the shadow radius with
the black hole horizon radius for different pressures and YM charges in Fig.
5(a), whose behaviors are similar to the case of $\gamma=1$. In this case, the
limited form of the shadow radius is exhibited in eq. (31). The behaviours for
the temperature as the functions of the horizon radius and the shadow radius
have been shown in Fig. 6, which are similar to that in the case of
$\gamma=1$.

(a) $~{}P=P_{c}-0.0005,~{}P_{c},~{}P_{c}+0.0005$

(b) $~{}P=P_{c}-0.0005,~{}P_{c},~{}P_{c}+0.0005$

(c) $~{}P=P_{c}-0.0005$

Figure 6: The parameters set to $\gamma=1.5,~{}q=1.9$. A static observer at
$r_{0}=100$.

(a) $\gamma=1$

(b) $\gamma=1$

(c) $\gamma=1.5$

(d) $\gamma=1.5$

Figure 7: The parameters set to $q=1.9$. And the temperature various from
$0.5T_{c}$ to $Tc$. A static observer at $r_{0}=100$.

In order to exhibit the relationship between the black hole phase transition
and the shadow radius in two cases of $\gamma=1,1.5$, the pictures of the
shadow radii for two-coexistent black hole phases as the functions of
temperature and horizon radius are shown in Fig. 7. For the big coexistent
phase, its shadow radius decreases with the increasing of temperature, while
it increases with the horizon radius. However for the small one its shadow
radii as the function of temperature and the horizon radius, both have a local
minimum value. At the local minimum point there maybe exists a new phase
transition.

##  IV Thermal profile of the EPYM AdS BH

Since the shape of a spherically symmetric black hole shadow is circular for
any observer Chang2020 , we will establish a thermal profile in the two-
dimensional plane to more intuitively present the relationship between the BH
phase structure and its shadow for the EPYM AdS black hole. According to Ref.
Eiroa2019 , the shadow boundary curve at the celestial coordinate reads as

$\displaystyle
x=\lim_{r\rightarrow\infty}\left(-r^{2}\sin\theta_{0}\frac{d\phi}{dr}\right)_{\theta_{0}\rightarrow\pi/2},~{}~{}y=\lim_{r\rightarrow\infty}\left(r^{2}\frac{d\theta}{dr}\right)_{\theta_{0}\rightarrow\pi/2}.$
(32)

As $\gamma=1$ and $1.5$, the shadow contours of the coexistent big and small
black hole phases with temperature for a static spatial infinity observer are
exhibited in Fig. 8.

(a) $~{}r_{+}=r_{2}$

(b) $~{}r_{+}=r_{1}$

(c) $~{}r_{+}=r_{2}$

(d) $~{}r_{+}=r_{1}$

Figure 8: The shadow pictures of the coexistent big and small black hole
phases with temperature. The parameters set to $q=1.9$ and $\gamma=1$ (in
Figs. 8(a) and 8(b)), $\gamma=1.5$ (in Figs. 8(c) and 8(d)). The temperature
various from $0.5T_{c}$ to $T_{c}$.

The results show that the size of the BH shadow depends on the temperature.
When $T<T_{c}$, the shadow radius for the coexistent big black hole phase
decreases monotonically with temperature until to the critical shadow and it
is in the large radius region, which correspond to the supercritical black
hole phases; the red thick curves stand for the critical shadow; for the small
black hole phases the shadow radii support in the small radius region; the
inner black disks stand for the coexistent big and small black hole phases
with different temperatures. It is obviously that for the coexistent small
phases, the shadow radius decreases gently with the lower temperature until to
the minimum value, then increases drastically with the higher temperature.
These characters are consistent with those exhibited in Fig. 7.

##  V Discussions and Conclusions

In this manuscript, we examined the relationship between the thermodynamic
phase transition for the four-dimensional charge Einstein-power-Yang-Mills
(EPYM) AdS black hole and its shadow. This will provide a new way to realize
the link between the gravity and thermodynamics of black holes.

First, we presented the characters of EPYM AdS black hole phase transition:
the critical point depend on the YM charge and non-linear YM charge parameter
with the condition $\gamma>1/2$ and $\gamma\neq 3/4$. Since the critical point
stands for the boundary of the coexistent phases, it is keys to probe the
black hole phase structure. Then we investigated the null geodesics of a
photon in the equatorial plane of EPYM AdS black hole background. By analyzing
the effective potential of photon orbits with the certain parameters, we
obtained the photon sphere radius and impact parameter (angular momentum of
the photon sphere). For an infinity static observer, the corresponding black
hole shadow can be expressed as the function of the photon sphere radius
directly, namely the shadow is related with the black hole horizon. The
results showed that the non-monotonic behaviors appear in $T-r_{s}$ and
$r_{s}-r_{+}$ diagrams. That indicated that there exists a certain
relationship between the shadow and the phase transition of this system. In
addition, we also presented the influence of non-linear YM charge parameter
$\gamma$ on the shadow.

Finally, we further explored the relationship between the shadow, temperature,
and two-existent black hole phases. The relevant conclusions can be summarized
as follows:

  * •

For a isobar process of this system, the shadow radius as the function of the
black hole horizon has the non-monotonic behavior with the certain parameters.
In the $r_{s}-r_{+}$ plane, the local stable minimum maybe indicate a new
phase transition, which is different from with the black hole phase
transition. Furthermore, for a given pressure, the limitations of the shadow
radius with $r_{+}\rightarrow 0,\infty$ are the same one. That means as
$r_{+}\rightarrow 0,\infty$ the limitation of the shadow radius is independent
with the charge information ($q$ and $\gamma$) carried by the system, only is
related with the pressure;

  * •

For a given pressure, which is less than the critical value, there exists two
extremal points in the $T-r_{s}$ plane. Until the pressure approaches to the
critical one, two extremal points coincide with each other. When the pressure
is larger than the critical one, there is no extremal point. Those behaviors
of the shadow radius are consistent with that of the black hole phase
transition. Therefore, the behavior of photon sphere can be regarded a probe
to reveal the thermodynamic phase transition information of black holes;

  * •

For the same set of parameters, the phase transition temperature from the
viewpoint of the shadow radius is slightly higher than that from the viewpoint
of the black hole horizon, which is maybe caused by the gravitational effect
and the non-linear YM charge term;

  * •

At the coexistent curve of $T-P$, the shadow radius for the coexistent big
black hole phase decreases monotonically with temperature until to the
critical shadow, which correspond to the supercritical black hole phases; for
the coexistent small black hole phases, the shadow radius decreases gently
with the lower temperature until to the minimum value, then increases
drastically with the higher temperature. These characters are consistent no
matter in the $r_{s_{1},s_{2}}-r_{1,2}$ planes and the thermal profiles.

Those analyze firmly enhance our conjecture that there exists the relationship
between the null geometry and thermodynamic phase transition for EPYM AdS
black holes. It further supports the link between the gravity and
thermodynamics of black holes, and also provides a possible way to describe
the strong gravitational effect from the thermodynamic side.

## Acknowledgements

We would like to thank Prof. Ren Zhao, Meng-Sen Ma, and Yu-Peng Zhang for
their indispensable discussions and comments. This work was supported by the
National Natural Science Foundation of China (Grant No. 12075143)

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