# Trajectories of photons around a rotating black hole with unusual
asymptotics

Yong-Zhuang Li1<EMAIL_ADDRESS>Xiao-Mei Kuang2<EMAIL_ADDRESS>1 Research center for theoretical physics, School of Science, Jiangsu
University of Science and Technology, Zhenjiang, China.  2Center for
Gravitation and Cosmology, College of Physical Science and Technology,
Yangzhou University, Yangzhou, 225009, China

###### Abstract

Most black hole solutions are characterized with asymptotically flat, or
asymptotically (anti) de-Sitter behaviors, but some black holes with unusual
asymptotics have also been constructed, which is believed to provide
remarkable insights into our understanding of the nature of gravity. In this
paper, focusing on a rotating black hole with unusual asymptotics in Einstein-
Maxwell-dilaton (EMD) theory, we innovatively analyze the photons’
trajectories around this black hole background, showing that the unusual
asymptotics has significant influences on the photons’ trajectories. We expect
that our analysis could give more insights in the scenario of black holes’
shadow and image.

##  I Introduction

At present, string theory and loop quantum theory appear as two most promising
candidates to unify gravity with the other fundamental interactions in nature
Rovelli2008LPG ; Oriti2009 ; Kiritsis2019 . However, due to technical
difficulties and conceptual ambiguities, the problem of reconciling quantum
theory and general relativity remains open. A complete quantum gravity by one
common representation still fascinates the theoretical physics communities.
Nevertheless, as the predictions of general relativity, black holes provide
natural laboratories for testing different approaches and shedding new light
on quantum gravity in the strong field regime. Benefiting from the advances in
observational techniques, the more we learn about black holes, the closer we
get to a comprehensive theory of quantum gravity.

Over the past few decades, searching the black hole configurations in string
and supergravity theory has attracted the attention of many physicists, see
Myers1986aop ; Myers1987npb ; Callan1989npb ; Garfinkle1991prd ; Witten1991prd
; Maldacena1996arxiv ; Strominger1996plb ; Callan1996npb ; Maldacena1996npb ;
Breckenridge1997plb ; Emparan2000prl ; Mohaupt2000cqg ; Maeda2011ptp ;
Shahbazi2013 ; Pieri2019  and many other important references. Especially,
dilaton black holes have always been a major focus of these investigations,
which arise in the low-energy string theory and couple in a nontrivial way to
other fields such as gauge fields. Einstein-Maxwell-dilaton (EMD) gravity,
where the dilaton is non-minimally coupled to the Maxwell field, then provides
a natural theoretical model to investigate the effects of the dilaton on the
properties of the black hole spacetime Garfinkle1991prd ; Gibbonsnpb1988 ;
Yoshimura1987plb ; Horowitz1992prd ; Rakhmanov1994prd ; Wiltshire1994prd ;
Mann1995npb ; Sermutlu1995cqg ; Soh1998prd ; Yazadjiev1999ijmpd ;
Yazadjiev2001arxiv ; Zhang2004prd ; Leygnac2004prd ; Wang2004prd ;
Yazadjiev2005cqg ; Yazadjiev2005prd ; Pakravan2006prd ; Aliev2006prd ;
Ghosh2007prd ; Sheykhi2008prd ; Sheykhi2008prd2 ; Soda2009prd ;
Allahverdizadeh2010grg ; Nedkova2016prd ; Wu2019jhep ; Stetsko2019epjc ;
Ebrahimkhas2020epjp ; Huang2021prd ; Ghezelbash2021epjc ; Qaemmaqami2022epjp ;
Fabris2022prd ; QiQi:2023nex .

Especially, Poletti and Wiltshire derived the global properties of static
spherically symmetric solutions to the EMD system in the presence of a
Liouville-type dilaton potential, showing that with the exception of a pure
cosmological constant ‘potential’, no asymptotically flat, asymptotically de
Sitter (dS) or asymptotically anti-de Sitter (AdS) solutions exist in these
models Wiltshire1994prd . Chan et al. then presented a new class of such black
hole solutions in $n\geq 4$-dim EMD gravity Mann1995npb . Some exact
topological black hole solutions in the EMD theory with a Liouville-type
dilaton potential were generalized by Cai et al. in Ref. Soh1998prd , also see
Ref. Yazadjiev2005cqg . For the case of rotating black holes, Clément and
Leygnac generated two classes of rotating dyonic black hole solutions with a
specialized dilaton coupling parameter $\alpha^{2}=3$ from the non-
asymptotically flat magnetostatic or electrostatic black holes in Ref.
Leygnac2004prd . With one or two Liouville-type dilaton potential(s) in an
arbitrary dimension, Cai and Wang looked for and analyzed in details some
(static, electrically or magnetically charged) exact solutions of EMD gravity,
showing that some of these neither asymptotically flat nor AdS/dS solutions
have higher dimensional origins which have well-behaved asymptotics
Wang2004prd . New 5-dim, non-asymptotically flat black holes with rotation in
one plane were derived by Yazadjiev as a limiting case of new rotating, non-
asymptotically flat black ring solutions in a 5-dim EMd theory arising from a
six-dimensional Kaluza-Klein theory Yazadjiev2005prd . Later, Fabris and
Marques exploited the method of gravitational anomaly to compute the
temperature of non-asymptotically flat dilatonic black holes Marques2012epjc ,
and the thermodynamics in the canonical and grand canonical ensembles of a
class of non-asymptotically flat black holes of the 4D EMD theory with
spherical symmetry were analysed in Ref. Marques2013grg .

Although such black hole solutions are not as physically well-established as
asymptotically flat or AdS/dS solutions, there are still sufficient
motivations to conduct detailed investigation and discussion on them as the
suggestions in previously mentioned references. On the one hand, it has shown
that the methods originally developed for asymptotically flat or AdS/dS black
holes, such as the quasi-local energy approach and gravitational anomaly,
could be applied to compute the mass, angular momentum or Hawking radiation of
non-asymptotically flat rotating black holes Leygnac2004prd ; Marques2012epjc
, see also Ref. Zhang2002prd  where Zhang et al. checked the mass bound
conjecture in de Sitter space with the topological de Sitter solution and its
dilatonic deformation, which provides the evidence in favor of the conjecture.
On the other hand, it has been widely believed that any consistent theory of
quantum gravity should be holographic Hooft1993arxiv ; Susskind1995jmp .
Indeed, string theory has beautifully demonstrated holography Witten1998arxiv
; Polyakov1998plb ; Witten1998arxiv2 ; Polyakov1998npb ; Maldacena1999ijtp .
As the low-energy limits of string theory, holography for EMD theories,
including solutions with unusual asymptotic behaviors, have also been set up
and investigated in different literature Kanitscheider2008jhep ;
Goldstein2010jhep ; Chen2010jhep ; Gubser2010prd ; Charmousis2010jhep ;
DeWolfe2011prd ; Lee2011prd ; Taylor2012jhep ; Cai2012jhep ; Kuang:2012tq ;
Ghosh2015ass ; Park2015prd ; Rougemont2017prd ; Swingle2018jhep ; Goto2019jhep
; Mahapatra2019jhep ; Jain2020prd ; Bayona2020prd ; Fu2021prd ;
Rougemont2023arxiv  and the references therein.

In this article, we will examine the appearance of the dark silhouette of the
slowly rotating EMD black holes with unusual asymptotic behaviors, which still
deserves an in-depth investigation. Since the publications of the observations
of the Event Horizon Telescope (EHT) on the shadow of the supermassive black
holes M87* and SgrA* Akiyama2019apjl ; Akiyama2022apjl , there have been
numerous articles investigating the nature of the shadows of various types of
black holes, wormholes, or black branes, which we cannot catalog all here. For
EMD theory, Maki and Shiraishi examined the motion of test particles with
arbitrary mass, electric charges and dilatonic charges around a static,
spherically-symmetric charged dilaton black Maki1994cqg ; The light paths of
normal and phantom EMD black holes were determined analytically by the
Weierstrass elliptic functions in Ref. Azreg2013prd ; Amarilla and Eiroa
studied the shadow produced by a spinning Kaluza-Klein black hole in EMD
theory Amarilla2013prd ; In Ref. Wei2013jcap , the shadow cast by EMD-Axion
black hole and naked singularity were studied, while the geodesic motion of
test particles and light in the EMD-Axion spacetime was studied by Flathmann
and Grunau Flathmann2015prd ; The timelike and null geodesics around the
static and the rotating (Kerr-Sen dilaton-axion) dilaton black holes were
considered and analytically solved in terms of Weierstrass elliptic functions
in Ref. Soroushfar2016prd ; The shadow images of charged wormholes in EMD
theory were constructed in Ref. Amir2019aop ; The light deflection of
EM(anti-)D black holes, using the optical geometry and the Gauss-Bonnet
theorem, was studied in Ref. Jusufi2019aop , together with the deflection of
light by a class of asymptotically flat phantom wormholes; The accretion
process in thin disks around static charged and slowly rotating charged
dilaton black holes, using the Novikov-Thorne model, was investigated in Ref.
Sepangi2020epjc ; For more investigations, see Sharma2020ijmpa ;
Qaemmaqami2022epjc ; Sepangi2022prd ; Eiroa2023prd . However, most of these
works are focusing on static spherically symmetric, stationary axisymmetric
spacetimes or spacetimes with well-behaved asymptotic, or slowly rotating
cases generated from the modified Newman-Jans algorithm. Therefore, there is a
need for an extended study of the geodesic properties of spacetimes with
unusual asymptotic.

Meanwhile, on the more theoretical side, the photon spheres play a crucial
role when one images Einstein rings of black holes in AdS spacetimes or
reflects the AdS boundary with the quasinormal modes, which have recently
gained significant attentions Murata2019prl ; Murata2020prd ; Nambu2020entropy
; Tsujimura2021jhep ; Zhang2023jhep ; Huot2023jhep ; Yonezawa2023prr ;
Li2023arxiv ; Yoda2023arxiv ; Sun2023arxiv . Investigations on the null
geodesic and photon sphere of slowly rotating EMD black holes with unusual
asymptotic could thus uncover a little piece of the new territory to the
gauge/gravity correspondence, especially when we have the possibilities to
experimentally observe the light path in the framework of analogue gravity
Visser2005lrr ; Genov2013nature .

This paper is organized as follows. In Sec. II, we start by a briefly
introduction to the slowly rotating Einstein-Maxwell-Dilaton black holes. In
Sec. III, by solving the geodesic equation, we study the trajectories of
photon on the equatorial plane and on a spherical surface, respectively.
Conclusions and discussions are offered in Sec. IV.

##  II Einstein-Maxwell-Dilaton theory

Following Refs. Ghosh2007prd ; Sheykhi2008prd ; Stetsko2019epjc ; Huang2021prd
, the action integral for a 4-dimensional gravity with dilaton and linear
Maxwell fields can be written in the form

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int_{\mathscr{M}}\text{d}^{4}x\sqrt{-g}$ (1)
$\displaystyle\times\left(R-2\nabla^{\mu}\Phi\nabla_{\mu}\Phi-V\left(\Phi\right)-e^{-2\alpha\Phi}F_{\mu\nu}F^{\mu\nu}\right)$
$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathscr{M}}\text{d}^{3}x\sqrt{-\gamma}\Theta(\gamma),$

where $R,\,\Phi,\,\alpha$ are the scalar curvature, the dilaton field and the
dilaton-electromagnetic coupling parameter, respectively. $V(\Phi)$ denotes
the potential which depends on the dilaton field and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic filed tensor where $A_{\mu}$ is a component of the
electromagnetic potential. The second term in the action (1) corresponds to
Gibbons-Hawking boundary term. $\partial\mathscr{M}$ denotes the boundary of
the manifold $\mathscr{M}$. $\gamma_{\mu\nu}$ is the induced metric on
$\partial\mathscr{M}$ and $\Theta$ is the trace of the extrinsic curvature
tensor $\Theta_{\mu\nu}$ on $\partial\mathscr{M}$. Varying the action with
respect to the gravitational field $g_{\mu\nu}$, the dilaton field $\Phi$ and
the electromagnetic potential $A_{\mu}$, one will obtain the field equations
in the following forms:

$\displaystyle
R_{\mu\nu}=\frac{g_{\mu\nu}}{2}\left(V(\Phi)-e^{-2\alpha\Phi}F_{\rho\sigma}F^{\rho\sigma}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+2\partial_{\mu}\Phi\partial_{\nu}\Phi+2e^{-2\alpha\Phi}F_{\mu\sigma}F^{\sigma}_{\nu};$
(2) $\displaystyle\nabla_{\mu}\nabla^{\mu}\Phi=\frac{1}{4}\frac{\partial
V}{\partial\Phi}-\frac{\alpha}{2}e^{-2\alpha\Phi}F_{\rho\sigma}F^{\rho\sigma};$
(3) $\displaystyle\nabla_{\mu}\left(e^{-2\alpha\Phi}F^{\mu\nu}\right)=0.$ (4)

The above field equations admits a slowly rotating black hole if some
conditions are satisfied:

$\displaystyle A_{\varphi}$ $\displaystyle=$ $\displaystyle
aqh(r)\sin^{2}\theta;$ (5) $\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle\Lambda_{0}e^{\tilde{\lambda}_{0}\Phi}+\Lambda
e^{\tilde{\lambda}\Phi},$ (6)

and $h(r)$ is a function to be defined. The metric is then given by

$\displaystyle\text{d}s^{2}$ $\displaystyle=$
$\displaystyle-W(r)\text{d}t^{2}+\frac{\text{d}r^{2}}{W(r)}-2af(r)\sin^{2}{\theta}\text{d}t\text{d}\phi$
(7)
$\displaystyle+r^{2}R^{2}(r)\left(\text{d}\theta^{2}+\sin^{2}{\theta}\text{d}\phi^{2}\right),$

where

$\displaystyle W(r)$ $\displaystyle=$ $\displaystyle-
mr^{2\gamma-1}+\frac{(1+\alpha^{2})}{(1-\alpha^{2})}\left(\frac{r}{b}\right)^{2\gamma}$
(8)
$\displaystyle-\frac{\Lambda(1+\alpha^{2})^{2}}{2(3-\alpha^{2})}\frac{r^{2(1-\gamma)}}{b^{-2\gamma}}+q^{2}(1+\alpha^{2})\frac{r^{2\gamma-2}}{b^{2\gamma}};$
$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
mb^{2\gamma}r^{2\gamma-1}+\frac{\Lambda}{2}\frac{(1+\alpha^{2})^{2}}{(3-\alpha^{2})}\frac{r^{2(1-\gamma)}}{b^{-2\gamma}}$
(9) $\displaystyle-q^{2}(1+\alpha^{2})\frac{r^{2\gamma-2}}{b^{2\gamma}};$
$\displaystyle\Phi(r)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{(1+\alpha^{2})}\ln{\left(\frac{b}{r}\right)};$
(10) $\displaystyle R(r)$ $\displaystyle=$ $\displaystyle e^{\alpha\Phi};$
(11) $\displaystyle h(r)$ $\displaystyle=$ $\displaystyle r^{-1}.$ (12)

The parameters $m,\,q,\,a$ are relating to the black hole’s mass, charge and
the angular momentum, respectively. Especially, the black hole’s physical mass
$M$ can be represented by

$M=\frac{2b^{2\gamma}\omega_{2}}{16\pi(1+\alpha^{2})}m,$ (13)

where $\omega_{2}$ is the surface area of a $2$-dimensional unit hypersphere.
For other parameters, one could take them in the following form:

$\displaystyle\Lambda_{0}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}}{b^{2}(\alpha^{2}-1)};$ (14)
$\displaystyle\tilde{\lambda}_{0}$ $\displaystyle=$
$\displaystyle\frac{2}{\alpha};$ (15) $\displaystyle\tilde{\lambda}$
$\displaystyle=$ $\displaystyle 2\alpha;$ (16) $\displaystyle\gamma$
$\displaystyle=$ $\displaystyle\frac{\alpha^{2}}{1+\alpha^{2}}.$ (17)

$b$ is an integration constant that can be set to be unity, while $\Lambda$ is
treated as an effective cosmological constant that appears due to the dilaton
potential (6). In the limit $\alpha\rightarrow 0$ one will recover the metric
functions for a $(n+1)$-dimensional slowly rotating charged Kerr-AdS black
hole Aliev2006prd ; Sheykhi2008prd2  while $\alpha=1$ corresponds to the low-
energy limit of string theory. In this article we will focus on the weak
coupling $\alpha=0.1$.

##  III Null geodesic

###  III.1 Null geodesics on the equatorial plane

As an example, in this section we will investigate the geodesic structure of
photons in a slowly rotating EMD spacetime (7). The Lagrangian is written as

$\mathscr{L}=\frac{1}{2}\left(g_{tt}\dot{t}^{2}+g_{rr}\dot{r}^{2}+2g_{t\phi}\dot{t}\dot{\phi}+g_{\phi\phi}\dot{\phi}^{2}+g_{\theta\theta}\dot{\theta}^{2}\right),$
(18)

where the dot represents the derivative with respect to an affine parameter
$\lambda$. Then two Killing vectors related to the time and azimuthal angle
translational invariance of the spacetime provide two conserved quantities,
the energy $E$ and the angular momentum $L_{\phi}$ of the photon:

$\displaystyle\frac{\partial\mathscr{L}}{\partial\dot{t}}$ $\displaystyle=$
$\displaystyle g_{tt}\dot{t}+g_{t\phi}\dot{\phi}\equiv-E,$
$\displaystyle\frac{\partial\mathscr{L}}{\partial\dot{\phi}}$ $\displaystyle=$
$\displaystyle g_{\varphi\phi}\dot{\phi}+g_{t\phi}\dot{t}\equiv L_{\phi}$ (19)

Figure 1:  The effective potential as a function of $r$ for different
$\Lambda$ with $b=1,a=0.1,m=1,q=0.1,\alpha=0.1$. The horizontal lines show
$V_{eff}(r\rightarrow\infty)$ for each $\Lambda$. The solid black line stands
for $V_{eff}(r)$ with vanishing $\Lambda$.

To proceed, one usually needs the third quantity, namely the Carter constant
to separate the Euler-Lagrange differential equations for geodesic motion
Carter1968pr . However, the Jacobi action is inseparable in the present
spacetime. To see this, we define the action as

$S=-\frac{1}{2}\mu^{2}\lambda-Et+L_{\phi}\phi+\tilde{S}(r,\theta),$ (20)

with

$\frac{\partial S}{\partial\lambda}=\frac{1}{2}g^{ij}\frac{\partial
S}{\partial x^{i}}\frac{\partial S}{\partial x^{j}},$

one will have

$\displaystyle\left(\frac{\partial\tilde{S}}{\partial\theta}\right)^{2}+L^{2}_{\phi}\csc{\theta}^{2}+r^{2}R(r)^{2}W(r)\left(\frac{\partial\tilde{S}}{\partial
r}\right)^{2}$ (21)
$\displaystyle=\frac{\left(aL_{\phi}f(r)-Er^{2}R(r)^{2}\right)^{2}}{a^{2}f(r)^{2}\sin(\theta)^{2}+r^{2}R(r)^{2}W(r)},$

which shows the indivisibility. Indeed, the metric (7) is a Petrov type I
spacetime, and violates the so-called Levi-Civita theorem Civita1904MathAnn ;
Benenti1980book ; Katanaev2023arxiv .

We now focus on the equatorial plane with $\theta=\pi/2$. In such case, the
geodesic equations for photons are given by

$\displaystyle\dot{r}^{2}$ $\displaystyle=$
$\displaystyle\frac{W(r)\left(E^{2}r^{2}R(r)^{2}-2aEL_{\phi}f(r)-L_{\phi}^{2}W(r)\right)}{a^{2}f(r)^{2}+r^{2}R(r)^{2}W(r)},$
$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{Er^{2}R(r)^{2}-aL_{\phi}f(r)}{a^{2}f(r)^{2}+r^{2}R(r)^{2}W(r)},$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{aEf(r)+L_{\phi}W(r)}{a^{2}f(r)^{2}+r^{2}R(r)^{2}W(r)}.$
(22)

By eliminating the affine parameter, the relation between $r$ and $\phi$ is
written as

$\left(\frac{dr}{d\phi}\right)^{2}+\frac{W(r)\left(2aBf(r)-r^{2}R(r)^{2}+B^{2}W(r)\right)\left(a^{2}f(r)^{2}+r^{2}R(r)^{2}W(r)\right)}{\left(af(r)+BW(r)\right)^{2}}=0,$
(23)

where $B\equiv L_{\phi}/E$ is the impact parameter. By setting a constant
radial motion in (III.1), the effective potential is defined as

$V_{eff}(r)=\frac{W(r)L_{\phi}}{-af(r)\pm\sqrt{a^{2}f(r)^{2}+r^{2}R(r)^{2}W(r)}},$
(24)

where $``+"$ for prograde motion and $``-"$ for retrograde motion,
respectively.

|  |
---|---|---

Figure 2:  The number of photon orbits $n$ (top panel) and trajectories of
photons (bottom panel) as a function of the impact parameter $B$ for the
slowly rotating EMD black holes with $\Lambda=0$ (left column), $\Lambda=-1$
(middle column) and $\Lambda=-10$ (right column). The black lines, gold lines,
and red lines correspond to $(n<3/4)$, $(3/4<n<5/4)$, and $(n>5/4)$,
respectively. The black disk and the dashed curves denote the event horizon
and photon sphere.

Fig. 1 shows the behavior of the effective potential as a function of $r$ for
$\Lambda=-1,\,-5$ and $-10$, respectively. Other parameters have been
specifically chosen as $b=1,a=0.1,m=1,q=0.1,\alpha=0.1$, considering the
slowly rotating case and weak coupling. The values of these parameters should
be restricted to guarantee the existence of black hole horizon, i.e., $W(r)$
should have at least one solution111Actually for the parameters we have
chosen, $W(r)=0$ has two solutions $r_{+}>r_{-}>0$. However, we only consider
the photon orbits with radius larger than $r_{+}$.. As a comparison, the solid
black line describes the behavior of the effective potential when $\Lambda$ is
zero but other parameters remain unchanged. Unsurprisingly, the nonzero
$\Lambda$ leads to a finite $V_{eff}$ at $r\rightarrow\infty$, while
$V_{eff}(r\rightarrow\infty)=0$ for $\Lambda=0$.

The existence of circular geodesics require $E=V_{eff}(r_{ph})$ and
$V_{eff}^{\prime}(r_{ph})=0$, where $r_{ph}$ is the radius of the photon
sphere. With the previous chosen parameters and the numerical method, we plot
the number of photon orbits $n$ and trajectories of photons as a function of
the impact parameter $B$ for the slowly rotating EMD black holes with
parameter $\Lambda=-10,\Lambda=-1$ and $\Lambda=0$ in Fig. 2. It should be
noted that in our numerical calculation, the light source is located at a very
distant position, which can be approximated as being infinity. Following the
definitions in Ref. Gralla:2019xty  the photons are classified into three
classes according to $n=\phi/2\pi$: the first class is defined as $n\leq 3/4$,
where the deflection angle of the photons is less than $3\pi/2$. The second
class with $3/4<n\leq 5/4$, where light rays deflect with angles
$3\pi/2<\phi\leq 5\pi/2$. The final class is that the light rays deflect with
$n>5/4$.

One can tell that due to the unusual asymptotics, the trajectories of photons
diffuse when the impact parameter $B$ increases. Especially, for the larger
absolute value of $\Lambda$, the diffusion is more obvious. Meanwhile, to make
sure the square is positive, $B$ is bounded by (23). This restriction could be
used to understand the repulsive curves shown in the third column of Fig. 2.
Indeed, for large enough $r$, the critical condition $dr/d\phi=0$ for (23)
could be approximately expressed as

$r_{min}^{\zeta}=\frac{\zeta\left[B\Lambda(1+\alpha^{2})^{2}(2a-B)+2\alpha^{2}-6\right]}{4B^{2}(3-\alpha^{2})},$
(25)

where $\zeta=-2+4\gamma$. $r_{min}$ is actually the closest point to the black
hole for each photon trajectory in the third class. Since $\zeta<0$ this
equation will finally give us fixed $B_{max}$ relating to nonzero $\Lambda$,
i.e., for nonzero $\Lambda$, the value of $B$ cannot be arbitrarily large. For
$\Lambda=0$, (25) is simply $r_{min}^{\zeta}=-\zeta/(2B^{2})$. With this in
hand, by transforming $(r,\phi)$ into the Cartesian coordinates $(x,y)$ we can
always find a “straight line” $y=-kx+\mathfrak{C}$ where $k>0$ and
$\mathfrak{C}=r(\phi=\pi/2)$. This will also lead us a specified $B_{crit}$
for each nonzero $\Lambda$ such that for $B<B_{crit}$ the trajectories bending
towards the black hole while for $B_{crit}<B<B_{max}$ the trajectories bending
away from the black hole. For the case $\Lambda=0$ there does not exist such a
$B_{crit}$, see Appendix A for more calculations. This analytical analysis
gives the rationality of the diffusion effect and even the repulsive
trajectories, but the physics behind these phenomena deserve deeper study.
Moreover, the central shadow region that captures the nearby photon and the
widths of $B$ for the second class photons as well as the third class photons
are all suppressed by larger absolute value of $\Lambda$.

Another point arises from the relation between $B$ and $a$, see Eq. (23) and
Fig. 3. For $B>a$, no singularities exist in Eq. (23) and the path of photons
can fall into the horizon. For $B<a$, the paths have two branches. One will
fall into the horizon and the other one will nonphysically “bounces off”. In
particular, for $\Lambda=0$, the situation occurs only for $B<0$, regardless
of the values of $a$. In Fig. 4, we plot the photon paths for positive and
negative impact parameter $B$ with $\Lambda=-1$ as an example. As expected,
the paths are different for $B$ with same absolute value but different sign,
which is understandable due to the drag effect of the rotating black hole.

Figure 3:  A contour-plot of the condition $af(r)+BW(r)=0$ for $a=0.1$ (red
dotdashed), $a=0.3$ (green dashed), $a=0.5$ (solid black), respectively. The
horizontal lines show the corresponded $a$ and we set $\Lambda=-1$. The curves
are similar for $\Lambda=-10$. Figure 4:  The photon paths for positive
(dashed lines) and negative (solid lines) $B$, respectively. The effective
cosmological constant $\Lambda=-1$. $B=\pm 0.1,\pm 1.77,\pm 2$.

###  III.2 Null geodesics on the spherical orbits

Since the black hole is assumed to rotate slowly, performing a perturbative
expansion of $a$ is reasonable. Discarding the limitation of $\theta=\pi/2$,
and utilizing the Hamilton-Jacobi technologies, the geodesic equations
governing the motion of lightlike particles are given by

$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{Er^{2}R(r)^{2}-aL_{z}f(r)}{r^{2}R(r)^{2}W(r)},$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{aEf(r)+L_{z}\csc^{2}{\theta}W(r)}{r^{2}R(r)^{2}W(r)},$
$\displaystyle r^{2}R(r)^{2}\dot{r}^{2}$ $\displaystyle=$ $\displaystyle
E^{2}r^{2}R(r)^{2}-2aEL_{z}f(r)-W(r)\mathscr{C},$ $\displaystyle=$
$\displaystyle\mathscr{R}(r),$ $\displaystyle r^{4}R(r)^{4}\dot{\theta}^{2}$
$\displaystyle=$
$\displaystyle\left(\mathscr{C}-L_{z}^{2}\csc^{2}(\theta)\right)=\Theta(\theta),$
(26)

where $\mathscr{C}$ is the Carter constant and we use $L_{z}$ for the angular
momentum corresponding to the symmetry axis with different $\theta$. Note for
the equatorial plane, $\dot{\theta}=0$, so $\mathscr{C}=L_{z}^{2}$.
Substituting the Carter constant in the third equation, one will recover Eq.
(III.1) with small $a$. The spherical orbits will require

$\mathscr{R}(r)=0,\quad\frac{d\mathscr{R}(r)}{dr}=0,$ (27)

which generates two equations:

$\displaystyle\frac{\mathscr{C}}{E^{2}}$
$\displaystyle=\frac{rR(r)\left(rR(r)f^{\prime}(r)-2f(r)\left(R(r)+rR^{\prime}(r)\right)\right)}{W(r)f^{\prime}(r)-f(r)W^{\prime}(r)},$
$\displaystyle\frac{L_{z}}{E}$
$\displaystyle=\frac{rR(r)\left(2W(r)\left(R(r)+rR^{\prime}(r)\right)-rR(r)W^{\prime}(r)\right)}{2a\left(W(r)f^{\prime}(r)-f(r)W^{\prime}(r)\right)}.$
(28)

Then one can define the effective inclination angle $i$ as

$\cos{i}=\frac{L_{z}}{\sqrt{L^{2}_{z}+\mathscr{C}}}.$ (29)

The radius of polar orbits can be calculated by setting $L_{z}=0$, i.e.,
$\cos{i}=0$. While for the equatorial plane, one has $\mathscr{C}=L^{2}_{z}$
so $\cos{i}=\pm 1/\sqrt{2}$. By eliminating $E$ in equations (III.2), one can
obtain the following equation:

$\mathscr{M}_{1}r^{2}+\mathscr{M}_{2}r+\mathscr{M}_{3}+\mathscr{M}_{4}r^{-1}+\mathscr{M}_{5}r^{-2}+\sum_{n=0}^{2}\mathscr{N}_{n}r^{\zeta-n}=0,$
(30)

where $\zeta=-2+4\gamma$, and the detailed expressions of
$\mathscr{M},\mathscr{N}$ are given in Appendix B. Thus the radii of generic
spherical photon orbits are provided by the positive real roots of this
equation.

Finding the analytical solutions to Eq. (30) is a difficult task. But one can
analytically solve the ordinary differential equations of $\theta$ and $\phi$
for each root of (30). With the definition
$\mathscr{Z}(\lambda)=\csc{\theta}$, the forth equation in (III.2) can be
rewritten as

$\left(\mathscr{Z}^{\prime}(\lambda)\right)^{2}=-\xi\mathscr{Z}(\lambda)^{2}+(\xi+\eta)\mathscr{Z}(\lambda)^{4}-\eta\mathscr{Z}(\lambda)^{6},$
(31)

with

$\xi=\frac{vL_{z}^{2}}{(1-v)r^{4}R(r)^{4}},\quad\eta=\frac{L_{z}^{2}}{r^{4}R(r)^{4}},$
(32)

where $v\equiv\sin^{2}{i}\in[1/2,1]$. Note that here $r$ is the root of (30).
The solution to Eq. (31) is

$\mathscr{Z}(\lambda)=\frac{\sqrt{\xi}\sqrt{\sec{[\sqrt{\xi}(\lambda-
c_{\theta})]}^{2}}}{\sqrt{\xi+\eta\tan{[\sqrt{\xi}(\lambda-
c_{\theta})]}^{2}}},$ (33)

where $c_{\theta}$ is a constant determined by initial conditions. If we set
the initial conditions as $\csc\theta_{0}=\mu$, then

$c_{\theta}=\pm\frac{1}{\sqrt{\xi}}\operatorname{arcsec}{\left(\frac{\mu}{\sqrt{\xi}}\sqrt{\frac{(\eta-\xi)\xi}{\eta\mu^{2}-\xi}}\right)}.$
(34)

Due to the oscillations of $\theta$ about equatorial plan, one can choose
either $+$ or $-$. The differential equation of $\phi$ can be recast into

$\phi^{\prime}(\lambda)=\delta+\zeta\mathscr{Z}(\lambda)^{2},$ (35)

with the solution

$\phi(\lambda)=c_{\phi}+\int^{\lambda}_{1}\left(\delta+\zeta\mathscr{Z}(y)^{2}\right)dy$
(36)

and

$\delta=\frac{aEf(r)}{r^{2}R(r)^{2}W(r)},\quad\zeta=\frac{L_{z}}{r^{2}R(r)^{2}}.$
(37)

$c_{\phi}$ is also determined by the initial conditions, and by setting
$\phi_{0}=0$ one will have

$\displaystyle c_{\phi}=\delta$ $\displaystyle+$
$\displaystyle\frac{\zeta}{\sqrt{\eta}}\arctan{\left[\frac{\sqrt{\eta}\tan{[\sqrt{\xi}(1+c_{\theta})]}}{\sqrt{\xi}}\right]}$
(38) $\displaystyle-$
$\displaystyle\frac{\zeta}{\sqrt{\eta}}\arctan{\left[\frac{\sqrt{\eta}\tan{(\sqrt{\xi}c_{\theta})}}{\sqrt{\xi}}\right]}.$

Without causing any confusion, we will set the start point at equatorial plan
so $\mu=0$, $c_{\theta}=0$ for our numerical calculations.

Eq. (33) expresses that the oscillation period of $\theta$ around the
equatorial plane is $T_{\theta}=2\pi/\sqrt{\xi}$. One can then determine the
effect of $\Lambda$ on this period using Eq. (30), which has been shown in
Fig. 5. It is clear that the period $T_{\theta}$ increases as the absolute
value of $\Lambda$ increasing. A diffusion-like behavior also happens when we
compare the period difference for $v=0.55$ and $0.95$ as $|\Lambda|$
increasing, see Sec. III.3 for more discussions. Especially, one can find a
specified root of (30) so $\theta$ and $\phi$ will have same period $2\pi$.

Figure 5:  The effect of $\Lambda$ on the period $T_{\theta}$ for
$v=0.95(top),\,0.75(middle),\,0.55(bottom)$, respectively. Figure 6:  The
bound null geodesics on the spherical orbits for $\Lambda=0$ (top),
$\Lambda=-1$ (middle) and $\Lambda=-10$ (bottom). For each row from left to
right $v=0.75,\,0.55,\,0.55$, which will fix the radius of the spherical
orbits by Eq. (30) together with other chosen parameters
$\alpha=0.1,a=0.1,m=1,q=0.1,E=0.2$. Especially, the first and second columns
correspond to prograde motion, and the third column is retrograde motion,
respectively.

Then we figure out the bound null geodesics on the spherical orbits for
$\Lambda=0$ (the top row), $-1$ (the middle row), $-10$ (the bottom row) with
$v=0.75$ and $v=0.55$, showing in Fig. 6. The differences among the plots in
each row indicate that the effect of unusual asymptotics is significant. We
can also tell that the drag effect reflects in the motions of prograde and
retrograde photons by comparing the trajectories between the second and third
columns.

###  III.3 Photon regions

The region accommodating the photon orbits that stay on a sphere constitute
the so-called photon region. Besides the conditions (27), the photon region
also needs the condition $\Theta(\theta)\geq 0$, which leads to

$rR(r)f^{\prime}(r)-2f(r)\frac{d(rR(r))}{dr}\leq\csc{\theta}^{2}\frac{\left(2rW(r)R^{\prime}(r)+2W(r)R(r)-rR(r)W^{\prime}(r)\right)^{2}}{4a^{2}\left(W(r)f^{\prime}(r)-f(r)W^{\prime}(r)\right)},$
(39)

where the prime denotes a derivative to $r$. The stable or unstable spherical
null geodesics in this region are determined by the sign of
$\mathscr{R}^{\prime\prime}(r)$. The condition
$\mathscr{R}^{\prime\prime}(r)>0$ indicates that the spherical null geodesics
are unstable, while $\mathscr{R}^{\prime\prime}(r)<0$ indicates the stable
spherical null geodesics. We consider only the photon region outside the black
hole horizon.

Fig. 7 shows the photon regions for $\Lambda=-1$ and $-10$. With smaller
horizon radius, the photon region becomes larger. In other words, a negative
$\Lambda$ with larger absolute value will cause a larger photon region. This
feature also explains the behavior in Fig. 5. In fact, the spherical orbits
are more closer to black hole horizon for larger $\Lambda$, resulting in a
shorter period for zenith angle $\theta$.

Figure 7:  The photon region outside the horizon for $\Lambda=-1$ (orange) and
$\Lambda=-10$ (yellow). The red dashed line and black solid line denote the
horizons for $\Lambda=-1$ and $\Lambda=-10$, respectively. The blue circles
and dark diamonds signify the photon spheres on the equatorial plan with
$\Lambda=-1$ and $\Lambda=-10$, respectively.

##  IV Closing remarks

In this article, we analytically and numerically studied the null geodesics
around a slowly rotating black hole with unusual asymptotics in Einstein-
Maxwell-Dilaton gravity. In order to disclose the effects of unusual
asymptotic behaviors on the trajectories of photons, we, as a first attempt,
trace the photons’ motions on the equatorial plane and on the spherical
surface, respectively.

Comparing to asymptotically flat spacetime, we found some interesting features
for the trajectories photons on the equatorial plane of the rotating black
hole with unusual asymptotics denoted by a effective cosmological constant
$\Lambda$. Firstly, the effective potential of the photons asymptotically
approaches a finite constant depending on $\Lambda$, instead of zero in the
asymptotically flat case. Secondly, the unusual asymptotics make the
trajectories of photons diffuse and this diffusion is more significant for the
larger absolute value of $\Lambda$, which corresponds to that the central
shadow region and the widths of $B$ for the second class photons as well as
the third class photons all become smaller as the absolute value of $\Lambda$
increases. In addition, we observed the repulsive trajectories of the photon
in the case with nonzero $\Lambda$, of which we give the analytical
understanding from the equation of motion. We believe that the deeper physics
behind these novel features deserve further investigation, and the work in
this direction is in progress.

Then, we extended the study into spherical orbits by performing a perturbative
expansion on $a$ so we can let $\theta$ be free. We found that it is tough to
analytically solve the compete null geodesic equations, but we can fix the
trajectories in a spherical surface and analytically solve the equations for
$\theta$ and $\phi$. So we managed to plot the bound trajectories of photons
on the spherical surface, where the prints of unusual asymptotics are
explicit. Finally, we figured out the photon regions of this rotating black
hole with unusual asymptotics. Though the black hole with a negative $\Lambda$
with larger absolute value have smaller event horizon, it has a wider photon
region due to the diffusion of light rays.

The study of null geodesic and the trajectories of photons around a central
black hole is the first step to explore the black hole shadow and images.
Based on our results, it is interesting to extend into the shadows related
physics for the black hole with unusual asymptotics in this scenario and
beyond. We also expect that this direction somehow has potential connection
with EHT observations.

###### Acknowledgements.

We appreciate Zi-Liang Wang for helpful discussions. This work is partly
supported by Natural Science Foundation of China under Grants No.12375054 and
Natural Science Foundation of Jiangsu Province under Grant No.BK20211601.

##  Appendix A Calculations for $B_{crit}$

We can use one coordinate transformation, two equations and three points to
determine $B_{crit}$.

  * •

To proceed, we first transform the polar coordinate $(r,\phi)$ into the
Cartesian coordinate $(x,y)$: $x=r\cos{\phi}$, $y=r\sin{\phi}$. Therefore, if
there really exist different types of null geodesics with different
deflections, there must be a critical photon trajectory that can be
represented in the Cartesian coordinates as $y=-kx+\mathfrak{C}$. Then one
will have the relation

$\frac{dy}{dx}=\frac{x+r^{\prime}(\phi)\sin{\phi}}{-y+r^{\prime}(\phi)\cos{\phi}}=-k.$
(A.1)

  * •

Suppose the observer is located at $\mathscr{S}_{o}$ on the $x$-axis, and the
impact parameter of the critical trajectory is $B$, then we have the first
relation

$B=-k\mathscr{S}_{o}+\mathfrak{C},$ (A.2)

where $\mathscr{S}_{o}$ is taken to be finite but large enough in our
calculations.

  * •

Consider $\phi=\pi/2$, which corresponds to $x=0$. Then
$y=\mathfrak{C}=r(\pi/2)$. Using (A.1) and (23) one will get the second
relation:

$-\frac{W(\mathfrak{C})\left(2aBf(\mathfrak{C})-\mathfrak{C}^{2}R(\mathfrak{C})^{2}+B^{2}W(\mathfrak{C})\right)\left(a^{2}f(\mathfrak{C})^{2}+\mathfrak{C}^{2}R(\mathfrak{C})^{2}W(\mathfrak{C})\right)}{\left(af(\mathfrak{C})+BW(\mathfrak{C})\right)^{2}}=k^{2}\mathfrak{C}^{2}.$
(A.3)

The left-hand side of this relation should be positive to make it true. So the
polynomial in the first bracket needs to be negative. As $B$ is limited for
nonzero $\Lambda$ so $\mathfrak{C}$ is also finite.

  * •

The condition $r^{\prime}(\phi)=0$ actually indicates the closet point
$r_{min}$ to the black hole, leading us to the third relation:

$2aBf(r_{min})-r_{min}^{2}R(r_{min})^{2}+B^{2}W(r_{min})=0.$ (A.4)

Then using (A.1) we can find the Cartesian coordinates for this point, i.e.,

$\displaystyle x_{min}$ $\displaystyle=$ $\displaystyle
r_{min}\cos{\phi_{min}}=\frac{k\mathfrak{C}}{1+k^{2}},$ $\displaystyle
y_{min}$ $\displaystyle=$ $\displaystyle
r_{min}\sin{\phi_{min}}=\frac{\mathfrak{C}}{1+k^{2}},$ (A.5)

which represents the fourth relation

$r_{min}=\frac{\mathfrak{C}}{\sqrt{1+k^{2}}}.$ (A.6)

Now we have four relations (A.2), (A.3), (A.4) and (A.6) for four unknown
parameters $(B,k,\mathfrak{C},r_{min})$.

However, for $\Lambda=0$, as been discussed in Sec. III.1, the relation (A.4)
will not provide an upper bound on $B$ for large enough $r$. In other words,
$B$ increases as $r$ increases. This feature violates the relation (A.3) as
$B$ could be arbitrarily large, which indicates that the assumed straight line
$y=-kx+\mathfrak{C}$ does not exist.

##  Appendix B Coefficients

With the definition $v=\sin^{2}{i}$, the coefficients of Eq.(30) are

$\displaystyle\mathscr{M}_{1}$ $\displaystyle=-4v,$ (B.1)
$\displaystyle\mathscr{M}_{2}$
$\displaystyle=4a^{2}m(v-1)(1+\alpha^{2})\Lambda-\frac{4mv(\alpha^{2}-3)}{\alpha^{2}+1},$
(B.2) $\displaystyle\mathscr{M}_{3}$
$\displaystyle=16q^{2}\left(\frac{a^{2}(v-1)(1+\alpha^{2})^{2}\Lambda}{\alpha^{2}-3}-v\right)-\frac{m^{2}v(\alpha^{2}-3)^{2}}{(1+\alpha^{2})^{2}},$
(B.3) $\displaystyle\mathscr{M}_{4}$
$\displaystyle=-\frac{8mq^{2}v(\alpha^{2}-3)}{a+\alpha^{2}},$ (B.4)
$\displaystyle\mathscr{M}_{5}$ $\displaystyle=-16vq^{4},$ (B.5)
$\displaystyle\mathscr{N}_{0}$
$\displaystyle=\frac{4a^{2}m^{2}(v-1)(\alpha^{2}-3)}{1-\alpha^{2}},$ (B.6)
$\displaystyle\mathscr{N}_{1}$
$\displaystyle=\frac{8a^{2}mq^{2}(v-1)(5-\alpha^{2})(1+\alpha^{2})}{1-\alpha^{2}},$
(B.7) $\displaystyle\mathscr{N}_{2}$
$\displaystyle=\frac{32a^{2}q^{4}(1-v)(1+\alpha^{2})^{2}}{1-\alpha^{2}}.$
(B.8)

## References

  * (1) C. Rovelli, “Loop Quantum Gravity,” Living Rev. Relativ. 11, 5 (2008)
  * (2) D. Oriti, ed., “Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter,” Cambridge University Press, Cambridge, U.K., 2009.
  * (3) E. Kiritsis, “String Theory in a Nutshell (Second Edition),” Princeton University Press, 2019.
  * (4) R. C. Myers and M. J. Perry, “Black Holes in Higher Dimensional Space-Times,” Annals of Physics 172, 304 (1986).
  * (5) R. Myers, “Superstring Gravity and Black Holes,” Nuclear Physics B 289, 701 (1987).
  * (6) C. G. Callan, R. C. Myers and M. J. Perry, “Black Holes in String Theory,” Nuclear Physics B 311, 673 (1989).
  * (7) D. Garfinkle, G. T. Horowitz and A. Strominger, “Charged Black Holes in String Theory,” Phys. Rev. D 43, 3140 (1991).
  * (8) E. Witten, “String Theory and Black Holes,” Physical Review D 44, 314 (1991).
  * (9) J. M. Maldacena, “Black Holes in String Theory,” arXiv:hep-th/9607235.
  * (10) A. Strominger and C. Vafa, “Microscopic Origin of the Bekenstein-Hawking Entropy,” Physics Letters B 379, 99 (1996).
  * (11) C. G. Callan and J. M. Maldacena, “D-Brane Approach to Black Hole Quantum Mechanics,” Nuclear Physics B 472, 591 (1996).
  * (12) J. Maldacena and L. Susskind, “D-Branes and Fat Black Holes,” Nuclear Physics B 475, 679 (1996).
  * (13) J. C. Breckenridge, R. C. Myers, A. W. Peet and C. Vafa, “D-Branes and Spinning Black Holes,” Physics Letters B 391, 93 (1997).
  * (14) R. Emparan, G. T. Horowitz and R. C. Myers, “Black Holes Radiate Mainly on the Brane,” Phys. Rev. Lett. 85, 499 (2000).
  * (15) T. Mohaupt, “Black Holes in Supergravity and String Theory,” Class. Quantum Grav. 17, 3429 (2000).
  * (16) K. Maeda and M. Nozawa, “Black hole solutions in string theory,” Prog. Theor. Phys. Suppl. 189, 310 (2011).
  * (17) C. S. Shahbazi, “Black Holes in Supergravity with Applications to String Theory,” arXiv:1307.3064.
  * (18) L. Pieri, “Black Holes in String Theory,” arXiv:1911.11670.
  * (19) G. W. Gibbons and K. Maeda, “Black Holes and Membranes in Higher-Dimensional Theories with Dilaton Fields,” Nuclear Physics B 298, 741 (1988).
  * (20) T. Koikawa and M. Yoshimura, “Dilaton Fields and Event Horizon,” Physics Letters B 189, 29 (1987).
  * (21) J. H. Horne and G. T. Horowitz, “Rotating Dilaton Black Holes,” Phys. Rev. D 46, 1340 (1992).
  * (22) M. Rakhmanov, “Dilaton Black Holes with Electric Charge,” Phys. Rev. D 50, 5155 (1994).
  * (23) S. J. Poletti and D. L. Wiltshire, “Global Properties of Static Spherically Symmetric Charged Dilaton Spacetimes with a Liouville Potential,” Phys. Rev. D 50, 7260 (1994).
  * (24) K. C. K. Chan, J. H. Horne and R. B. Mann, “Charged Dilaton Black Holes with Unusual Asymptotics,” Nucl. Phys. B 447, 441 (1995).
  * (25) M. Gürses and E. Sermutlu, “Static Spherically Symmetric Solutions to Einstein-Maxwell Dilaton Field Equations in D Dimensions,” Class. Quantum Grav. 12, 2799 (1995).
  * (26) R.-G. Cai, J.-Y. Ji and K.-S. Soh, “Topological Dilaton Black Holes,” Phys. Rev. D 57, 6547 (1998).
  * (27) S. Yazadjiev, “EXACT STATIC SOLUTIONS IN FOUR-DIMENSIONAL EINSTEIN–MAXWELL-DILATON GRAVITY,” Int. J. Mod. Phys. D 08, 635 (1999).
  * (28) S. S. Yazadjiev, “Exact Static Solutions in Einstein-Maxwell-Dilaton Gravity with Arbitrary Dilaton Coupling Parameter,” arXiv:gr-qc/0101078.
  * (29) C. J. Gao and S. N. Zhang, “Dilaton Black Holes in the de Sitter or Anti–de Sitter Universe,” Phys. Rev. D 70, 124019 (2004).
  * (30) G. Clément and C. Leygnac, “Non-Asymptotically Flat, Non-AdS Dilaton Black Holes,” Phys. Rev. D 70, 084018 (2004).
  * (31) R.-G. Cai and A. Wang, “Nonasymptotically AdS/dS Solutions and Their Higher Dimensional Origins,” Phys. Rev. D 70, 084042 (2004).
  * (32) S. S. Yazadjiev, “Rotating Nonasymptotically Flat Black Rings in Charged Dilaton Gravity,” Phys. Rev. D 72, 104014 (2005).
  * (33) S. S. Yazadjiev, “Non-Asymptotically Flat, Non-DS/AdS Dyonic Black Holes in Dilaton Gravity,” Class. Quantum Grav. 22, 3875 (2005).
  * (34) A. Sheykhi, M. H. Dehghani, N. Riazi and J. Pakravan, “Thermodynamics of Rotating Solutions in (n + 1)-Dimensional Einstein-Maxwell-Dilaton Gravity,” Phys. Rev. D 74, 084016 (2006).
  * (35) A. N. Aliev, “Rotating Black Holes in Higher Dimensional Einstein-Maxwell Gravity,” Phys. Rev. D 74, 024011 (2006).
  * (36) T. Ghosh and S. SenGupta, “Slowly rotating dilaton black hole in anti-de Sitter spacetime,” Phys. Rev. D 76, 087504 (2007).
  * (37) A. Sheykhi, “Rotating black holes in Einstein-Maxwell-dilaton gravity,” Phys. Rev. D 77, 104022 (2008).
  * (38) A. Sheykhi and M. Allahverdizadeh, “Higher Dimensional Slowly Rotating Dilaton Black Holes in AdS Spacetime,” Phys. Rev. D 78, 064073 (2008).
  * (39) C. Charmousis, B. Goutéraux and J. Soda, “Einstein-Maxwell-Dilaton Theories with a Liouville Potential,” Phys. Rev. D 80, 024028 (2009).
  * (40) A. Sheykhi and M. Allahverdizadeh, “Higher Dimensional Charged Rotating Dilaton Black Holes,” Gen. Relativ. Gravit.42, 367 (2010).
  * (41) C. Knoll and P. Nedkova, “Charged Rotating Dilaton Black Holes with Kaluza-Klein Asymptotics,” Phys. Rev.D 93, 064052 (2016).
  * (42) H. Lü, P. Mao and J.-B. Wu, “Asymptotic Structure of Einstein-Maxwell-Dilaton Theory and Its Five Dimensional Origin,” J. High Energ. Phys. 2019, 5 (2019).
  * (43) M. M. Stetsko, “slowly Rotating Einstein–Maxwell-Dilaton Black Hole and Some Aspects of Its Thermodynamics,” Eur. Phys. J. C 79, 244 (2019).
  * (44) Y. Younesizadeh, A. H. Ahmed, A. A. Ahmad, Y. Younesizadeh, and M. Ebrahimkhas, “Modified BTZ Black Hole and Some Thermodynamical Properties in Dilaton/Scalar Gravity Model, ”Eur. Phys. J. Plus 135, 686 (2020).
  * (45) Y. Huang and H. Zhang, “Quasibound States of Charged Dilatonic Black Holes,” Physical Review D 103, 044062 (2021).
  * (46) B. H. Fahim and M. Ghezelbash, “New Class of Exact Solutions to Einstein–Maxwell-Dilaton Theory on Four-Dimensional Bianchi Type IX Geometry,” Eur. Phys. J. C 81, 587 (2021).
  * (47) Y. Younesizadeh, F. Younesizadeh and M. M. Qaemmaqami, “Gravitational Lensing and Shadow of Charged Black Holes in the Low-Energy Limit of String Theory,” Eur. Phys. J. Plus 137, 76 (2022).
  * (48) M. G. Richarte, É. L. Martins and J. C. Fabris, “Scattering and Absorption of a Scalar Field Impinging on a Charged Black Hole in the Einstein-Maxwell-Dilaton Theory,” Phys. Rev. D 105, 064043 (2022).
  * (49) Q. Qi, Y. Meng, X. J. Wang and X. M. Kuang, “Gravitational lensing effects of black hole with conformally coupled scalar hair,” Eur. Phys. J. C 83, no.11, 1043 (2023).
  * (50) J. C. Fabris and G. T. Marques, “Hawking Radiation for Non-Asymptotically Flat Dilatonic Black Holes Using Gravitational Anomaly,” Eur. Phys. J. C 72, 2214 (2012).
  * (51) M. E. Rodrigues and G. T. Marques, “Thermodynamics of a Class of Non-Asymptotically Flat Black Holes in Einstein–Maxwell–Dilaton Theory,” Gen. Relativ. Gravit. 45, 1297 (2013).
  * (52) R.-G. Cai, Y. S. Myung and Y.-Z. Zhang, “Check of the Mass Bound Conjecture in de Sitter Space,” Phys. Rev. D 65, 084019 (2002).
  * (53) G. ’t Hooft, “Dimensional Reduction in Quantum Gravity,” arXiv:gr-qc/9310026.
  * (54) L. Susskind, “The World as a Hologram,” J. Math. Phys. 36, 6377 (1995).
  * (55) E. Witten, “Anti De Sitter Space And Holography,” arXiv:hep-th/9802150.
  * (56) S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge Theory Correlators from Non-Critical String Theory,” Physics Letters B 428, 105 (1998).
  * (57) L. Susskind and E. Witten, “The Holographic Bound in Anti-de Sitter Space,” arXiv:hep-th/9805114.
  * (58) A. M. Polyakov, “String Theory and Quark Confinement,” Nucl.Phys.Proc.Suppl. 68, 1 (1998).
  * (59) J. Maldacena, “The Large-N Limit of Superconformal Field Theories and Supergravity,” International Journal of Theoretical Physics 38, 1113 (1999).
  * (60) I. Kanitscheider, K. Skenderis and M. Taylor, “Precision Holography for Non-Conformal Branes,” J. High Energy Phys. 2008, 094 (2008).
  * (61) K. Goldstein, S. Kachru, S. Prakash and S. P. Trivedi, “Holography of Charged Dilaton Black Holes,” J. High Energ. Phys. 2010, 78 (2010).
  * (62) C.-M. Chen and D.-W. Pang, “Holography of Charged Dilaton Black Holes in General Dimensions,” J. High Energ. Phys. 2010, 93 (2010).
  * (63) S. S. Gubser and F. D. Rocha, “Peculiar Properties of a Charged Dilatonic Black Hole in $AdS_{5}$,” Phys. Rev. D 81, 046001 (2010).
  * (64) C. Charmousis, B. Goutéraux, B. Soo Kim, E. Kiritsis and R. Meyer, “Effective Holographic Theories for Low-Temperature Condensed Matter Systems,” J. High Energ. Phys. 2010, 151 (2010).
  * (65) O. DeWolfe, S. S. Gubser and C. Rosen, “A Holographic Critical Point,” Phys. Rev. D 83, 086005 (2011).
  * (66) B.-H. Lee, S. Nam, D.-W. Pang and C. Park, “Conductivity in an Anisotropic Background,” Phys. Rev. D 83, 066005 (2011).
  * (67) B. Goutéraux, J. Smolic, M. Smolic, K. Skenderis and M. Taylor, “Holography for Einstein-Maxwell-Dilaton Theories from Generalized Dimensional Reduction,” J. High Energ. Phys. 2012, 89 (2012).
  * (68) R.-G. Cai, S. He and D. Li, “A hQCD Model and Its Phase Diagram in Einstein-Maxwell-Dilaton System,” J. High Energ. Phys. 2012, 33 (2012).
  * (69) X. M. Kuang, B. Wang and J. P. Wu, “Dynamical gap from holography in the charged dilaton black hole,” Class. Quant. Grav. 30, 145011 (2013)
  * (70) T. Ghosh and S. SenGupta, “Ads-CFT Correspondence in Dilaton Coupled $n$ Dimensional Black Holes,” Astrophys. Space Sci. 357, 2 (2015).
  * (71) C. Park, “Holographic Entanglement Entropy in the Nonconformal Medium,” Phys. Rev. D 91, 126003 (2015).
  * (72) R. Rougemont, R. Critelli, J. Noronha-Hostler, J. Noronha and C. Ratti, “Dynamical versus Equilibrium Properties of the QCD Phase Transition: A Holographic Perspective,” Phys. Rev. D 96, 014032 (2017).
  * (73) B. Swingle and Y. Wang, “Holographic Complexity of Einstein-Maxwell-Dilaton Gravity,” J. High Energ. Phys. 2018, 106 (2018).
  * (74) K. Goto, H. Marrochio, R. C. Myers, L. Queimada and B. Yoshida, “Holographic Complexity Equals Which Action?,” J. High Energ. Phys. 2019, 160 (2019).
  * (75) S. Mahapatra, “Interplay between the Holographic QCD Phase Diagram and Mutual & N-Partite Information,” J. High Energ. Phys. 2019, 137 (2019).
  * (76) P. Jain and S. Mahapatra, “Mixed State Entanglement Measures as Probe for Confinement,” Phys. Rev. D 102, 126022 (2020).
  * (77) A. Ballon-Bayona, H. Boschi-Filho, E. F. Capossoli and D. M. Rodrigues, “Criticality from Einstein-Maxwell-Dilaton Holography at Finite Temperature and Density,” Phys. Rev. D 102, 126003 (2020).
  * (78) G. Fu, P. Liu, H. Gong, X.-M. Kuang and J.-P. Wu, “Holographic Informational Properties for a Specific Einstein-Maxwell-Dilaton Gravity Theory,” Phys. Rev. D 104, 026016 (2021).
  * (79) R. Rougemont, J. Grefa, M. Hippert, J. Noronha, J. Noronha-Hostler, I. Portillo and C. Ratti, “Hot QCD Phase Diagram From Holographic Einstein-Maxwell-Dilaton Models,” arXiv:2307.03885.
  * (80) The Event Horizon Telescope Collaboration, K. Akiyama, et.al. “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” ApJL 875, L1 (2019).
  * (81) The Event Horizon Telescope Collaboration, K. Akiyama, et.al. “First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way,” ApJL 930, L2 (2019).
  * (82) T. Maki and K. Shiraishi, “Motion of Test Particles around a Charged Dilatonic Black Hole,” Class. Quantum Grav. 11, 227 (1994).
  * (83) M. Azreg-Aïnou, “Light Paths of Normal and Phantom Einstein-Maxwell-Dilaton Black Holes,” Phys. Rev. D 87, 024012 (2013).
  * (84) L. Amarilla and E. F. Eiroa, “Shadow of a Kaluza-Klein Rotating Dilaton Black Hole,” Phys. Rev. D 87, 044057 (2013).
  * (85) S.-W. Wei and Y.-X. Liu, “Observing the Shadow of Einstein-Maxwell-Dilaton-Axion Black Hole,” J. Cosmol. Astropart. Phys. 2013, 063 (2013).
  * (86) K. Flathmann and S. Grunau, “Analytic Solutions of the Geodesic Equation for Einstein-Maxwell-Dilaton-Axion Black Holes,” Phys. Rev. D 92, 104027 (2015).
  * (87) S. Soroushfar, R. Saffari and E. Sahami, “Geodesic Equations in the Static and Rotating Dilaton Black Holes: Analytical Solutions and Applications,” Phys. Rev. D 94, 024010 (2016).
  * (88) M. Amir, A. Banerjee and S. D. Maharaj, “Shadow of Charged Wormholes in Einstein–Maxwell–Dilaton Theory,” Annals of Physics 400, 198 (2019).
  * (89) A. Övgün, G. Gyulchev and K. Jusufi, “Weak Gravitational Lensing by Phantom Black Holes and Phantom Wormholes Using the Gauss–Bonnet Theorem,” Annals of Physics 406, 152 (2019).
  * (90) M. Heydari-Fard, M. Heydari-Fard and H. R. Sepangi, “Thin Accretion Disks and Charged Rotating Dilaton Black Holes,” Eur. Phys. J. C 80, 351 (2020).
  * (91) S. Kala, Saurabh, H. Nandan and P. Sharma, “Deflection of Light and Shadow Cast by a Dual-Charged Stringy Black Hole,” Int. J. Mod. Phys. A 35, 2050177 (2020).
  * (92) Y. Younesizadeh, F. Younesizadeh and M. M. Qaemmaqami, “Gravitational Lensing and Shadow of Charged Black Holes in the Low-Energy Limit of String Theory,” Eur. Phys. J. Plus 137, 76 (2022).
  * (93) M. Heydari-Fard, M. Heydari-Fard and H. R. Sepangi, “Null Geodesics and Shadow of Hairy Black Holes in Einstein-Maxwell-Dilaton Gravity,” Phys. Rev. D 105, 124009 (2022).
  * (94) J. Badía and E. F. Eiroa, “Shadows of Rotating Einstein-Maxwell-Dilaton Black Holes Surrounded by a Plasma,” Phys. Rev. D 107, 124028 (2023).
  * (95) K. Hashimoto, S. Kinoshita and K. Murata, “Einstein Rings in Holography,” Phys. Rev. Lett. 123, 031602 (2019).
  * (96) K. Hashimoto, S. Kinoshita and K. Murata, “Imaging Black Holes through the AdS / CFT Correspondence,” Phys. Rev.D 101, 066018 (2020).
  * (97) J. Tsujimura and Y. Nambu, “Null Wave Front and Ryu–Takayanagi Surface,” Entropy 22, 1297 (2020).
  * (98) Y. Kaku, K. Murata and J. Tsujimura, “Observing Black Holes through Superconductors,” J. High Energ. Phys. 2021, 138 (2021).
  * (99) Y. Liu, Q. Chen, X.-X. Zeng, H. Zhang and W. Zhang, “Holographic Einstein Ring of a Charged AdS Black Hole,” J. High Energ. Phys. 2022, 189 (2022).
  * (100) S. Caron-Huot, “Holographic Cameras: An Eye for the Bulk,” J. High Energ. Phys. 2023, 47 (2023).
  * (101) K. Hashimoto, D. Takeda, K. Tanaka and S. Yonezawa, “Spacetime-Emergent Ring toward Tabletop Quantum Gravity Experiments,” Phys. Rev. Research 5, 023168 (2023).
  * (102) X.-X. Zeng, K.-J. He, J. Pu and G. Li, “Holographic Einstein Rings of a Gauss-Bonnet AdS Black Hole,” arXiv:2302.03692.
  * (103) K. Hashimoto, K. Sugiura, K. Sugiyama and T. Yoda, “Photon Sphere and Quasinormal Modes in AdS/CFT,” arXiv:2307.00237.
  * (104) M. Riojas and H.-Y. Sun, “The Photon Sphere and the AdS/CFT Correspondence,” arXiv:2307.06415.
  * (105) C. Barceló, S. Liberati and M. Visser, “Analogue Gravity,” Living Rev. Relativity, 8, 12 (2005);
  * (106) C. Sheng, H. Liu, Y. Wang, S. N. Zhu and D. A. Genov, “Trapping Light by Mimicking Gravitational Lensing,” Nature Photon 7, 902 (2013).
  * (107) B. Carter, ”Global Structure of the Kerr Family of Gravitational Fields,” Phys. Rev. 174, 1559 (1968).
  * (108) T. Levi-Civita, “Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili,” Math. Ann. 59, 383-397 (1904).
  * (109) S. Benenti and M. Francaviglia, “The Theory of Separability of the Hamilton-Jacobi Equation and Its Applications to General Relativity,” in General Relativity and Gravitation: one Hundred Years After the Birth of Albert Einstein, Vol.1, edited by A. Held, 1980,PLENUM PRESS, New York and London.
  * (110) M. O. Katanaev, “Complete separation of variables in the geodesic Hamilton-Jacobi equation,” arXiv:2305.02222v2.
  * (111) S. E. Gralla, D. E. Holz and R. M. Wald, “Black Hole Shadows, Photon Rings, and Lensing Rings,” Phys. Rev. D 100, 024018 (2019).