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# Photon orbits and phase transition for Non-Linear charged Anti-de Sitter
black holes

Yun-Zhi Du, Huai-Fan Li111the corresponding author, Fang Liu, Li-Chun Zhang
Department of Physics, Shan xi Da tong University, Da tong 037009, China
Institute of Theoretical Physics, Shan xi Da tong University, Da tong 037009,
China

###### Abstract

In this work, we investigate the relationship between the photon sphere radius
and the first-order phase transition for the charged EPYM AdS black hole.
Through the analysis, we find with a certain condition there exist the non-
monotonic behaviors between the photon sphere radius, the impact parameter,
the non-linear YM charge parameter, temperature, and pressure. And both the
changes of photon sphere radius and impact parameter before and after phase
transition can be regarded as the order parameter, their critical exponents
near the critical point are equal to the same value $1/2$, just like the
ordinary thermal systems. These indicate that there maybe exists a universal
relationship of gravity nearby the critical point for a black hole
thermodynamical system. Furthermore, the effect of impact parameter on the
deflect angle is also investigated.

##  I Introduction

Since the Hawking-Page phase transition Hawking1983  of an AdS spacetime was
proposed by Hawking and Page in $1983$, which was explained to a
confinement/deconfinement phase transition in gauge theory in Ref. Witten1998
and could also be understood as a solid/liquid phase transition Altamirano2013
by regarding the cosmological constant as pressure
$P=-\frac{\Lambda}{8\pi}=\frac{(n-1)(n-2)}{16\pi l^{2}}$, black hole
thermodynamics has attracted lots of attention. Subsequently the phase
transition in the extended AdS/dS phase space has been widely considered
Hendi2017a ; Hennigar2017a ; Frassin ; Kubiznak2012 ; Cai2013 ; Ma2017 ;
Mir2017b ; Banerjee2017 ; Simovic2019 ; Mbarek2019 ; Li2017a ; Ma2018 ;
Zhang2016 ; Dinsmore2020 ; Cai1306 ; Caldarelli2000 ; Mann1207 ; Wei2015 ;
Banerjee1109 ; Hendi1702 ; Bhattacharya2017 ; Zeng2017 ; Hendi1803 ; Zhang1502
; Cheng1603 ; Dolan2014 ; Altamirano2014 ; Du2021 ; Zhang2020 . Based on the
similarity of thermodynamic behaviors for black holes and ordinary systems,
the microstructure of black holes system was investigated by the Ruppeiner
scalar curvature Wei2015  following the Ruppeiner geometry Rupperiner1996 .
The number density of the speculative black hole molecules was introduced to
examine its phase transition and microstructure. This method was quickly
generalized to other black holes Ruppeiner2018 ; Miao2019 . Furthermore from
the view point of the Gibbs free energy landscape, the dynamic properties of
various black hole phase transitions were studied in Refs. Li2020 ; Wei2021 ;
Yang2021 ; Kumara2021 ; Mo2021 . Since the phase transition of black holes is
more important in black hole thermodynamics, there exists a natural question
how to probe the information of the black hole phase transition.

So far although experimental evidences of black hole thermodynamics have not
been found, people still expect that black hole thermodynamics should have
some observational signatures, for example the quasinormal modes (QNMs), which
may be detected from astronomical observations in the future. Recently, the
authors in Refs. Jing2008 ; Berti2008 ; He2008 ; Shen2007 ; Rao2007 ; Liu2014
; Mahapatra2016 ; Chabab2016 ; Prasia2017 ; Zou2017 ; Liang2019  had found
that QNMs indeed exhibit characteristic behaviours along the black hole phase
transition. Thus QNMs can be use to probe the phase transition of black holes
in AdS/dS spacetime. Especially there is a dramatic change of the slope of
QNMs along with the black hole phase transition Liu2014 . These results
indicate that there is a close connection between the dynamical and
thermodynamical aspects of black holes. In addition, as we well known that
QNMs are intimately linked to the unstable photon orbits Goebel1972 ;
Ferrari1984 ; Mashhoon1985 , then they are playing an important role in strong
gravitational phenomena (such as the shadow, lensing, and gravitational waves)
Bozza2002 ; Cardoso2009 ; Stefanov2010 ; Wei2011 ; Hod2011 ; Wei2014 ;
Raffaelli2016 ; Cardoso2016 ; Konoplya2017 . In short, QNMs can be regarded as
the vibration frequencies of photon orbits and can be derived by the photon
orbits Dolan2009 . Based on these observations, it is natural to construct the
relation between the photon orbits and black hole phase transitions and to
check out whether properties of photon orbits can signal the black hole phase
transition.

Firstly, people proposed that the photon orbits signals a possible York-
Hawking-Page phase transition in Refs. Cvetic2016 ; Gibbons2008 . Then the
more clearly relation between photon orbits and phase transitions of Reissner-
Nordstrom-AdS black hole and rotating Kerr-AdS black hole were shown in Ref.
Wei2018 . Namely, the photon orbit radius and the minimum impact parameter
along with the temperature are both exhibiting non-monotonic behaviours for
the low pressure, whose behaviours are indeed the characters of the phase
transition existence. Particularly, the changes of those two quantities near
the critical point could be treated as the order parameters of phase
transition, and they have an universal exponent $1/2$. Subsequently, this
issue has been extended to other black holes Kynara2020  and gravity theory
(Gauss-Bonnet case Han2018 , massive gravity case Chabab2019 , Born-Infeld
case Xu2019 ; Li2020NPB ). A general proof of the existence for these
relations between photon orbits and phase transitions were shown in Ref.
Zhang2019 .

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 . An interesting non-linear generalization of charged black holes
involves a Yang-Mill field exponentially coupled to Einstein gravity, which
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 Du2021 ;
Zhang2015 ; Moumni2018 .

Inspired by these, we mainly investigate the relation between photon orbits
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 the second-order phase transition of charged EPYM
AdS black hole. The mull geodesics of this system and the deflection angle are
also analyzed. In Sec. III, we present the behaviour of the photon orbit
radius and minimum impact parameter with the temperature nearby the phase
transition point and explore the influence of non-linear YM charge parameter
on the phase transition. A universal exponent for the changes of photon orbit
radius and minimum impact parameter is also obtained. A brief summary is given
in Sec. IV.

##  II Null geodesics of Non-Linear charged AdS black holes

The action for four-dimensional Einstein-power-Yang-Mills (EPYM) gravity with
a cosmological constant $\Lambda$ was given by Zhang2015 ; Corda2011 ;
Mazharimousavi2009 ; Lorenci2002

$\displaystyle I=\frac{1}{2}\int
d^{4}x\sqrt{g}\left(R-2\Lambda-[Tr(F^{(a)}_{{\mu\nu}}F^{{(a)\mu\nu}})]^{\gamma}\right)$
(1)

with the Yang-Mills (YM) field

$\displaystyle
F_{\mu\nu}^{(a)}=\partial_{\mu}A_{\nu}^{(a)}-\partial_{\nu}A_{\mu}^{(a)}+\frac{1}{2\xi}C_{(b)(c)}^{(a)}A_{\mu}^{b}A_{\nu}^{c}$
(2)

Here,
$Tr(F^{(a)}_{\mu\nu}F^{(a)\mu\nu})=\sum^{3}_{a=1}F^{(a)}_{\mu\nu}F^{(a)\mu\nu}$,
$R$ and $\gamma$ are the scalar curvature and a positive real parameter,
respectively; $C_{(b)(c)}^{(a)}$ represents the structure constants of three
parameter Lie group $G$; $\xi$ is the coupling constant; and $A_{\mu}^{(a)}$
represents the $SO(3)$ gauge group Yang-Mills (YM) potentials.

For this system, the EPYM black hole solution with the negative cosmological
constant $\Lambda$ in the four-dimensional spacetime is obtained by adopting
the following metric Yerra2018 :

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

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 parameter $M$ represents the ADM mass of the black hole and
it reads

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

And in our set up it is associated with the enthalpy of the system. The black
hole temperature, entropy, and volume were given by Zhang2015

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

The YM potential $\Psi$ was given by Anninos2009

$\displaystyle\Psi=\frac{\partial M}{\partial
q^{2\gamma}}=\frac{r_{+}^{3-4\gamma}2^{\gamma-2}}{4\gamma-3}.$ (7)

The above thermodynamic quantities satisfy the first law

$\displaystyle dM=TdS+\Psi dq^{2\gamma}+VdP.$ (8)

The equation of state $P(V,T)$ for canonical ensemble (fixed YM charge $q$)
can be obtained from the expression of temperature as

$\displaystyle
P=\left(\frac{4\pi}{3V}\right)^{1/3}\left[\frac{T}{2}-\frac{1}{8\pi}\left(\frac{4\pi}{3V}\right)^{1/3}+\frac{\left(2q^{2}\right)^{\gamma}}{16\pi}\left(\frac{4\pi}{3V}\right)^{\frac{1-4\gamma}{3}}\right].$
(9)

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

$\displaystyle r_{c}^{4\gamma-2}$ $\displaystyle=$
$\displaystyle\left(2q^{2}\right)^{\gamma}f(1,\gamma),\quad
f(1,\gamma)=\gamma(4\gamma-1),~{}S_{c}=\pi\left(2q^{2}\right)^{\frac{\gamma}{2\gamma-1}}f^{2}(1,\gamma),$
(10) $\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi\left(2q^{2}\right)^{\gamma/(4\gamma-2)}f^{1/(4\gamma-2)}(1,\gamma)}\frac{2\gamma-1}{4\gamma-1},$
(11) $\displaystyle P_{c}$ $\displaystyle=$
$\displaystyle\frac{2\gamma-1}{16\pi\gamma\left(2q^{2}\right)^{\gamma/(2\gamma-1)}f^{1/(2\gamma-1)}(1,\gamma)}.$
(12)

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. We can obtain a interesting relation, which is
only related with the non-linear YM charge parameter, from the above
quantities as

$\displaystyle
S_{c}^{2}T_{c}^{2}P_{c}=\frac{(2\gamma-1)^{3}}{16\pi\gamma(4\gamma-1)}.$ (13)

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

$\displaystyle
H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=\frac{1}{2}\left(-f^{-1}p_{t}^{2}+fp_{r}^{2}+r^{-2}p_{\phi}^{2}\right),$
(14)

where the dot represents the derivative to the affine parameter and $p_{\mu}$
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

$\displaystyle-E=p_{t}=-f(r)\dot{t},~{}~{}~{}~{}~{}~{}L=p_{\phi}=r^{2}\dot{\phi}.$
(15)

The light rays are the solutions to Hamilton’s equations

$\displaystyle\dot{p}_{\mu}=-\frac{\partial H}{\partial
x^{\mu}},~{}~{}~{}~{}\dot{x}^{\mu}=\frac{\partial H}{\partial p_{\mu}},$ (16)

which read

$\displaystyle\dot{p}_{t}$ $\displaystyle=$ $\displaystyle
0,~{}~{}\dot{p}_{\phi}=0,~{}~{}\dot{p}_{\theta}=0,$ $\displaystyle\dot{p}_{r}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{f^{\prime}p_{t}^{2}}{f^{2}}+f^{\prime}p_{r}^{2}+2fp_{r}p_{r}^{\prime}-\frac{2p_{\phi}^{2}}{r^{3}}\right),$
$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle-\frac{p_{t}}{f},~{}~{}\dot{r}=fp_{r},~{}~{}\dot{\phi}=\frac{p_{\phi}}{r^{2}}.$
(17)

With $H=0$, we have

$\displaystyle\dot{r}^{2}+V_{eff}=0,~{}~{}~{}V_{eff}=\frac{L^{2}f}{r^{2}}-E^{2}.$
(18)

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 because of the spherically
symmetric static black hole. In the following, we mainly discuss the relation
between the photon sphere radius and phase transition of the EPYM AdS black
hole.

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.

The unstable circular photon sphere is determined by

$\displaystyle
V_{eff}=0,~{}~{}\frac{dV_{eff}}{dr}=0,~{}~{}\frac{d^{2}V_{eff}}{dr^{2}}<0.$
(19)

From the second equation, we find that the radius $r_{ph}$ satisfies

$\displaystyle 2f(r)\mid_{r_{ph}}=rf^{\prime}(r)\mid_{r_{ph}}.$ (20)

For given the metric function, we also obtain the radius from the above
equation. Solving the first equation, the impact parameter or the angular
momentum of the photon is

$\displaystyle\mu_{ph}\equiv\frac{L}{E_{c}}=\frac{r}{\sqrt{f(r)}}\bigg{|}_{r_{ph}}.$
(21)

From this relation, we have the behaviors of the non-linear YM charge
parameter and the impact parameter with the photon sphere radius, which are
shown in Fig. 2.

(a) $q=1.9,~{}\mu_{ph}^{2}=1,10,21.6,27,28,28.5$

(b) $q=2,~{}\gamma=0.85,0.95,0.978,1,1.005,1.01$

Figure 2: $\gamma$ and $\mu_{ph}$ vs. $r_{ph}$ with $P=0.003,~{}M=2$. Figure
3: The angle as the function of impact parameter with the parameters
$\gamma=1,~{}P=0.0003,~{}q=1.9,~{}M=2$. The observer’s position varies from
$1$ to $1000$ from the red line to the black one.

It is obviously that there exist the critical curves when given the pressure
and mass of black hole. The corresponding critical values are exhibited in
Table. 1. The critical impact parameter, radius, and non-linear YM charge
parameter are both increasing with the YM charge.

Table 1: The critical quantities in $r_{ph}-\gamma$ diagram for different YM charges and $M=2$. $q$ | $0.85$ | $1$ | $1.5$ | $1.9$ | $2$
---|---|---|---|---|---
$\mu_{phc}^{2}$ | 7.351515 | 10.18365 | 17.91095 | 21.5645 | 22.216
$r_{phc}$ | 1.06611 | 1.33639 | 2.18652 | 2.74 | 2.86303
$\gamma_{c}$ | 0.804019 | 0.821639 | 0.893342 | 0.960512 | 0.978177

Next, we will investigate the effect of the angular momentum on the observer’s
angle, especially the critical angle with the maximum impact parameter.
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. Note that
the light rays in the photon sphere are unstable with respect to radial
perturbations.

We consider a light ray that is sent from the observer’s position at $r_{0}$
into the past under an deflection angle $\alpha$ with respect to the radial
direction. The deflection angle is given by

$\displaystyle\cot{\alpha}=\sqrt{\frac{g_{rr}}{g_{\phi\phi}}}\frac{dr}{d\phi}\bigg{|}_{r_{0}}=\sqrt{\frac{1}{f(r)r^{2}}}\frac{dr}{d\phi}\bigg{|}_{r_{0}}.$
(22)

From eqs. (14) and (17), we have the orbit equation

$\displaystyle\frac{dr}{d\phi}=\sqrt{f(r)r^{2}}\sqrt{\frac{r^{2}\mu^{2}}{f(r)}-1}$
(23)

with the impact parameter or the angular momentum $\mu\equiv\frac{L}{E}$. Thus
the deflection angle $\alpha$ satisfies

$\displaystyle\sin^{2}{\alpha}=\frac{f(r_{0})}{r_{0}^{2}\mu^{2}}.$ (24)

Especially, the critical angle reads
$\sin^{2}{\alpha_{c}}=\frac{f(r_{0})}{\mu^{2}_{ph}r_{0}^{2}}$. That indicates
that $\mu$ has a close relation with the critical deflection angle and the
observer’s position, which is just the phenomenon of the black hole lensing.
For the light ray with the large impact parameter, its deflection angle is
small. Gradually decreasing $\mu$ to $\mu_{ph}$, the deflection angle will be
larger and larger, until to the bounded one $\alpha_{c}$. These behaviors are
presented in Fig. 3. That is consistent with that showed in Ref. Bozza2002 .

As what we have shown in the above, there are two important quantities
$r_{ph}$ and $\mu_{ph}$ for a photon sphere. In the following, we will probe
the detailed behaviors of these two quantities near phase transition, and
discuss whether they carry the phase transition information.

##  III Behaviors of $r_{ph}$ and $\mu_{ph}$ near phase transition point

For the EPYM AdS black hole with the non-linear source, the condition of the
first-order phase transition was presented in our work Du2021 as

$\displaystyle\chi\frac{2\gamma-1}{\gamma^{1/(4\gamma-2)}(4\gamma-1)^{(4\gamma-1)/(4\gamma-2)}}=\frac{1}{f^{1/(4\gamma-2)}(x,\gamma)}\left(1+x-\frac{1-x^{4\gamma}}{2f(x,\gamma)(1-x)x^{4\gamma-2}}\right),~{}~{}\frac{(2q^{2})^{\gamma}}{r_{2}^{4\gamma-2}}=\frac{1}{f(x,\gamma)}$
(25)

with

$\displaystyle f(x,\gamma)=\frac{(3-4\gamma)(1+x)(1-x^{4\gamma})+8\gamma
x^{2}(1-x^{4\gamma-3})}{2x^{4\gamma-2}(3-4\gamma)(1-x)^{3}},~{}~{}x=\frac{r_{1}}{r_{2}}.$
(26)

And we also investigated the effect of the non-linear YM charge parameter
$\gamma$ on the phase transition from two different viewpoints. Firstly, from
the viewpoint of the coexistent curve ($P-T$), when fixed the YM electric $q$
we found that for a given temperature ($T<T_{c}$) or a given pressure
($P<P_{c}$), the pressure or the temperature and the radius of the coexistent
big black hole phase increase with the increasing of $\gamma$; while the
radius of the coexistent small black hole phase decreases with $\gamma$. Here
the two-phases coexistent curves in the $T-P$ plane with different values of
the non-linear YM charge parameter $\gamma$ are shown in Fig. 4. Secondly,
from the viewpoint of the geometry scalar curvature, the absolute values of
the scalar curvature for the coexistent big and small black holes in the range
of $3/4<\gamma\leq 1$ both decrease with the increasing of $\gamma$, while
their behavior of the scalar curvature are just opposite in the range of
$1\leq\gamma$. In the following, we only exhibit the behaviors of the photon
sphere radius as the function of temperature in two cases of $\gamma=1$ (the
analytical solution) and $\gamma=1.5$ (the numerical Solution), respectively.
Since the solutions of the photon sphere radius with other values of $\gamma$
are difficult to present, so we cannot give the relevant analysis and
conclusions.

(a) $q=1.9$

(b) $q=1.9$

Figure 4: The coexistent curve of $P-T$.

For $\gamma=1$, the photon sphere radius of this system can be expressed as

$\displaystyle r_{ph}=\frac{1}{2}\left(3M+\sqrt{9M^{2}-8q^{2}}\right).$ (27)

It is clear that this result is exactly equal to that of the asymptotically
flat charged black hole, in which the orbit radius depends on the pressure.
However, we need to note that the mass of eq. (5) is related with the
pressure. Substituting it into above equation, we have the photon sphere
radius

$\displaystyle r_{ph}=\frac{3\pi q^{2}+S(3+8PS)+\sqrt{[3\pi
q^{2}+S(3+8PS)]^{2}-32\pi q^{2}S}}{4\sqrt{\pi S}}.$ (28)

At the critical point given in eqs. (10) and (12), the critical radius reads

$\displaystyle r_{phc}=(2+\sqrt{6})q.$ (29)

The temperature in eq. (6) can be rewritten as

$\displaystyle T(S,P,q)=\frac{1}{4\sqrt{\pi S}}\left(1+8PS-\frac{\pi
q^{2}}{S}\right).$ (30)

Combining eqs. (28) and (30), we give the behavior of temperature as a
function of the photon sphere radius with the fixed pressure, which is shown
in Fig. 5(a). It is obviously that for $P<P_{c}$, there will be a non-
monotonic behavior. However, the temperature will become only a monotone
increasing function of the photon sphere radius when the pressure is bigger
than the critical one. At the critical pressure, there is a deflection point
at $r_{ph}=r_{phc}$. That behavior is very similar to the isobar of the vdW’s
system in $T-S$ plane, which means there exists the first-order phase
transition. Thought constructing the Maxwell’s equal-area law, we also obtain
the phase transition point.

(a) $r_{ph}-T~{}~{}\gamma=1$

(b) $\mu_{ph}^{2}-T~{}~{}\gamma=1$

Figure 5: The YM charge set to $q=1.9$. The pressure varies from
$P_{c}-0.0002$ to $P_{c}+0.00004$ from bottom to top.

(a) $\gamma=1$

(b) $\gamma=1.5$

Figure 6: The parameters set to $q=1.9$ and $P=P_{c}-0.0002$.

The behavior of the temperature as a function of the impact parameter square
in Figs 5(b). Interestingly, it also confirms a non-monotonic property for
$P<P_{c}$. And it becomes monotonous for $P\leq P_{c}$. Note that these non-
monotonic behaviors in $r_{ph}-T$ and $\mu_{ph}^{2}-T$ diagrams only just
indicate the existence of the first-order phase transition, but not the first-
order one between high potential black hole and low potential black hole. In
order to illustrate the phase transition relationship between the radius of
the photon sphere and horizon, the phase transition diagrams in the $r_{+}-T$
and $r_{ph}-T$ planes are exhibited in Fig. 6(a). It is obviously that the
phase transition temperature in $r_{ph}-T$ curve is higher than the one in
$r_{+}-T$. And from the viewpoint of the photon sphere radius, the horizon
radii corresponding to the big- and small-coexistent black holes are only just
extended correspondingly.

Furthermore, when $\gamma=1.5$ we also present the numerical results of the
temperature with the photon sphere radius and impact parameter for different
pressures in Fig. 7. Furthermore, the phase transition relationship from two
different viewpoints of the photon sphere radius and the horizon radius is
also exhibited in Fig. 6(b). Those behaviors in this case are similar to that
in the case of $\gamma=1$. In other words, in both two cases of $\gamma=1$ and
$\gamma=1.5$ $r_{ph}$ and $\mu^{2}_{ph}$ have the non-monotonic behavior and
exhibit the first-order phase transition under a certain condition. And from
the viewpoint of the photon sphere radius, its radius value and corresponding
phase transition temperature are both larger than them from the viewpoint of
the horizon radius. These results are the same for this two cases. While the
differences shown in Fig. 8 are that: the phase transition temperature with
$\gamma=1.5$ is high than the one with $\gamma=1$; the photon sphere radii and
impact parameter of the two coexistent phases are different in this two cases.

Note that for the fixed quantities $q$, $\gamma$, and $P$ ($P<P_{c}$), there
also exist the non-monotonic behaviors in the $r_{ph}-T$ and $\mu^{2}_{ph}-T$
planes; for different points of the $r_{ph}-T$ and $\mu^{2}_{ph}-T$ curves,
the mass parameter of the EPYM AdS black hole is different, which is
completely different from the comments discussed in the second part (the mass
parameter was fixed). So far, since the behaviors of the photon sphere radius
from two different viewpoints have been exhibited, the corresponding results
in two parts cannot be directly compared. In the further work, we will focus
on the relationship between the observable quantity, the shadow radius and the
thermodynamic phase transition of this system in order to realize the
experimental detection of the black hole thermodynamic properties when the
phase transition occurs.

(a) $r_{ph}-T~{}~{}\gamma=1.5$

(b) $\mu_{ph}^{2}-T~{}~{}\gamma=1.5$

Figure 7: The YM charge set to $q=1.9$. The pressure varies from
$P_{c}-0.0004$ to $P_{c}+0.00003$ from bottom to top.

(a) $r_{ph}-T$ (b) $\mu_{ph}^{2}-T$

Figure 8: The pressures set to $P_{c}-0.0002$ and $q=1.9$.

In order to probe the changes of the photon sphere radius and the impact
parameter during the phase transition process, we present the temperature as a
function of the photon sphere radius with different pressure ($P<P_{c}$) in
Fig. 9. The dashed lines stand for the coexistent phases, the corresponding
phase transition temperature is determined by eq. (30). With the increasing of
photon sphere radius, the phase transitions will emerge at the dashed lines.
And there existences the sudden change of photon sphere radius. Further
increasing the pressure or temperature to the critical values, the sudden
change of photon sphere radius disappears: $\Delta r_{ph}\equiv
r_{ph2}-r_{ph1}=0$. Thus if the photon sphere radius has a sudden change, the
black hole must be undergoing a first-order phase transition. The similar
conclusions hold for the case between the impact parameter and temperature.
With the method of numerical drawing, we exhibit the changes of the reduced
photon sphere radius and reduced impact parameter ($\Delta r_{ph}/r_{phc}$ and
$\Delta\mu_{ph}/\mu_{phc}$) as the functions of the reduced temperature
($T/T_{c}$) with the parameters $\gamma=1,~{}q=1.9$ in Fig. 10. It is
obviously that both the values of $\Delta r_{ph}/r_{phc}$ and
$\Delta\mu_{ph}/\mu_{phc}$ decrease with the reduced temperature, and they are
approach to zero until $\frac{T}{T_{c}}=1$. At the same time the first-order
phase transition becomes the second-order one. Therefore the changes of
reduced photon sphere radius and reduced impact parameter can be regarded the
order parameters for the black hole phase transition.

Figure 9: The parameters set to $q=1.9$ and $\gamma=1$. The pressure varies
from $P_{c}-0.00023$ to $P_{c}-0.0002$ from bottom to top.

It is well known that the critical exponent of the order parameter for a
ordinary thermodynamic system nearby the critical point equals to $1/2$,
whether these two order parameters have the same value of critical exponent.
Using the numerical fitting method near $T_{c}$, we find

$\displaystyle\Delta r_{ph}/r_{phc}$ $\displaystyle\sim$ $\displaystyle
3.935\sqrt{1-T/T_{c}},$ (31) $\displaystyle\Delta\mu_{ph}/\mu_{phc}$
$\displaystyle\sim$ $\displaystyle 1.1423\sqrt{1-T/T_{c}}.$ (32)

This result exactly confirms that both the critical exponents of $\Delta
r_{ph}/r_{phc}$ and $\Delta\mu_{ph}/\mu_{phc}$ equal to $1/2$ at the critical
point. That indicates from the viewpoint of thermodynamic black holes have
more similar behaviors as the ordinary thermal systems, and they maybe also
have the similar microstructures from the perspective of the statistics
Wei2015 . Those conclusions strongly imply there exist a relationship between
the photon sphere radius and phase transition of black holes.

(a) $\Delta r_{ph}/r_{phc}-T/T_{c}~{}~{}\gamma=1$

(b) $\Delta\mu_{ph}/\mu_{phc}-T/T_{c}~{}~{}\gamma=1$

Figure 10: The YM charge set to $q=1.9$.

##  IV Discussions and Conclusions

In this manuscript, we examined the relationship between the photon sphere
radius the thermodynamic phase transition for the four-dimensional charge
Einstein-power-Yang-Mills (EPYM) AdS black hole. 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). The results showed that the non-monotonic behaviors appear
in $\gamma-r_{ph}$ and $\mu_{ph}-r_{ph}$ diagrams. That indicated that there
exists a certain relationship between the photon sphere radius, impact
parameter, and EPYM AdS black hole phase transitions. In addition, we also
presented the influence of impact parameter on the reflected angle.

Finally, we further explored the relationship between the unstable photon
sphere radius, impact parameter, and thermodynamic phase transition. It mainly
contains three aspects as the following:

  * •

For the isobar and isotherm processes of this system, the photon sphere radius
and impact parameter have the non-monotonic behaviors with the certain
parameters. When the pressure or temperature is less than the corresponding
critical value, these two quantities both have two extremal points. Until the
pressure or temperature approaches to the critical one, two extremal points
coincide with each other. When the pressure or temperature is more than the
critical one, there is no extremal point. Thus, the photon sphere radius and
impact parameter are the non-monotonic functions with the pressure or
temperature. Those behaviors of the photon sphere radius and impact parameter
are consistent with that of the black hole thermodynamics. Therefore, the
behavior of photon sphere can be regarded a probe to reveal the thermodynamic
phase transition information of black holes;

  * •

In both two cases of $\gamma=1$ and $\gamma=1.5$, from the viewpoint of the
photon sphere radius the corresponding phase transition temperature is higher
than that from the viewpoint of the horizon radius. Compared with the horizon
radius, the photon sphere radii corresponding to the two coexistent phases are
only expanded accordingly;

  * •

At the first-order phase transition point, the photon sphere radius and impact
parameter both have a sudden change. The changes of them before and after
first-order phase transition can act as order parameters to describe EPYM AdS
black hole phase transitions. More importantly, at the critical point, the
fitting result displays that they have a same critical exponent of $1/2$.

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 Si-Jiang Yang for
their indispensable discussions and comments. This work was supported by the
National Natural Science Foundation of China (Grant No. 11705106, 11475107,
12075143), the Natural Science Foundation of Shanxi Province, China (Grant
No.201901D111315), the Natural Science Foundation for Young Scientists of
Shanxi Province, China (Grant No.201901D211441), and the Scientific Innovation
Foundation of the Higher Education Institutions of Shanxi Province (Grant Nos.
2020L0471, 2020L0472, 2016173).

## References

  * (1) S. W. Hawking and Don N. Page, Thermodynamics of Black Holes in Anti-de Sitter Space, Commun. Math. Phys. 87, 577-588 (1983).
  * (2) E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2, 505 (1998), arXiv:hep-th/9803131.
  * (3) N. Altamirano, D. Kubiznak, and R. B. Mann, Reentrant phase transitions in rotating anti-de Sitter black holes, Phys. Rev. D 88 (2013) 101502, arXiv:1306.5756.
  * (4) R. A. Hennigar, E. Tjoa, and R. B. Mann, Thermodynamics of hairy black holes in Lovelock gravity, J. High Energ. Phys (2017) 70, arXiv:1612.06852.
  * (5) R. A. Hennigar and R. B. Mann, Superfluid black hole, Phys. Rev. Lett. 118, 021301 (2017), arXiv:1609.02564.
  * (6) A. M. Frassino, D. Kubiznak, R. B. Mann, and F. Simovic, Multiple Reentrant Phase Transitions and Triple Points in Lovelock Thermodynamics, J. High Energ. Phys, 09 (2014) 080, arXiv:1406.7015.
  * (7) D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes, J. High Energ. Phys 1207 (2012) 033, arXiv:1205.0559.
  * (8) R.-G Cai, L.-M Cao, L. Li, and R.-Q Yang, P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space, J. High Energ. Phys, 2013 (9), arXiv:1306.6233.
  * (9) M.-S Ma, R. Zhao, Y.-S Liu, Phase transition and thermodynamic stability of topological black holes in Horava-Lifshitz gravity, Classical and Quantum Gravity, 34 (2017) 165009, arXiv:1604.06998.
  * (10) M. Mir and R. B. Mann, Charged Rotating AdS Black Holes with Chern-Simons coupling, Phys. Rev. D 95, 024005 (2017), arXiv:1610.05281.
  * (11) R. Banerjee, B. R. Majhi, and S. Samanta, Thermogeometric phase transition in a unified framework, Phys. Lett. B 767 (2017) 25 C28, arXiv:1611.06701.
  * (12) F. Simovic and R. B. Mann, Critical Phenomena of Charged de Sitter Black Holes in Cavities, Class. Quant. Grav. 36 (2019) 014002, arXiv:1807.11875
  * (13) S. Mbarek and R. B. Mann, Reverse Hawking-Page Phase Transition in de Sitter Black Holes, J. High Energ. Phys 02 (2019) 103, arXiv:1808.03349.
  * (14) X.-Y Guo, H.-F Li, L.-C Zhang, and R. Zhao, Microstructure and continuous phase transition of RN-AdS black hole, Phys. Rev. D 100, 064036 (2019).
  * (15) Y.-B Ma, Y. Zhang, L.-Ch Zhang, L. Wu, Y. Gao, Sh. Cao, and Y. Pan, Phase transition and entropic force of de Sitter black hole in massive gravity, Eur. Phys. J. C 81 (2021) 42.
  * (16) L.-C Zhang, R. Zhao, M.-S Ma, Entropy of Reissner-Nordstrom-de Sitter black hole, Phys. Lett. B 761 (2016) 74-76.
  * (17) J. Dinsmore, P. Draper, D. Kastor, Y. Qiu, and J. Traschen, Schottky Anomaly of deSitter Black Holes, Class. Quant. Grav. 37 (2020) 5, arXiv:1907.00248.
  * (18) R.-G Cai, L.-M Cao, L. Li, and R.-Q Yang, P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space, J. High Energ. Phys, 9 (2013) 1, arXiv:1306.6233.
  * (19) M. M. Caldarelli, G. Cognola, and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal filed hteories, Class. Quant. Grav. 17 (2000) 399, arXiv:9908022.
  * (20) D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes, J. High Energ. Phys 1207 (2012) 033, arXiv:1205.0559.
  * (21) S.-W Wei and Y.-X Liu, Insight into the microscopic structure of an AdS black hole from a thermodynammical phase transition, Phys. Rev. Lett 115 (2015) 111302.
  * (22) R. Banerjee, and D. Roychowdhury, Thermodynamics of phase transition in higher dimensional AdS black holes, J. High Energ. Phys 11 (2011) 004, arXiv:1109.2433.
  * (23) S. H. Hendi, R. B. Mann, S. Panahiyan, and B. Eslam Panah, van der Waals like behavior of topological AdS black holes in massive gravity, Phys. Rev. D 95 (2017) 021501, arXiv:1702.00432.
  * (24) K. Bhattacharya, B. R. Majhi, and S. Samanta, van der Waals criticality AdS black holes: A phenomenological study, Phys. Rev. D 96, 084037 (2017).
  * (25) X.-X Zeng and L.-F Li, van der Waals phase transition in the framework of holography, Phys. Lett. B 764 (2017) 100.
  * (26) S. H. Hendi, Z. S. taghadomi, and C. Corda, New aspect of critical nonlinearly charged black hole, Phys. Rev. D 97, 084039 (2018), arXiv:1803.10767.
  * (27) J.-L Zhang, R.-G Cai, and H.-W Yu, Phase transition and Thermodynamical geometry of Reissner-Nordström-AdS Black Holes in Extended Phase Space, Phys. Rev. D 91, 044028 (2015), arXiv:1502.01428.
  * (28) P. Cheng, S.-W Wei, and Y.-X Liu, Critical phenomena in the extended phase space of Kerr-Newman-AdS black holes, Phys. Rev. D 94, 024025 (2016), arXiv:1603.08694.
  * (29) B. P. Dolan, On the thermodynamic stability of rotating black holes in higher dimensions-a comparision of thermodynamic ensembles, Class. Quant. Grav. 31 (2014) 135012, arXiv:1308.2672.
  * (30) N. Altamirano, D. Kubiznak, R. B. Mann, and Z. Sherkatghanad, Kerrr-AdS analogue of thriple point and solid/liquid/gas phase transition, Class. Quant. Grav. 31 (2014) 042001, arXiv:1308.2672.
  * (31) Y.-Z Du, H.-F Li, F. Liu, R. Zhao, and L.-C Zhang, Phase transition of non-linear charged Anti-de Sitter blackh holes, Chin. Phys. C 45 (2021) 11, arXiv:2112.10403.
  * (32) Y. Zhang, W.-Q Wang, Y.-B Ma, and J. Wang, Phase Transition and Entropy Force between Two Horizons in ($n+2$)-Dimensional de Sitter Space, Adv. H. E. Phys. 2020 (2020) 7263059, arXiv:2004.06796.
  * (33) G. Ruppeiner, Riemannian geometry in thermodynamic flucturation theory,Re. Mod. Phys. 67, 605 (1995). G. Ruppeiner, Thermodynamic curvature and black holes, Springer Proc. Phys, 153 (2014) 179-203, arXiv:1309.0901. G. Ruppeiner, Thermodynamic curvature and phase transitions in Kerr-Newman black holes, Phys. Rev. D 78, 024016 (2008), arXiv:0802.1326. G. Ruppeiner, Thermodynamic curvature: pure fluids to black holes, J. Phys.: Conf. Series 410, 012138 (2013), arXiv:1210.2011.
  * (34) G. Ruppeiner, Thermodynamic Black Holes, Entropy 20, 460 (2018), arXiv:1803.08990.
  * (35) Y.-G Miao and Z.-M Xu, Microscopic structures and thermal stability of black holes conformally coupled to scalar fields, Nucl. Phys. B 942 (2019) 205-220, arXiv:1711.01757. Y.-G Miao and Z.-M Xu, Validity of Maxwell equal area law for black holes conformall coupled to scalar fields in $AdS_{5}$ spacetime, Eur. Phys. J. C 77 (2017) 403, arXiv:1610.01769.
  * (36) R. Li and J. Wang, Thermodynamics and kinetics of Hawking-Page phase transition, Phys. Rev. D 102 (2020) 024085. R. Li, K. Zhang, and J. Wang, The kinetics and its turnover of Hawking-Page phase transition under the black hole evaporation, arXiv:2105.00229. R. Li, K. Zhang, and J. Wang, Thermal dynamic phase transition of Reissner-Nordström Anti-de Sitter black holes on free energy landscape, J. High Energ. Phys 10 (2020) 090, arXiv:2008.00495.
  * (37) S.-W Wei, Y.-X Liu, and Y.-Q Wang, Dynamic properties of thermodynamic phase transition for five-dimensional neutral Gauss-Bonnet AdS black hole on free energy landscape, arXiv:2009.05215. S.-W Wei, Y.-Q Wang, Y.-X Liu, and R. B. Mann, Observing dynamic oscillatory behavior of triple points among black hole thermodynamic phase transitions, Sci. China Phys. Mech. Astron. 64 (2021) 7, 270411, arXiv:2102.00799.
  * (38) S.-J Yang, R. Zhou, S.-W Wei, and Y.-X Liu, Dynamics and kinetics of phase transition for Kerr AdS black hole on free energy landscape, arXiv:2105.00491.
  * (39) A. N Kumara, S. Punacha, K. Hegde, C. L. Ahmed Rizwan, and K. M Ajith, Dynamics and kinetics of phase transition for regular AdS black holes in general relativity coupled to non-linear electrodynamics, arXiv:2106.11095.
  * (40) J.-X Mo and S.-Q Lan, Dynamic phase transition of charged dilaton black holes, arXiv:2105.00868.
  * (41) J. Jing and Q. Pan, Quasinormal modes and second order thermodynamic phase transition for Reissner-Nordstrom black hole, Phys. Lett. B 660 (2008) 13, arXiv:0802.0043 [gr-qc].
  * (42) E. Berti and V. Cardoso, Quasinormal modes and thermodynamic phase transitions, Phys. Rev. D 77 (2008) 087501, arXiv:0802.1889 [hep-th].
  * (43) X. He, B. Wang, S. Chen, R.-G Cai, and C.-Y Lin, Quasinormal modes in the background of charged Kaluza-Klein black hole with squashed horizons, Phys. Lett. B 665 (2008) 392, arXiv:0802.2449 [hep-th].
  * (44) J. Shen, B. Wang, C.-Y Lin, R.-G Cai, and R.-K Su, The phase transition and the quasi-normal modes of black holes, J. High Energy Phys. 0707 (2007) 037, arXiv:hep-th/0703102.
  * (45) X.-P Rao, B. Wang, G.-H Yang, Quasinormal modes and phase transition of black holes, Phys. Lett. B 649 (2007) 472, arXiv:0712.0645 [gr-qc].
  * (46) Y. Liu, D.-C Zou, and B. Wang, Signature of the Van der Waals like small-large charged AdS black hole phase transition in quasinormal modes, J. High Energy Phys. 1409 (2014) 179, arXiv:1405.2644 [hep-th].
  * (47) S. Mahapatra, Thermodynamics, phase transition and quasinormal modes with Weyl corrections, J. High Energy Phys. 1604 (2016) 142, arXiv:1602.03007 [hep-th].
  * (48) M. Chabab, H. El Moumni, S. Iraoui, and K. Masmar, Behavior of quasinormal modes and high dimension RN CAdS black hole phase transition, Eur. Phys. J. C 76 (2016) 676, arXiv:1606.08524 [hep-th].
  * (49) P. Prasia and V. C. Kuriakose, Quasinormal modes and thermodynamics of linearly charged BTZ black holes in massive gravity in (anti) de Sitter space-time, Eur. Phys. J. C 77 (2017) 27, arXiv:1608.05299 [gr-qc].
  * (50) D.-C Zou, Y. Liu, and R.-H Yue, Behavior of quasinormal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity, Eur. Phys. J. C 77 (2017) 365, arXiv:1702.08118 [gr-qc].
  * (51) B. Liang, S.-W Wei, and Y.-X Liu, Quasinormal modes and Van der Waals like phase transition of charged AdS black holes in Lorentz symmetry breaking massive gravity, Int. J. Mod. Phys. D 28 (09) (2019) 1950113, arXiv:1712.01545 [gr-qc].
  * (52) C. G. Goebel, Comments on the vibrations of a black hole, Astrophys. J. 172 (1972) L95.
  * (53) V. Ferrari and B. Mashhoon, New approach to the quasinormal modes of a black hole, Phys. Rev. D 30 (1984) 295.
  * (54) B. Mashhoon, Stability of charged rotating black holes in the Eikonal approximation, Phys. Rev. D 31 (1985) 290.
  * (55) V. Bozza, Gravitational lensing in the strong field limit, Phys. Rev. D 66 (2002) 103001, arXiv:gr-qc/0208075.
  * (56) V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Zanchin, Geodesic stability, Lyapunov exponents and quasinormal modes, Phys. Rev. D 79 (2009) 064016, arXiv:0812.1806 [hep-th].
  * (57) I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev, Connection between black-hole quasinormal modes and lensing in the strong deflection limit, Phys. Rev. Lett. 104 (2010) 251103, arXiv:1003.1609 [gr-qc].
  * (58) S.-W Wei, Y.-X Liu, and H. Guo, Relationship between high-energy absorption cross section and strong gravitational lensing for black hole, Phys. Rev. D 84 (2011) 041501, arXiv:1103.3822 [hep-th].
  * (59) S. Hod, The fastest way to circle a black hole, Phys. Rev. D 84 (2011) 104024, arXiv:1201.0068 [gr-qc].
  * (60) S.-W Wei and Y.-X Liu, Establishing a universal relation between gravitational waves and black hole lensing, Phys. Rev. D 89 (2014) 047502, arXiv:1309.6375 [gr-qc].
  * (61) B. Raffaelli, Strong gravitational lensing and black hole quasinormal modes: towards a semiclassical unified description, Gen. Relativ. Gravit. 48 (2) (2016) 16, arXiv:1412.7333 [gr-qc].
  * (62) V. Cardoso, E. Franzin, and P. Pani, Is the gravitational-wave ringdown a probe of the event horizon? Phys. Rev. Lett. 116 (17) (2016) 171101, Erratum: Phys. Rev. Lett. 117 (2016) 089902, arXiv:1602.07309 [gr-qc].
  * (63) R. A. Konoplya and Z. Stuchlik, Are Eikonal quasinormal modes linked to the unstable circular null geodesics? Phys. Lett. B 771 (2017) 597, arXiv:1705.05928 [gr-qc].
  * (64) S. R. Dolan and A. C. Ottewill, On an expansion method for black hole quasinormal modes and regge poles, Class. Quantum Gravity 26 (2009) 225003, arXiv:0908.0329 [gr-qc].
  * (65) M. Cvetic, G. W. Gibbons, and C. N. Pope, Photon spheres and sonic horizons in black holes from supergravity and other theories, Phys. Rev. D 94 (2016) 106005, arXiv:1608.02202 [gr-qc].
  * (66) G. W. Gibbons, C. M. Warnick, and M. C. Werner, Light-bending in Schwarzschild-de-Sitter: projective geometry of the optical metric, Class. Quantum Gravity 25 (2008) 245009, arXiv:0808.3074 [gr-qc].
  * (67) S.-W Wei and Y.-X Liu, Photon orbits and thermodynamic phase transition of d-dimensional charged AdS black holes, Phys. Rev. D 97 (2018) 104027, arXiv:1711.01522 [gr-qc]. S.-W Wei, Y.-X Liu, and Y.-Q Wang, Probing the relationship between the null geodesics and thermodynamic phase transition for rotating Kerr-AdS black holes, Phys. Rev. D 99 (2019) 044013, arXiv:1807.03455 [gr-qc].
  * (68) A. N. Kumara, C. L. Ahmed Rizwan, S. Punacha, and K. M. Ajith, Photon orbits and thermodynamics phase transition of regular AdS black holes, Phys. Rev. D 102 (2020) 084059.
  * (69) S.-Z Han, J. Jiang, M. Zhang, and W.-B Liu, Photon orbits and thermodynamic phase transition in Gauss-Bonnet AdS black holes, arXiv:1812.11862 [gr-qc].
  * (70) M. Chabab, H. El Moumni, S. Iraoui, and K. Masmar, Probing correlation between photon orbits and phase structure of charged AdS black hole in massive gravity background, arXiv:1902.00557 [hep-th].
  * (71) Y.-M Xu, H.-M Wang, Y.-X Liu, and S.-W Wei, Photon sphere and reentrant phase transition of charged Born-Infeld-AdS black holes, Phys. Rev. D 100 (2019) 104044.
  * (72) H.-T Li, Y. C, and S.-J Zhang, Photon orbits and phase transitions in Born-Infeld-dilaton black holes, Nuclear Phys. B 954 (2020) 114975.
  * (73) M. Zhang, S.-Z Han, J. Jiang, and W.-B Liu, Circular orbit of a test particle and phase transition of a black hole, Phys. Rev. D 99 (2019) 065016, arXiv:1903.08293 [hep-th].
  * (74) A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Holography, thermodynamics, and fluctuations of charged AdS black holes, Phys. Rev. D 60, 104026 (1999). A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Charged AdS Black Holes and Catastrophic Holography, Phys. Rev. D 60 (1999) 064018, arXiv:hep-th/9902170.
  * (75) C. V. Johnson, Critical Black Holes in a Large Charge Limit, Mod. Phys. Lett. A 33, 1850175 (2018), arXiv:1705.01154. C. V. Johnson, An Exact Model of the Power/Efficiency Trade-Off While Approaching the Carnot Limit, Phys. Rev. D 98, 026008 (2018), arXiv:1703.06119.
  * (76) M. Born, On the Quantum Theory of the Electromagnetic Field, Proc. Roy. Soc. Lond. A 143 1934 (849), 399-410, http://www.jstor.org/stable/96100.
  * (77) Y. Kats, L. Motl, and M. Padi, Higher-order corrections to mass-charge relation of extremal black holes, Journal of High Energy Physics, 2007 (12) 068, https://doi.org/10.1088/1126-6708/2007/12/068.
  * (78) D. Anninos and G. Pastras, Thermodynamics of the Maxwell-Gauss-Bonnet Anti-de Sitter black hole with higher derivative gauge corrections, Journal of High Energy Physics 07 (2009) 030, arXiv:0807.3478.
  * (79) R.-G. Cai, Z.-Y. Nie, and Y.-W. Sun, Shear Viscosity from Effective Couplings of Gravitons, Phys. Rev. D 78 (2008) 126007, arXiv:0811.1665.
  * (80) N. Seiberg and E. Witten, Journal of High Energy Physics 09, 032 (1999), arXiv:hep-th/9908142.
  * (81) P. Dirac, Lectures on Quantum Mechanics, Dover Books on Physics (Dover Publications, 2013).
  * (82) Z. Bialynicka-Birula and I. Bialynicki-Birula, Phys. Rev. D 2, 2341 (1970).
  * (83) H. El Moumni, Phys. Lett. B 776, 124 (2018).
  * (84) M. Zhang, Z.-Y. Yang, D.-C. Zou, W. Xu, and R.-H. Yue, Gen. Rel. Grav. 47, 14 (2015), arXiv:1412.1197.
  * (85) C. Corda and H. J. Mosquera Cuesta, Astropart. Phys. 34, 587 (2011), arXiv:1011.4801 [physics.gen-ph].
  * (86) S. H. Mazharimousavi and M. Halilsoy, Phys. Lett. B 681, 190 (2009), arXiv:0908.0308.
  * (87) V. A. De Lorenci, R. Klippert, M. Novello, and J. M. Salim, Phys. Rev. D 65, 063501 (2002).
  * (88) P. K. Yerra and C. Bhamidipati, A Note on Critical Nonlinearly Charged Black Holes, Mod. Phys. Lett. A 34 (2019) 27, arXiv:1806.08226.
  * (89) Y.-Z Du, H.-F Li, F. Liu, and L.-C Zhang, Dynamic Property of Phase Transition for Non-Linear Charged Anti-de Sitter black holes, Chin. Phys. C 46 (2022) 5, arXiv:2112.10398.