††institutetext: aSchool of Science, Xihua University, Chengdu 610039, China

# Angular momentum and chaos bound of charged particles around Einstein-Euler-
Heisenberg AdS black holes

Deyou Chen a,2 Chuanhong Gao<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate influences of the charge and angular momentum of
a particle around a charged Einstein-Euler-Heisenberg AdS black hole on a
Lyapunov exponent, and find spatial regions where the chaos bound is violated.
Positions of circular orbits are gotten by fixing the charge and angular
momentum of the particle, respectively. The positions gradually move away from
the event horizon with the increase of the angular momentum when the charge is
fixed and with the decrease of the charge when the angular momentum is fixed.
For certain values of the charge, angular momentum and Euler-Heisenberg
parameter, the spatial regions where the bound is violated are found. When the
charge is fixed and the Euler-Heisenberg parameter is large, a small angular
momentum causes the violation. Although the angular momentum is small, the
corresponding spatial region is not small. An interesting discovery is that
the bound is violated by the black hole when the particle’s charge is less
than 1 and $\Lambda=0$, but this requires the black hole’s charge to be large
enough. This violation may be related to the dynamical stability of the black
hole. The backreaction of the particle on the background spacetime isn’t
considered in the investigation.

###### Keywords:

Chaos bound, Euler-Heisenberg parameter, angular momentum.

##  1 Introduction

Motions of particles around black holes, especially in their equatorial
planes, attract much attention. The reason is that these motions convey
important information on the background spacetimes. For example, null
geodesics are useful to explain quasinormal modes (QNMs) of a black hole for
the test fields CMBWZ ; KZS1 ; KZS2 . The real part of QNMs is determined by
the angular velocity of a particle at the unstable orbit, and the imaginary
part is related to the unstable timescale of the orbit. Unstable circular
geodesics affect the optical appearance of a gravitationally collapsed star,
and then can explain the star’s luminosity AT . Moreover, the spins and merger
of black holes can also be explained by circular motions of particles PK .

The motions of particles may cause chaos. Chaos is an important phenomenon in
nonlinearly dynamic systems. In a classical chaos, a particle’s trajectory
deviates exponentially from it’s initial location due to a small change of
initial conditions. In quantum systems, an important probe of chaotic
behaviors is out-of-time ordered correlators (OTOCs) HS1 ; HS2 ; RRP ; JKY ;
LR . It was first computed by Shenker and Stanford using the eikonal
approximation. They found that the Lyapunov exponent is related to the
temperature of the black hole HS2 . Recently, Maldacena, Shenker and Stanford
conjectured that there is a universal upper bound for the exponent of the
chaos in the thermal quantum systems with a large number of degrees of
freedom,

$\displaystyle\lambda\leq\frac{2\pi T}{\hbar},$ (1)

where $\lambda$ denotes the Lyapunov exponent, and $T$ is the temperature of
the system MSS . This conjecture plays an important role in black hole physics
and gravity duality. After this seminal work was put forward, it immediately
attracted much attention and was verified by a large number of work PR ; MS ;
CFCZZ ; HT ; HT1 ; HT2 ; KS ; HG ; BHM ; FK ; HRY ; CSSTW ; BPK ; LLW ; WHH ;
GSS ; CRYZ ; LX ; CJLY ; WWT ; CDP ; CDP2 ; NNT ; CLC ; EA ; JMW ; CCB ; BCR ;
GJT ; HSBV ; EFST ; BD ; DMM2 ; ACO ; LG1 ; MPD ; GR ; CCKM1 ; BL ; MRS ; LTW1
; LTW2 ; PP1 ; PP2 ; PP3 ; CWJ1 ; MWZ1 ; LW ; CKR1 ; CKR2 ; HY ; CFL ; GZK1 ;
GZK2 ; ADNT ; ADNT1 ; ADNT2 ; ADNT3 ; QCL . The study of the chaos in the
Sachdev-Ye-Kitaev (SYK) model is a successful verification. It was found that
the SYK model saturates the chaos bound, which leads to a speculation that the
saturation of the bound is a necessary condition for Einstein gravity dual.
For a single-particle system, considering that a particle moves radially and
is close to a black hole by a strong enough external forces without falling
into it, Hashimoto and Tanahashi found that the exponent is not related to the
species and strength of the forces, but only to the black hole’s surface
gravity, namely $\lambda\leq\kappa$, where $\kappa$ is the surface gravity HT
. From the relation between the surface gravity and temperature, this result
is consistent with the conjecture in MSS .

However, there are cases that the chaos bound is violated. When a probe
particle moves around a charged black hole, the particle can be in equilibrium
in a circular orbit by centrifugal or Lorentz forces. For a neutral particle,
a force on it is a centrifugal force, and this force is related to the
particle’s angular momentum. The angular momentum causes the change of the
effective potential, but it is not sufficient to make the particle’s orbit
closer to the event horizon. For a charged particle, one can adjust its
charge-to-mass ratio to make it close to the horizon. Based on this, Zhao et
al expanded the exponent at the horizon and studied the chaos bound of the
charged particles near the horizons of the charged black holes ZLL . They
found that the bound was satisfied by the Reissner-Nordström (RN) and
Reissner-Nordström anti-de Sitter (RN AdS) black holes, and violated by a
large number of black holes. They obtained the exponent by using the effective
potential method. The contribution of the angular momenta of the particles was
neglected. In fact, the angular momenta not only affect the effective
potentials, but also increases the magnitudes of the chaotic behaviors. When
this influence was considered, the auhors discussed the bound of the charged
particles around the rotating charged black holes, and found that the bound is
violated KG1 ; KG2 . In this work, the exponent was also obtained by the
effective potentials. In LG2 , Lei and Ge got the exponent by solving the
eigenvalue of the Jacobian matrix. The bound in the RN and RN AdS black holes
was studied by the expansions of the exponent at the horizon. When the angular
momentum’s influence is taken into account, the bound is violated.

In this paper, we investigate influences of the charge and angular momentum of
a particle around a charged Einstein-Euler-Heisenberg (EEH) AdS black hole on
a Lyapunov exponent, and find spatial regions where the chaos bound is
violated. The influence of the Euler-Heisenberg (EH) parameter on the exponent
is discussed in detail. The exponent is obtained by solving a determination of
eigenvalues of a Jacobian matrix in a phase space. The particle’s charge is
fixed to obtain the exponent’s expression in our calculation, which is
different from that in LG2 . At the same time, we don’t expand the exponent at
the event horizon. Based on this derivation, we numerically discuss the
exponent by fixing the charge and angular momentum, respectively. The angular
momentum plays an important role in the investigation. It affects not only the
value of the exponent, but also positions of circular orbits. We find the
positions and spatial regions where the bound is violated by fixing the
particle’s charge and changing its angular momentum.

Nonlinear electrodynamics (NED) is a generalization of Maxwell’s theory and
can be viewed as a low-energy limit from string theory or D-brane physics,
where Abelian and non-Abelian NED Lagrangians could be produced YCLW . In BI ,
Born and Infeld put forward a theory where the central singularity of the
electromagnetic field of a point charge could be removed to study the NED.
Based on this theory, some interesting black hole solutions were gotten and
their properties were studied. In HE , Euler and Heisenberg first constructed
the Lagrangian of the NED by using the Dirac electron-positron theory. This
Lagrangian has the high-order terms of the nonlinear electromagnetic field,
and was reconstructed by Schwinger in the frame of quantum electrodynamics
(QED) JSS1 . The reconstructed Lagrangian carries the main features of the EH
NED and characterizes the phenomenon of vacuum polarization. If the strength
of an electric field is higher than the critical value ($m^{2}c^{3}/e\hbar$),
the QED effect leads to the emergence of particle pairs in the vacuum. A more
general formulation of the NED which admits an arbitrary function of the
electromagnetic invariants was later developed in JP1 . Based on the four-
dimensional action of general relativity coupled to the EH NED1 JP2 ; SGP 111A
short review of the EH effective Lagrangian was given in RWX2 . A summary of
the EH NED and the static spherically symmetric solution of the EEH equations
were given in RWX5 ., some black hole solutions with or without the
cosmological constant have been obtained RWX1 ; RWX2 ; RWX3 ; RWX4 ; AM ; MB ;
RWX5 ; RWX6 . The EEH AdS black hole is one of them. In addition, the EH
action is approximated to the supersymmetric system of the minimum coupling
spin-1/2, -0 particles with appropriate parameters. When the field has high
intensity, this is a more accurate QED classical approximation than Maxwell’s
theory RWX1 . QED provides a solid foundation for the electromagnetic
interaction. Moreover, it has been experimentally verified RVX . Therefore, it
is very important to study QED effects in black hole physics. On the other
hand, an important feature of AdS black holes is the existence of the phase
transitions and critical phenomenon, where the cosmological constant is
regarded as a thermodynamic pressure and its conjugate quantity is a
thermodynamic volume. Recent developments in the AdS/CFT correspondence
greatly stimulate the study of the AdS black holes.

The rest is organized as follows. In the next section, we obtain the Lyapunov
exponent by solving the eigenvalues of the Jacobian matrix in a phase space
$(\pi_{r},r)$. In Section 3, the positions of circular orbits of the particle
around the EEH AdS black hole are obtained. The values of the exponent are
calculated by considering the influences of the angular momentum and charge,
and the spatial regions where the bound is violated are found. The last
section is devoted to our conclusions and discussions.

##  2 Lyapunov exponent in EEH AdS black holes

The four-dimensional action of general relativity with the cosmological
constant coupled to the NED is JP2 ; SGP

$\displaystyle
S=\frac{1}{4\pi}\int_{M^{4}}d^{4}x\sqrt{-g}\left[\frac{1}{4}(R-2\Lambda)-\mathcal{L}(F,G)\right],$
(2)

where $R$ is the Ricci scalar, and $\Lambda$ is the cosmological constant.
$\mathcal{L}(F,G)$ is the NED Lagrangian which depends on the electromagnetic
invariants, $F=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$,
$G=\frac{1}{4}F_{\mu\nu}$${}^{\star}F^{\mu\nu}$, $F_{\mu\nu}$ is the
electromagnetic field strength tensor and its dual is
${}^{\star}F^{\mu\nu}=\frac{\epsilon_{\mu\nu\sigma\rho}F^{\sigma\rho}}{2\sqrt{-g})}$.
$\epsilon_{\mu\nu\sigma\rho}$ is the antisymmetric tensor and satisfies
$\epsilon_{\mu\nu\sigma\rho}\epsilon^{\mu\nu\sigma\rho}=-4!$. The Lagrangian
density for the EH NED is HE

$\displaystyle\mathcal{L}(F,G)=-F+\frac{a}{2}F^{2}+\frac{7a}{8}G^{2},$ (3)

where $a=\frac{8\alpha^{2}}{45m^{4}}$ is the EH parameter, $\alpha$ is the
fine structure constant and $m$ is the electron mass. In MB , Magos and Bretón
got a static spherically symmetric solution from the action (2). The solution
is given by

$\displaystyle
ds^{2}=-F(r)dt^{2}+\frac{1}{F(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (4)

with an electromagnetic potential

$\displaystyle A_{t}=\frac{Q}{r}-\frac{aQ^{3}}{10r^{5}},$ (5)

where

$\displaystyle F(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda
r^{2}}{3}-\frac{aQ^{4}}{20r^{6}},$ (6)

$M$ and $Q$ are the mass and electric charge of the black hole, respectively.
The cosmological constant $\Lambda$ can be positive or negative values. In
this paper, we focus our attention on the AdS spacetime and then the constant
is negative. When $\Lambda=0$, the metric describes a charged EEH black hole
gotten by Amaro and Macias AM . When $a=0$, the metric describes RN AdS black
holes. Due to the existence of the EH parameter, there is an extra term
compared to the metric of the RN AdS black holes and its effect is of
reinforcing the gravitational attraction. Therefore, the behavior of this
system is more like Schwarzschild AdS black holes than RN AdS black holes.
There is the opposite sign of the charge terms in the solution, and the minus
sign makes the divergences of the solution opposite to those of the RN AdS
black holes. The term $\frac{Q^{2}}{r^{2}}$ is always bigger than
$\frac{aQ^{4}}{20r^{6}}$ outside the event horizon when $0\leq
a\leq\frac{32Q^{2}}{7}$. This condition ensure that the dominant behavior of
the solution is given by the linear theory. The event horizon $r_{+}$ is
determined by $F(r)=0$. The surface gravity is

$\displaystyle\kappa=\frac{1}{2r_{+}}\left(1-\frac{Q^{2}}{r_{+}^{2}}-\Lambda
r_{+}^{2}+\frac{aQ^{4}}{4r_{+}^{6}}\right).$ (7)

A Lyapunov exponent denotes the average rate of contraction and expansion of
two adjacent orbits in a classical phase space. It can be derived by the
effective potential method and the matrix method. Here we review the Lyapunov
exponent which has been obtained by the matrix method in CMBWZ ; PP1 ; PP2 ;
PP3 ; LG1 ; LG2 . From the equations of motion of a particle,

$\displaystyle\frac{dX_{i}(t)}{dt}=F_{i}(X^{j}),$ (8)

we use a certain orbit and linearize the equations,

$\displaystyle\frac{d\delta X_{i}(t)}{dt}=K_{ij}(t)\delta X_{j}(t),$ (9)

where $K_{ij}(t)$ is a linear stability matrix defined by
$K_{ij}(t)=\left.\frac{\partial F_{i}}{\partial X_{j}}\right|_{X_{i}(t)}$. The
solution of Eq. (9) satisfies

$\displaystyle\delta X_{i}(t)=L_{ij}(t)\delta X_{j}(0),$ (10)

where $L_{ij}(t)$ is a evolution matrix and obeys
$\dot{L}_{ij}(t)=K_{im}L_{mj}(t)$ and $L_{ij}(0)=\delta_{ij}$. The exponent is
obtained from the determination of the eigenvalues of the matrix $L_{ij}$. It
can be written as

$\displaystyle\lambda=\lim_{t\to\infty}\frac{1}{t}ln\left(\frac{L_{jj}(t)}{L_{jj}(0)}\right).$
(11)

The positive exponent implies a chaos in the system. When we focus our
attention on a circular orbit in an equatorial plane of a black hole, the
classical phase space is expressed as $(p_{r},r)$. Let’s further linearize the
equation of motion with $X_{i}(t)=(p_{r},r)$ about the orbit of constant $r$,
and then get $K_{11}=K_{22}=0$,
$K_{12}=\frac{d}{dr}\left(\dot{t}^{-1}\frac{\delta\mathcal{L}}{\delta
r}\right)$ and $K_{21}=-(\dot{t}g_{rr})^{-1}$, where $\mathcal{L}$ is the
Lagrangian for geodesic motion. Thus, the exponent is gotten as

$\displaystyle\lambda=\sqrt{K_{12}K_{21}}.$ (12)

When a test particle with charge $q$ moves in a circle around the equator of
the black hole, its Lagrangian is CMBWZ ; PP1 ; PP2 ; LG1 ; LG2 ; WL1

$\displaystyle\mathcal{L}=\frac{1}{2}\left(-F\dot{t}^{2}+\frac{\dot{r}^{2}}{F}+r^{2}\dot{\phi}^{2}\right)-qA_{t}\dot{t},$
(13)

where $\dot{x^{\mu}}=\frac{dx^{\mu}}{d\tau}$ and $\tau$ is a proper time. From
the definition of the generalized momentum
$\pi_{\mu}=\frac{\partial\mathcal{L}}{\partial\dot{x}}$, we get

$\displaystyle\pi_{t}=-F\dot{t}-qA_{t}=-E,\quad\quad\pi_{r}=\frac{\dot{r}}{F},\quad\quad\pi_{\phi}=r^{2}\dot{\phi}=L.$
(14)

In the above derivation, $E$ is the energy of the particle, and $L$ denotes
the angular momentum. Then the Hamiltonian is

$\displaystyle
H=\frac{-(\pi_{t}+qA_{t})^{2}+\pi_{r}^{2}F^{2}+\pi^{2}_{\phi}r^{-2}F}{2F}.$
(15)

The motion equations of the particle are gotten from the Hamiltonian, which
are

$\displaystyle\dot{t}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial\pi_{t}}=-\frac{\pi_{t}+qA_{t}}{F},\quad\dot{\pi_{t}}=-\frac{\partial
H}{\partial t}=0,\quad\dot{r}=\frac{\partial H}{\partial\pi_{r}}=\pi_{r}F,$
$\displaystyle\dot{\pi_{r}}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial
r}=-\frac{1}{2}\left[\pi^{2}_{r}F^{\prime}-\frac{2qA^{\prime}_{t}(\pi_{t}+qA_{t})}{F}+\frac{(\pi_{t}+qA_{t})^{2}F^{\prime}}{F^{2}}-\pi^{2}_{\phi}(r^{-2})^{\prime}\right],$
$\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial\pi_{\phi}}=\frac{\pi_{\phi}}{r^{2}},\quad\dot{\pi_{\phi}}=-\frac{\partial
H}{\partial\phi}=0.$ (16)

In the above equations, "$\prime$" denotes the derivative with respect to $r$.
From the equations of motion, the relations between the radial coordinate and
time, and between the radial momentum and time are obtained.

$\displaystyle\frac{dr}{dt}$ $\displaystyle=$
$\displaystyle\frac{\dot{r}}{\dot{t}}=-\frac{\pi_{r}F^{2}}{\pi_{t}+qA_{t}},$
$\displaystyle\frac{d\pi_{r}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\dot{\pi_{r}}}{\dot{t}}=-qA^{\prime}_{t}+\frac{1}{2}\left[\frac{\pi^{2}_{r}FF^{\prime}}{\pi_{t}+qA_{t}}+\frac{(\pi_{t}+qA_{t})F^{\prime}}{F}-\frac{\pi^{2}_{\phi}(r^{-2})^{\prime}F}{\pi_{t}+qA_{t}}\right].$
(17)

For convenience, we define $F_{1}=\frac{dr}{dt}$ and
$F_{2}=\frac{d\pi_{r}}{dt}$. The normalization of the four-velocity of a
particle is given by $g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\eta$, where
$\eta=0$ describes a case of a massless particle and $\eta=-1$ describes a
case of a massive particle. A charged particle is considered, and then the
normalization yields a constrain condition,
$\pi_{t}+qA_{t}=-\sqrt{F(1+\pi_{r}^{2}F+\pi_{\phi}^{2}r^{-2})}$. We use this
constrain and rewrite Eq. (17) as

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\frac{\pi_{r}F^{2}}{\sqrt{F(1+\pi_{r}^{2}F+\pi_{\phi}^{2}r^{-2})}},$
$\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle-
qA^{\prime}_{t}-\frac{(2\pi^{2}_{r}F+1)F^{\prime}}{2\sqrt{F(1+\pi_{r}^{2}F+\pi_{\phi}^{2}r^{-2})}}-\frac{\pi^{2}_{\phi}(r^{-2}F)^{\prime}}{2\sqrt{F(1+\pi_{r}^{2}F+\pi_{\phi}^{2}r^{-2})}}.$
(18)

The effective potential of the particle plays an important role in the
acquisition of the exponent ZLL ; KG1 ; KG2 . Here the exponent is derived by
the eigenvalues of a Jacobian matrix in the phase space $(r,\pi_{r})$. In this
phase space, the Jacobian matrix is defined by $K_{ij}$ with the elements

$\displaystyle K_{11}=\frac{\partial F_{1}}{\partial r},\quad\quad
K_{12}=\frac{\partial F_{1}}{\partial\pi_{r}},\quad\quad K_{21}=\frac{\partial
F_{2}}{\partial r},\quad\quad K_{22}=\frac{\partial F_{2}}{\partial\pi_{r}}.$
(19)

When the particle is in equilibrium, its trajectory satisfies

$\displaystyle\pi_{r}=\frac{d\pi_{r}}{dt}=0.$ (20)

Then the exponent of the chaos of the charged particle in an circular orbit is
obtained by calculating the eigenvalues of the matrix, which is

$\displaystyle\lambda^{2}=\frac{1}{4}\left[\frac{F^{\prime}+\pi_{\phi}^{2}(r^{-2}F)^{\prime}}{1+\pi_{\phi}^{2}r^{-2}}\right]^{2}-\frac{1}{2}F\frac{F^{\prime\prime}+\pi_{\phi}^{2}(r^{-2}F)^{\prime\prime}}{1+\pi_{\phi}^{2}r^{-2}}-\frac{qA_{t}^{\prime\prime}F^{2}}{\sqrt{F(1+\pi_{\phi}^{2}r^{-2})}}.$
(21)

At the event horizon, $F(r_{+})=0$. We get

$\displaystyle\lambda^{2}=\frac{1}{4}F^{\prime}=\kappa^{2},$ (22)

which is saturated at the horizon. This result is consistent with that
obtained in the spherically symmetric black holes ZLL ; LG2 . In the
derivation of Eq. (21), to find the range of the angular momentum and spatial
region where the bound is violated, we fixed the particle’s charge in the
calculation process, and didn’t expand the exponent at the event horizon. When
the charge is expressed in terms of $\pi_{\phi}$, $r$ and $A_{t}$, the
expression of the exponent in LG2  is easily recovered. It is clearly that the
exponent is affected by the particle’s angular momentum and charge. When the
angular momentum is neglected and the expression of the charge is adopted, the
exponent obtained in ZLL  is recovered. In the following, we investigate the
influences of the charge and angular momentum on the exponent, and find
spatial regions where the bound is violated.

##  3 Chaos bound and its violation by EEH AdS black holes

When a charged particle moves around a charged black hole, the position of the
particle’s circular orbit is affected by its charge and angular momentum. One
can let the particle infinitely close to the horizon by adjusting the charge-
to-mass ratio and angular momentum. The chaos bound in the near-horizon
regions has been studied by the expansions of the Lyapunov exponent at the
horizon. Here we investigate the influences of the angular momentum and charge
on the exponent and find spatial regions where the bound is violated. The
region we are concerned about is not limited to the near-horizon region, but
also at a certain distance from the horizon. The cosmological constant is a
very small quantity in physics. In some papers, people often took a small
value of the constant to study various problems WL1 . We also found that the
constant’s value in some papers is relatively large, such as LG2 ; KG2 ; MB ,
etc. In KG2 , the authors ordered the constant to be $-0.5$ and $-1$ for the
RN AdS and Kerr-Newman AdS black holes. In LG2 , the authors ordered the black
holes’ charge to be $1$, and the constant to be $-3$ for the RN AdS black
holes. In MB , the black holes’ mass is $1$, the charge is $0.8$, and the
constant is $-3$ for the RN AdS and EEH AdS black holes. In view of these
values of the constant, we order $\Lambda=-3$ and $\Lambda=0$ in this section,
respectively.

We first find the positions of the orbits by numerical calculations. When
$\Lambda=-3$, Eq. (6) takes form

$\displaystyle
F(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+r^{2}-\frac{aQ^{4}}{20r^{6}},$ (23)

and then the surface gravity is
$\kappa=\frac{1}{2r_{+}}\left(1-\frac{Q^{2}}{r_{+}^{2}}+3r_{+}^{2}+\frac{aQ^{4}}{4r_{+}^{6}}\right)$.
When $\Lambda=0$, Eq. (6) becomes

$\displaystyle
F(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{aQ^{4}}{20r^{6}},$ (24)

and the surface gravity is
$\kappa=\frac{1}{2r_{+}}\left(1-\frac{Q^{2}}{r_{+}^{2}}+\frac{aQ^{4}}{4r_{+}^{6}}\right)$.
Inserting Eqs. (23) and (24) into Eq. (18) and using Eq. (20), we get the
expressions of the orbits. Due to their complexities, we let the parameters of
the black hole and particle be certain values to obtain the positions $r_{0}$
of some certain orbits.

###  3.1 Fixing particle’s charge and changing its angular momentum

In this subsection, we order $M=1$ and $q=15$. The positions $r_{0}$ of the
circular orbits are gotten by changing the angular momentum and listed in
Table 1-Table 5. The horizon radius is numerically gotten.

| L | 0 | 1 | 2 | 3 | 5 | 10 | 20
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | a=1.0 | 0.907044 | 0.920979 | 0.954082 | 0.994850 | 1.080621 | 1.273581 | 1.548244
9 |  |  |  |  |  |  |  |
| a=0.8 | 0.904490 | 0.916975 | 0.947887 | 0.987382 | 1.072733 | 1.267649 | 1.544991
9 |  |  |  |  |  |  |  |
| a=0.6 | 0.902236 | 0.913399 | 0.942085 | 0.980113 | 1.064787 | 1.261615 | 1.541707
9 |  |  |  |  |  |  |  |
| a=0.3 | 0.899289 | 0.908722 | 0.934129 | 0.969656 | 1.052787 | 1.252361 | 1.536719
9 |  |  |  |  |  |  |  |
| a=0 | 0.896733 | 0.904721 | 0.927047 | 0.959819 | 1.040741 | 1.242852 | 1.531654

Table 1. Positions of circular orbits of a charged particle around the EEH AdS
black hole, where $\Lambda=-3$ and $Q=0.6$. The event horizon is located at
$r_{+}=0.894048$ when $a=1.0$, at $r_{+}=0.893291$ when $a=0.8$, at
$r_{+}=0.892526$ when $a=0.6$, at $r_{+}=0.891364$ when $a=0.3$ and at
$r_{+}=0.890184$ when $a=0$.

| L | 0 | 1 | 2 | 3 | 5 | 10 | 20
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | a=1.0 | 0.857517 | 0.871160 | 0.900337 | 0.933774 | 1.001922 | 1.159156 | 1.402398
9 |  |  |  |  |  |  |  |
| a=0.8 | 0.851938 | 0.863014 | 0.888933 | 0.920650 | 0.988113 | 1.147949 | 1.395847
9 |  |  |  |  |  |  |  |
| a=0.6 | 0.847348 | 0.856198 | 0.878595 | 0.908001 | 0.974043 | 1.136302 | 1.389141
9 |  |  |  |  |  |  |  |
| a=0.3 | 0.841572 | 0.847857 | 0.865193 | 0.890328 | 0.952625 | 1.117915 | 1.378767
9 |  |  |  |  |  |  |  |
| a=0 | 0.836492 | 0.841000 | 0.854103 | 0.874625 | 0.931193 | 1.098282 | 1.367983

Table 2. Positions of circular orbits of a charged particle around the EEH AdS
black hole, where $\Lambda=-3$ and $Q=0.7$. The event horizon is located at
$r_{+}=0.844860$ when $a=1.0$, at $r_{+}=0.842699$ when $a=0.8$, at
$r_{+}=0.840469$ when $a=0.6$, at $r_{+}=0.836986$ when $a=0.3$ and at
$r_{+}=0.833318$ when $a=0$.

| L | 0 | 1 | 2 | 3 | 5 | 10 | 20
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | a=1.0 | 1.447628 | 1.449130 | 1.453559 | 1.460695 | 1.481730 | 1.556220 | 1.707025
9 |  |  |  |  |  |  |  |
| a=0.8 | 1.445805 | 1.447240 | 1.451475 | 1.458321 | 1.478635 | 1.551775 | 1.702627
9 |  |  |  |  |  |  |  |
| a=0.6 | 1.443969 | 1.445339 | 1.449389 | 1.455954 | 1.475562 | 1.547319 | 1.698174
9 |  |  |  |  |  |  |  |
| a=0.3 | 1.441187 | 1.442465 | 1.446252 | 1.452416 | 1.470992 | 1.540618 | 1.691391
9 |  |  |  |  |  |  |  |
| a=0 | 1.438368 | 1.439561 | 1.443102 | 1.448886 | 1.466463 | 1.533900 | 1.684475

Table 3. Positions of circular orbits of a charged particle around the EEH
black hole, where $Q=0.9$. The event horizon is located at $r_{+}=1.444450$
when $a=1.0$, at $r_{+}=1.442783$ when $a=0.8$, at $r_{+}=1.441094$ when
$a=0.6$, at $r_{+}=1.438518$ when $a=0.3$ and at $r_{+}=1.435890$ when $a=0$.

| L | 0 | 1 | 2 | 3 | 5 | 10 | 20
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | a=1.0 | 1.334404 | 1.335576 | 1.339041 | 1.344660 | 1.361467 | 1.423745 | 1.561036
9 |  |  |  |  |  |  |  |
| a=0.8 | 1.330490 | 1.331560 | 1.334770 | 1.339985 | 1.355750 | 1.415822 | 1.552969
9 |  |  |  |  |  |  |  |
| a=0.6 | 1.326463 | 1.327457 | 1.330410 | 1.335242 | 1.349995 | 1.407772 | 1.544650
9 |  |  |  |  |  |  |  |
| a=0.3 | 1.320185 | 1.321061 | 1.323673 | 1.327969 | 1.341261 | 1.395443 | 1.531650
9 |  |  |  |  |  |  |  |
| a=0 | 1.313580 | 1.314350 | 1.316651 | 1.320454 | 1.332360 | 1.382786 | 1.517948

Table 4. Positions of circular orbits of a charged particle around the EEH
black hole, where $Q=0.95$. The event horizon is located at $r_{+}=1.332304$
when $a=1.0$, at $r_{+}=1.328569$ when $a=0.8$, at $r_{+}=1.324707$ when
$a=0.6$, at $r_{+}=1.318655$ when $a=0.3$ and at $r_{+}=1.312250$ when $a=0$.

| L | 0 | 1 | 2 | 3 | 5 | 10 | 20
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | Q=0.76 | 0.784386 | 0.787263 | 0.795811 | 0.809757 | 0.851533 | 0.995822 | 1.256175
9 |  |  |  |  |  |  |  |
| Q=0.74 | 0.803891 | 0.807291 | 0.817319 | 0.833445 | 0.880289 | 1.032450 | 1.295411
9 |  |  |  |  |  |  |  |
| Q=0.72 | 0.821077 | 0.825019 | 0.836556 | 0.854864 | 0.906611 | 1.066345 | 1.332510
9 |  |  |  |  |  |  |  |
| Q=0.70 | 0.836492 | 0.841000 | 0.854103 | 0.874625 | 0.931193 | 1.098282 | 1.367983

Table 5. Positions of circular orbits of a charged particle around the RN AdS
black hole, where $\Lambda=-3$. The event horizon is located at
$r_{+}=0.782618$ when $Q=0.76$, at $r_{+}=0.801691$ when $Q=0.74$, at
$r_{+}=0.818410$ when $Q=0.72$ and at $r_{+}=0.833318$ when $Q=0.70$.

From the tables, it is found that when the angular momentum of the particle
increases, the positions of the orbits gradually move away from the horizon,
but they finally tend to the certain positions. For the fixed angular momenta
and charges, the orbits gradually move away from the horizon with the increase
of the value of the EH parameter. In Table 1, there is $r_{0}=1.0145r_{+}$ for
$a=1$ and $r_{0}=1.0088r_{+}$ for $a=0.3$ when $L=0$, which shows that the
orbit is close to the horizon of the EEH AdS black hole when the angular
momentum is zero and the value of $a$ is small enough.

Figure 1: The influence of the angular momentum of the charged particle around
the EEH AdS black hole on the Lyapunov exponent, where $\Lambda=-3$ and
$Q=0.6$. The violation for the chaos bound occurs at $0.63<L<6.62$ (The
corresponding spatial region is $1.0210r_{+}<r_{0}<1.2836r_{+}$.) when $a=1$,
at $0.94<L<6.12$ ($1.0250r_{+}<r_{0}<1.2535r_{+}$) when $a=0.8$ and at
$1.36<L<5.38$ ($1.0332r_{+}<r_{0}<1.2111r_{+}$) when $a=0.6$. Figure 2: The
influence of the angular momentum of the charged particle around the EEH AdS
black hole on the Lyapunov exponent, where $\Lambda=-3$ and $Q=0.7$. The
violation for the chaos bound occurs at $0\leq L<11.20$ $(1.0150r_{+}\leq
r_{0}<1.4123r_{+})$ when $a=1$, at $0\leq L<11.05$ $(1.0110r_{+}\leq
r_{0}<1.3985r_{+})$ when $a=0.8$, at $0\leq L<11.15$ $(1.0082r_{+}\leq
r_{0}<1.3926r_{+})$ when $a=0.6$ and at $0\leq L<10.05$ $1.0055r_{+}\leq
r_{0}<1.3375r_{+})$ when $a=0.3$.  Figure 3: The influence of the angular
momentum of the charged particle around the RN AdS black hole on the Lyapunov
exponent, where $\Lambda=-3$. There are violations for the chaos bound when
$Q=0.72,0.74,0.76$. The violation occurs at $4.01<L<39.08$
$(1.0590r_{+}<r_{0}<1.9956r_{+})$ when $Q=0.76$, at
$4.59<L<22.09$$(1.0845r_{+}<r_{0}<1.6703r_{+})$ when $Q=0.74$ and at
$5.76<L<15.00$ $(1.1360r_{+}<r_{0}<1.4811r_{+})$ when $Q=0.72$. Figure 4: The
influence of the angular momentum of the charged particle around the EEH black
hole on the Lyapunov exponent, where $Q=0.9$. There are violations for the
chaos bound when $a=0,0.3,0.6,0.8,1.0$. The violation occurs at $3.03<L<22.16$
$(1.0114r_{+}<r_{0}<1.2007r_{+})$ when $a=1$, at $3.52<L<22.13$
$(1.0139r_{+}<r_{0}<1.1989r_{+})$ when $a=0.8$, at $4.24<L<22.07$
$(1.0181r_{+}<r_{0}<1.1969r_{+})$ when $a=0.6$, at $6.28<L<21.91$
$(1.0333r_{+}<r_{0}<1.1931r_{+})$ when $a=0.3$ and at $11.55<L<21.42$
$(1.0851r_{+}<r_{0}<1.1862r_{+})$ when $a=0$. Figure 5: The influence of the
angular momentum of the charged particle around the EEH black hole on the
Lyapunov exponent, where $Q=0.95$. There are violations for the chaos bound
when $a=0,0.3,0.6,0.8,1.0$. When $a=1$, the violation occurs at $1.93<L<85.40$
$(1.0048r_{+}<r_{0}<1.4486r_{+})$.

The values of the Lyapunov exponent at the orbits $r=r_{0}$ and the surface
gravity are numerically calculated by using Eqs. (7), (21), (23) and (24). The
violation of the chaos bound in specific regions is analyzed in Figure
1-Figure 5. For the EEH AdS black hole with a fixed charge, there are
violations for the chaos bound when the value of $a$ is relatively large. In
Figure 1, the range of the angular momentum and spatial region where the chaos
bound is violated increase with the increase of the value of $a$. In Figure 2,
the violation occurs when $a=1$, $0\leq L<11.20$ $(1.0150r_{+}\leq
r_{0}<1.4123r_{+})$, and when $a=0.8$, $0\leq L<11.05$ $(1.0110r_{+}\leq
r_{0}<1.3985r_{+})$. $L=0$ means that the particle is stationary relative to
the black hole. This implies that as long as the values of the charge and EH
parameter are large enough, the angular momentum is small enough, or even is
zero, the bound can be violated. However, when the angular momentum is large,
there is no violation. Therefore, the spatial region for the violation is
affected by the EH parameter, charge and angular momentum. When $a=0.3$, the
violation occurs when $Q=0.7$, and does not occur when $Q=0.6$. This shows
that the large charge of a black hole is more likely to cause the violation.
When $a=0$, the metric (4) describes the RN AdS black hole and there is no
violation for the bound in Figure 1 and Figure 2. However, it does not imply
that there is no violation for the chaos bound of a charged particle around
the black hole. This is because the value of the black hole’s charge is
relatively small. When the charge is large enough, the violation is found in
Figure 3. From the figure, it is shown again that the spatial region where the
bound is violated is large when the charge of the black hole is large.

When $\Lambda=0$, the chaos bound of the particle around the EEH black hole
with a fixed charge is described in Figure 4 and Figure 5. From the figures,
it is found that the bound is violated for all the values of $a$ when the
charge is large. For the fixed EH parameter, the violation occurs in a large
range of the angular momentum (spatial region) when the charge is relatively
large. When the charge takes the different values, the changes of the values
of $\lambda^{2}-\kappa^{2}$ are significantly different. For the fixed charge
and large value of the EH parameter, the values of $\lambda^{2}-\kappa^{2}$
increase faster when the angular momentum is less than $20$, and decrease
faster when the angular momentum is greater than $20$. This is more evident in
Figure 5. However, the angular momentum region and spatial region where the
bound is violated decrease with the increase of the value of $a$ in Figure 5.
This is different from the first four pictures.

Therefore, for the certain values of the EH parameter, charge and cosmological
constant, the chaos bound is violated in the near-horizon region and at a
certain distance from the horizon. The angular momentum of the particle plays
an important role in the violation of the bound. Because other parameters are
fixed, the value of the angular momentum determines the positions of the
orbits and the region where the bound is violated. When the value of the
angular momentum is small, the orbit is very close to the horizon. With the
increase of the value of the angular momentum, the orbits gradually move away
from the horizon and tend to constant positions.

###  3.2 Fixing particle’s angular momentum and changing its charge

In this subsection, we order $M=1$ and $L=5$. The positions $r_{0}$ of the
circular orbits are derived by changing the charge of the particle and listed
in Table 6-Table 9. In the tables, "$\ast\ast$$\ast\ast$" indicates that
circular orbits don’t exist. The locations of the horizons are also obtained
by numerical calculations. A common feature in these tables is that when the
particle’s charge increases, the orbits gradually approach the horizons. In
Table 6, there is no circular orbit around the EEH AdS black hole when $q\leq
3$. When $a=0$, the EEH AdS black hole is reduced to the RN AdS black hole.
There is also no circular orbit for $q\leq 2$ in Table 7. When $\Lambda=0$,
circular orbits around the EEH black hole are found for $q>1$ and $q\leq 1$ in
Table 8 and 9.

| $q$ | 0.5 | 1 | 2 | 3 | 4 | 5 | 10
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | a=1.0 | **** | **** | **** | **** | 1.499239 | 1.355364 | 1.091599
9 |  |  |  |  |  |  |  |
| a=0.8 | **** | **** | **** | **** | 1.490236 | 1.345668 | 1.078504
9 |  |  |  |  |  |  |  |
| a=0.6 | **** | **** | **** | **** | 1.481070 | 1.335704 | 1.064915
9 |  |  |  |  |  |  |  |
| a=0.3 | **** | **** | **** | **** | 1.466991 | 1.320205 | 1.043546
9 |  |  |  |  |  |  |  |
| a=0 | **** | **** | **** | **** | 1.452472 | 1.303962 | 1.020926

Table 6. Positions of circular orbits of a particle around the EEH AdS black
hole, where $\Lambda=-3$ and $Q=0.7$. The event horizon is located at
$r_{+}=0.844860$ when $a=1.0$, at $r_{+}=0.842699$ when $a=0.8$, at
$r_{+}=0.840469$ when $a=0.6$, at $r_{+}=0.836986$ when $a=0.3$ and at
$r_{+}=0.833318$ when $a=0$.

| $q$ | 0.5 | 1 | 2 | 3 | 4 | 5 | 10
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | Q=0.76 | **** | **** | **** | 1.537130 | 1.299666 | 1.174371 | 0.924467
9 |  |  |  |  |  |  |  |
| Q=0.74 | **** | **** | **** | 1.621182 | 1.350527 | 1.219018 | 0.958992
9 |  |  |  |  |  |  |  |
| Q=0.72 | **** | **** | **** | 1.734581 | 1.401095 | 1.261953 | 0.990881
9 |  |  |  |  |  |  |  |
| Q=0.70 | **** | **** | **** | **** | 1.452472 | 1.303962 | 1.020926

Table 7. Positions of circular orbits of a particle around the RN AdS black
hole, where $\Lambda=-3$. The event horizon is located at $r_{+}=0.782618$
when $Q=0.76$, at $r_{+}=0.801691$ when $Q=0.74$, at $r_{+}=0.818410$ when
$Q=0.72$ and at $r_{+}=0.833318$ when $Q=0.70$.

| $q$ | 0.5 | 1 | 2 | 3 | 4 | 5 | 10
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | a=1.0 | 2.011500 | 1.842216 | 1.518311 | 1.485960 | 1.404786 | 1.352891 | 1.254538
9 |  |  |  |  |  |  |  |
| a=0.8 | 2.007721 | 1.835878 | 1.606614 | 1.470208 | 1.386966 | 1.334511 | 1.239276
9 |  |  |  |  |  |  |  |
| a=0.6 | 2.003875 | 1.829340 | 1.594156 | 1.453004 | 1.367308 | 1.314264 | 1.222888
9 |  |  |  |  |  |  |  |
| a=0.3 | 1.997973 | 1.819123 | 1.573722 | 1.423532 | 1.332973 | 1.278960 | 1.195276
9 |  |  |  |  |  |  |  |
| a=0 | 1.991903 | 1.808352 | 1.550538 | 1.387280 | 1.288877 | 1.233642 | 1.161352

Table 8. Positions of circular orbits of a particle around the EEH black hole,
where $Q=0.99$. The event horizon is located at $r_{+}=1.206347$ when $a=1.0$,
at $r_{+}=1.196578$ when $a=0.8$, at $r_{+}=1.185690$ when $a=0.6$, at
$r_{+}=1.166388$ when $a=0.3$ and at $r_{+}=1.141067$ when $a=0$.

| $q$ | 0.5 | 1 | 2 | 3 | 4 | 5 | 10
---|---|---|---|---|---|---|---|---
9 |  |  |  |  |  |  |  |
$r_{0}$ | a=1.0 | 2.166321 | 2.003486 | 1.785054 | 1.651170 | 1.565267 | 1.507993 | 1.392599
9 |  |  |  |  |  |  |  |
| a=0.8 | 2.164209 | 2.000086 | 1.779213 | 1.643572 | 1.556737 | 1.499199 | 1.385459
9 |  |  |  |  |  |  |  |
| a=0.6 | 2.162080 | 1.996641 | 1.773234 | 1.635746 | 1.547954 | 1.409159 | 1.378258
9 |  |  |  |  |  |  |  |
| a=0.3 | 2.158853 | 1.991383 | 1.763986 | 1.623539 | 1.534233 | 1.476095 | 1.367328
9 |  |  |  |  |  |  |  |
| a=0 | 2.155584 | 1.986014 | 1.754373 | 1.610700 | 1.519779 | 1.461362 | 1.356214

Table 9. Positions of circular orbits of a particle around the EEH black hole,
where $Q=0.95$. The event horizon is located at $r_{+}=1.332304$ when $a=1.0$,
at $r_{+}=1.328569$ when $a=0.8$, at $r_{+}=1.324707$ when $a=0.6$, at
$r_{+}=1.318655$ when $a=0.3$ and at $r_{+}=1.312250$ when $a=0$.

Figure 6: The influence of the charge of the particle around the EEH AdS black
hole on the Lyapunov exponent, where $\Lambda=-3$ and $Q=0.7$. Figure 7: The
influence of the charge of the particle around the RN AdS black hole on the
Lyapunov exponent, where $\Lambda=-3$. Figure 8: The influence of the charge
of the particle around the EEH black hole on the Lyapunov exponent, where
$Q=0.99$. Figure 9: The influence of the charge of the particle around the EEH
black hole on the Lyapunov exponent, where $Q=0.95$.

The values of the exponent and surface gravity are also numerically calculated
by using Eqs. (7), (21), (23) and (24), and their relations are plotted in
Figure 6-Figure 9. In Figure 6, the values of $\lambda^{2}-\kappa^{2}$
increase with the increase of the particle’s charge. When the charge increases
to a certain value, the violation occurs. Its increase speed is related to the
EH parameter. The larger the parameter, the greater the increase speed. For
$L=5$, the large parameter is more likely to violate the bound. When $a=0$,
there is no violation in the figure. However, this does not mean that the
bound isn’t violated. The reason for this is that the black hole’s charge is
not large enough. When this charge is large enough, the violation appears. For
convenience of comparison, we let $a=0$ and plot a picture in Figure 7. This
figure shows that a large charge of the black hole is more likely to cause the
violation. When the cosmological constant is 0, the violation in the EEH
spacetime is described in Figure 8 and 9. Comparing Figure 8 and 9, we find
that when the black hole’s charge is large enough (for example, $Q=0.99$ ),
the violation appears for $q<1$. However, this phenomenon does not exist when
$Q=0.95$. Figure 8 and 9 reflects that the range of the value of $q$
corresponding to the violation increases with the increase of $a$.

##  4 Conclusions and discussions

In this paper, we investigated the influences of the angular momentum and
charge of the particle around the EEH AdS black hole on the Lyapunov exponent.
For the specific parameters of the black hole, the spatial regions where the
chaos bound is violated were found by fixing the particle’s charge and
changing its angular momentum.

When the charge is fixed at $q=15$, the positions of the circular orbits were
gotten by changing the angular momentum. These positions gradually move away
from the horizons with the increase of the angular momentum. The spatial
region and angular momentum’s range where the violation occurs increase with
the increase of the values of the EH parameter. For the fixed cosmological
constant and EH parameter, the large charge of the black hole is more likely
to cause the violation. When the value of the EH parameter is large, a small
angular momentum causes the violation. Although the angular momentum’s value
is small, its corresponding spatial region is not small. For example, when
$\Lambda=-3$, $Q=0.7$ and $a=1$, the violation occurs in the range $0\leq
L<11.20$. The corresponding spatial region is $1.0150r_{+}\leq
r_{0}<1.4123r_{+}$. When $\Lambda=0$, $Q=0.95$ and $a=1$, the violation occurs
in the range $1.93<L<85.40$ and the spatial region is
$1.0048r_{+}<r_{0}<1.4486r_{+}$. When the angular momentum is fixed at $L=5$,
the circular orbits were obtained by changing the particle’s charge. As the
charge increases, the positions gradually approach the horizons. The violation
of the bound was found. For $\Lambda=-3$, the bound is violated when the
particle’s charge is relatively large. An interesting discovery is that the
bound is violated by the black hole when the particle’s charge is less than 1
and $\Lambda=0$, but this requires the black hole’s charge to be large enough.
In KG1 ; KG2 , the violation was found when the particle’s mass is 1 and its
charge is 10. The weak gravity conjecture asserts that, for the lightest
charged particle along the direction of a basis vector in charge space, the
charge-to-mass ratio is larger than those for extremal black holes. For an
extremal Kerr-Newman black hole, when its rotation parameter is very small,
the charge-to-mass ratio of the black hole approaches 1. Therefore, the bound
can be violated without the premise of the weak gravity conjecture.

There are two explanations for the violation of the bound. In LTW1 ; LTW2 ,
the authors discussed the minimal length effects on the chaotic motion of the
particle, and found that the bound is violated. They believed that the result
does not necessarily implies the violation for the bound conjectured in MSS ,
and the bound should be corrected due to the minimal length in bulk. On the
other hand, Lei and Ge found that the bound is violated by the RN and RN AdS
black holes LG2 . They perceived that this violation should be affected by
some properties of the black hole and related to the dynamical stability of
the black hole.

Although we found that the bound is violated, the backreaction of the particle
on the background spacetime wasn’t considered. When it is taken into account,
the value of the exponent should be changed, and its relation with the bound
needs to be further studied. This is our next work and may lead to an
interesting result. If the bound is still violated, we will understand and
study this violation from the stability of black holes. The study on the
stability of the black holes may help us to understand the exceeding of the
bound.

###### Acknowledgements.

We are grateful to Haitang Yang, Peng Wang and Jun Tao for useful discussions.
This work is supported by the NSFC (Grant No. 12105031) and Tianfu talent
project.

## References

  * (1) 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.
  * (2) R.A. Konoplya and Z. Stuchlík, _Are eikonal quasinormal modes linked to the unstable circular null geodesics?_ _Phys. Lett._ B 771 (2017) 597.
  * (3) R.A. Konoplya, A.F. Zinhailo and Z. Stuchlík, _Quasinormal modes, scattering and Hawking radiation in the vicinity of Einstein-dilaton-Gauss-Bonnet black hole_ , _Phys. Rev._ D 99 (2019) 124042.
  * (4) W.L. Ames and K.S. Thorne, _The optical appearance of a star that is collapsing through its gravitational radius_ , _Astrophys. J._ 151 (1968) 659.
  * (5) F. Pretorius and D. Khurana, _Black hole mergers and unstable circular orbits_ , _Class. Quant. Grav._ 24 (2007) S83–S108.
  * (6) S.H. Shenker and D. Stanford, _Black holes and the butterfly effect_ , _JHEP_ 1403 (2014) 067.
  * (7) S.H. Shenker and D. Stanford, _Stringy effects in scrambling_ , _JHEP_ 1405 (2015) 132.
  * (8) R.R. Poojary, _BTZ dynamics and chaos_ , _JHEP_ 2003 (2020) 048.
  * (9) V. Jahnkea, K.Y. Kim and J. Yoon, _On the chaos bound in rotating black holes_ , _JHEP_ 1905 (2019) 037.
  * (10) Y. Liu, A. Raju, _Quantum chaos in topologically massive gravity_ , _JHEP_ 2012 (2020) 027.
  * (11) J. Maldacena, S.H. Shenker and D. Stanford, _A bound on chaos_ , _JHEP_ 1608 (2016) 106.
  * (12) J. Polchinski and V. Rosenhaus, _The spectrum in the Sachdev-Ye-Kitaev model_ , _JHEP_ 1604 (2016) 001.
  * (13) J. Maldacena and D. Stanford, _Remarks on the Sachdev-Ye-Kitaev model_ , _Phys. Rev._ D 94 (2016) 106002.
  * (14) X. Chen, R.H. Fan, Y.M. Chen, H. Zhai and P.F. Zhang, _Competition between chaotic and nonchaotic phases in a quadratically coupled Sachdev-Ye-Kitaev Model_ , _Phys. Rev. Lett._ 119 (2017) 207603.
  * (15) K. Hashimoto and N. Tanahashi, _Universality in chaos of particle motion near black hole horizon_ , _Phys. Rev._ D 95 (2017) 024007.
  * (16) K. Hashimoto and K. Sugiura, _Causality bounds chaos in geodesic motions_ , [arXiv:2205.13818[hep-th]].
  * (17) K. Hashimoto, K. Murata, N. Tanahashi and R. Watanabe, _A bound on energy dependence of chaos_ , [arXiv:2112.11163 [hep-th]].
  * (18) A. Kitaev and S.J. Suh, _The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual_ , _JHEP_ 1805 (2018) 183.
  * (19) Y.C. Huang and Y.F. Gu, _Eigenstate entanglement in the Sachdev-Ye-Kitaev model_ , _Phys. Rev._ D 100 (2019) 041901(R).
  * (20) E. Berkowitz, M.i Hanada and J. Maltz, _Chaos in matrix models and black hole evaporation_ , _Phys. Rev._ D 94 (2016) 126009.
  * (21) A.L. Fitzpatrick and J. Kaplan, _A quantum correction to chaos_ , _JHEP_ 1605 (2016) 070.
  * (22) P. Hosur, X.L. Qi, D.A. Roberts and B. Yoshida, _Chaos in quantum channels_ , _JHEP_ 1602 (2016) 004.
  * (23) P. Caputa, J. Simón, A. Štikonas, T. Takayanagi and K. Watanabe,, _Scrambling time from local perturbations of the eternal BTZ black hole_ , _JHEP_ 1508 (2015) 011.
  * (24) T. Rakovszky, F. Pollmann and C.W. von Keyserlingk, _Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation_ , _Phys. Rev._ X 8 (2018) 031057.
  * (25) W.J. Li, P. Liu and J.P Wu, _Weyl corrections to diffusion and chaos in holography_ , _JHEP_ 1804 (2018) 115.
  * (26) W.H. Huang, _Holographic butterfly velocities in brane geometry and Einstein-Gauss-Bonnet gravity with matter_ , _Phys. Rev._ D 97 (2018) 066020.
  * (27) S. Grozdanov, K. Schalm and V. Scopelliti, _Black hole scrambling from hydrodynamics_ , _Phys. Rev. Lett._ 120 (2018) 231601.
  * (28) R.G. Cai, S.M Ruan, R.Q. Yang and Y.L. Zhang, _The string worldsheet as the holographic dual of SYK state_ , arXiv:1709.06297[hep-th].
  * (29) Y. Ling and Z.Y. Xian, _Holographic butterfly effect and diffusion in quantum critical region_ , _JHEP_ 1709 (2017) 003.
  * (30) J. Cotler, N. Hunter-Jones, J.Y. Liu and B. Yoshida, _Chaos, complexity, and random matrices_ , _JHEP_ 1711 (2017) 048.
  * (31) S.F. Wu, B. Wang, X.H. Ge and Y. Tian, _Holographic RG flow of thermoelectric transport with momentum dissipation_ , _Phys. Rev._ D 97 (2018) 066029.
  * (32) J.S. Cotler, D.W. Ding, G.R. Penington, _Out-of-time-order operators and the butterfly effect_ , _Annals Phys._ 396 (2018) 318
  * (33) A.A. Patel and S. Sachdev, _Quantum chaos on a critical Fermi surface_ , _Proc. Nat. Acad. Sci._ 114 (2017) 1844.
  * (34) L.M. Nie, M. Nozaki, S. Ryu and M.T. Tan, _Signature of quantum chaos in operator entanglement in 2d CFTs_ , _J. Stat. Mech._ 1909 (2019) 093107.
  * (35) A. Chan, A.D. Luca and J.T. Chalkere, _Eigenstate correlations, thermalization and the butterfly effect_ , _Phys. Rev. Lett._ 122 (2019) 220601.
  * (36) E. Altman, _Many-body localization and quantum thermalization_ , _Nature Phys._ 14 (2018) 979.
  * (37) R.A. Jalabert, I. García-Mata, D.A. Wisniacki, _Semiclassical theory of out-of-time-order correlators for low-dimensional classically chaotic systems_ , _Phys. Rev._ E 98 (2018) 062218.
  * (38) J. Chávez-Carlos, B. López-del-Carpio, M.A. Bastarrachea-Magnani, P. Stránský, S. Lerma-Hernández, L.F. Santos and J.G. Hirsch, _Quantum and classical Lyapunov exponents in atom-field interaction systems_ , _Phys. Rev. Lett._ 122 (2019) 024101.
  * (39) P.D. Bergamasco, G.G. Carlo and A.M.F. Rivas, _OTOC, complexity and entropy in bi-partite systems_ , _Phys. Rev. Research._ 1 (2019) 033044.
  * (40) G.J. Turiaci, _An inelastic bound on chao_ , _JHEP_ 1907 (2019) 099.
  * (41) S.S. Hegde, V. Subramanyan, B. Bradlyn and S. Vishveshwara, _Quasinormal modes and the Hawking-Unruh effect in quantum Hall systems: lessons from black hole phenomena_ , _Phys. Rev. Lett._ 123 (2019) 156802.
  * (42) R. Emparan, A.M. Frassino, M. Sasieta, M. Tomašević, _Holographic complexity of quantum black holess_ , _2202_ (2022) 204
  * (43) M. Blake and R.A. Davison, _Chaos and pole-skipping in rotating black holes_ , _2202_ (2022) 013.
  * (44) S. Dalui, B.R. Majhi and P. Mishra, _Induction of chaotic fluctuations in particle dynamics in a uniformly accelerated frame_ , _Int. J. Mod. Phys._ A 35 (2020) 2050081.
  * (45) A. Addazi, S. Capozziello and S.D. Odintsov, _Chaotic solutions and back hole shadow in $f(R)$ gravity_, _Phys. Lett._ B 816 (2021) 136257.
  * (46) M.H. Muoz-Arias, P.M. Poggi and I.H. Deutsch, _Nonlinear dynamics and quantum chaos of a family of kicked p-spin models_ , _Phys. Rev._ E 103 (2021) 052212.
  * (47) D.J. Gross and V. Rosehips, _Chaotic scattering of highly excited strings_ , _JHEP_ 2105 (2021) 048.
  * (48) D. Chandorkar, S.D. Chowdhury, S, Kundu and S. Minwalla, _Bounds on Regge growth of flat space scattering from bounds on chaos_ , _JHEP_ 2105 (2021) 143
  * (49) M. Blake and H. Liu, _On systems of maximal quantum chaos_ , _JHEP_ 2105 (2021) 229.
  * (50) M. Mondal, F. Rahaman and K.N. Singh, _Lyapunov exponent ISCO and Kolmogorov Senai entropy for Kerr Kiselev black hole_ , _Eur. Phys. J._ C 81 (2021) 84.
  * (51) X.B. Guo, K.K. Liang, B.R. Mu, P. Wang and H.T. Yang, _Minimal length effects on motion of a particle in Rindler space_ , _Chin. Phys._ C 45 (2021) 023115.
  * (52) F.H. Lu, J. Tao and P. Wang, _Minimal length effects on chaotic motion of particles around black hole horizon_ , _JCAP_ 1812 (2018) 036.
  * (53) Y.Q. Lei, X.H. Ge and C. Ran, _Chaos of particle motion near a black hole with quasitopological electromagnetism_ , _Phys. Rev._ D 104 (2021) 046020.
  * (54) P.P. Pradhan, _Lyapunov exponent and charged Myers Perry spacetimes_ , _Eur. Phys. J._ C 73 (2013) 2477.
  * (55) P. Pradhan, _Stability analysis and quasinormal modes of Reissner-Nordstrom space-time via Lyapunov exponent_ , _Pramana_ 87 (2016) 5.
  * (56) P. Pradhan, _Circular geodesics in tidal charged black hole_ , _Int. J. Geom. Meth. Mod. Phys._ 15 (2017) 1850011.
  * (57) S.B. Chen, M.Z. Wang and J.L. Jing, _Chaotic motion of particles in the accelerating and rotating black holes spacetime_ , _JHEP_ 1609 (2016) 082.
  * (58) Y. Ling, P. Liu and J.P. Wu, _Note on the butterfly effect in holographic superconductor models_ , _Phys. Lett._ B 768 (2017) 288.
  * (59) D. Li and X. Wu, _Chaotic motion of neutral and charged particles in a magnetized Ernst-Schwarzschild spacetime_ , _Eur. Phys. J. Plus_ 134 (2019) 96.
  * (60) B. Craps, M. De Clerck, D. Janssens, V. Luyten and C. Rabideau, _Lyapunov growth in quantum spin chains_ , _Phys. Rev._ B 101 (2020) 174313.
  * (61) B. Craps, S. Khetrapal and C. Rabideau, _Chaos in CFT dual to rotating BTZ_ , _JHEP_ 2111 (2021) 105.
  * (62) X.L. Han and Z.D. Yu, _Quantum chaos of the Bose-Fermi Kondo model at the intermediate temperature_ , _Phys. Rev._ B 104 (2021) 085139.
  * (63) P. Colangelo, F. De Fazio and N. Losacco, _Chaos in a Q $\overline{Q}$ system at finite temperature and baryon densitye_, _Phys. Rev._ D 102 (2020) 074016.
  * (64) D. Giataganas, L.A.P. Zayas and K. Zoubos, _On marginal deformations and non-integrability_ , _JHEP_ 1401 (2014) 129.
  * (65) D. Giataganas and K. Zoubos, _Non-integrability and chaos with unquenched flavor_ , _JHEP_ 1710 (2017) 042.
  * (66) M. Alishahiha, A. Davody, A. Naseh and S.F. Taghavi, _On butterfly effect in higher derivative gravities_ , _JHEP_ 1611 (2016) 032.
  * (67) L.A. Pando Zayas and C.A. Terrero-Escalante, _Chaos in the gauge/gravity correspondence_ , _JHEP_ 1009 (2010) 094.
  * (68) S. Dalui, B.R. Majhi and P. Mishra, _Presence of horizon makes particle motion chaotic_ , _Phys. Lett._ B 788 (2019) 486.
  * (69) S. Dalui, B.R. Majhi and P. Mishra, _Horizon induces instability locally and creates quantum thermality_ , _Phys. Rev._ D 102 (2020) 044006.
  * (70) L.C. Qu, J. Chen and Y.X. Liu, _Chaos and complexity for inverted Harmonic oscillators_ , _Phys. Rev._ D 105 (2022) 126015.
  * (71) Q.Q. Zhao, Y.Z. Li and H. Lü, _Static equilibria of charged particles around charged black holes: Chaos bound and its violations_ , _Phys. Rev._ D 98 (2018) 124001.
  * (72) N. Kan and B. Gwak, _Bound on the Lyapunov exponent in Kerr-Newman black holes via a charged particle_ , _Phys. Rev._ D 105 (2022) 026006.
  * (73) B. Gwak, N. Kan, B.H. Lee and H. Lee, _Violation of bound on chaos for charged probe in Kerr-Newman-AdS black hole_ , _JHEP_ 2209 (2022) 026.
  * (74) Y.Q. Lei and X.H. Ge, _Circular motion of charged particles near a charged black hole_ , _Phys. Rev._ D 105 (2022) 084011.
  * (75) X. Ye, Z.Q. Chen, M.D. Li and S.W. Wei, _QED effects on phase transition and Ruppeiner geometry of Euler-Heisenberg-AdS black holes_ , [arXiv:2202.09053[gr-qc]].
  * (76) M. Born and L. Infeld, _Foundations of the new field theory_ , _Nature_ 132 (1933) 1004.
  * (77) W. Heisenberg and H. Euler, _Consequences of Dirac’s theory of positrons_ , _Z. Phys._ 98 (1936) 714.
  * (78) J. Schwinger, _On gauge invariance and vacuuwn polarization_ , _Phys. Rev._ 82 (1951) 664.
  * (79) J. Plebanski, _Non-linear electrodynamics — a study_ (C.I.E.A. del I.P.N., Mexico City, 1966).
  * (80) J. Plebanski, _Lectures on nonlinear electrodynamics_. (Nordita, Copenhagen 1970).
  * (81) H. Salazar, A.G. D. and J. Plebanski, _Duality rotations and type D solutions to Einstein equations with nonlinear electromagnetic sources_ , _J. Math. Phys._ 28 (1987) 2171.
  * (82) R. Ruffini, Y.B. Wu and S.S. Xue, _Einstein-Euler-Heisenberg theory and charged black holes_ , _Phys. Rev._ D 88 (2013) 085004.
  * (83) N. Bretón and L.A. López, _Birefringence and quasinormal modes of the Einstein-Euler-Heisenberg black hole_ , _Phys. Rev._ D 104 (2021) 024064.
  * (84) H. Yajima and T. Tamaki, _Black hole solutions in Euler-Heisenberg theory_ , _Phys. Rev._ D 63 (2001) 064007.
  * (85) N. Bretón, C. L$\ddot{a}$mmerzah and A. Macías, _Rotating black holes in the Einstein–Euler–Heisenberg theory_ , _Class. Quant. Grav._ 36 (2019) 235022.
  * (86) M. Guerrero and D. Rubiera-Garcia, _Nonsingular black holes in nonlinear gravity coupled to Euler-Heisenberg electrodynamics_ , _Phys. Rev._ D 102 (2020) 024005.
  * (87) D. Amaro and A. Macías, _Geodesic structure of the Euler-Heisenberg static black hole_ , _Phys. Rev._ D 102 (2020) 104054.
  * (88) D. Magos and N. Bretón, _Thermodynamics of the Euler-Heisenberg-AdS black hole_ , _Phys. Rev._ D 102 (2020) 084011.
  * (89) G.R. Li, S. Gu and E.W. Liang, _High-order QED correction impacts on phase transition of the Euler-Heisenberg AdS black hole_ , _Phys. Rev._ D 106 (2022) 064011.
  * (90) R. Ruffini, G. Vereshchagin and S.S. Xue, _Electron-positron pairs in physics and astrophysics: from heavy nuclei to black holes_ , _Phys. Rept._ 487 (2010) 1.
  * (91) 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 .