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

# Spatial regions, chaos bound and its violation

Zeqiang Wanga111E-mail<EMAIL_ADDRESS>Deyou Chen, a222E-mail:
<EMAIL_ADDRESS>
###### Abstract

The influence of the angular momentum of the particle on the Lyapunov exponent
has been studied. In this paper, we investigate influences of the charge and
angular momentum of a particle around non-extremal and extremal Reissner-
Nordström black holes with a scalar hair on the exponent, and find spatial
regions where the chaos bound is violated for certain values of the black
holes’ parameters. The exponent is gotten by calculating the eigenvalue of a
Jacobian matrix in a phase space. The violation occurs from the nea-horizon
regions to certain distances from the event horizons. When the charge or
angular momentum of the particle are fixed at certain values, the spatial
regions increase with the increase of the hair parameter’s value when the
black hole’s charge is fixed, and with the increase of the black hole’s charge
when the hair parameter is fixed. For the extremal black hole, the violation
can occur very close to the horizon when the particle’s charge is large
enough.

###### Keywords:

Chaos bound, spatial regions, angular momentum.

##  1 Introduction

Chaos is an important physical phenomenon in nonlinear dynamic systems, and is
very sensitive to initial conditions. Its sensitivity is characterized by a
Lyapunov exponent. In classical systems, the exponent can be obtained by
calculating the eigenvalue of a Jacobian matrix in a phase space and an
effective potential approach, respectively. Inspired by AdS/CFT
correspondence, out-of-time-order correlators were used to diagnose the chaos
in quantum systems SS1 ; SS2 ; SS3 ; SS4 . In the study of the chaos in the
vicinity of the event horizons of black holes, a large number of calculations
show that the exponent is related to a system’s temperature,
$\lambda_{L}=\frac{2\pi}{\beta}$, where $\lambda_{L}$ is the exponent and
$\beta$ is the inverse temperature of the system. This value is conjectured as
a upper bound on the exponent in the work which was gotten by Maldacena,
Shenker and Stanford in thermal quantum systems with a large number of degrees
of freedom MSS . This inspiring work builds a bridge between quantum chaos and
black hole physics. After the conjecture was proposed, it immediately
attracted much attention and was supported by a lot of work PR ; MS ; CFCZZ ;
KS ; HG ; GR1 ; JKY ; RRP1 ; RRP2 ; RRP3 ; RRP4 ; DMM1 ; DMM2 ; CKR1 ; CKR2 ;
ACO ; MPD ; HY ; GR ; CCKM1 ; BL ; MRS ; QCL ; DG ; HMY1 ; HST1 ; GLMP ; SK1 ;
SK2 ; SK3 ; SK4 ; SK5 ; SK6 ; SK7 ; SK8 ; SK9 ; SK10 ; SK11 ; LLW ; HZ ; HZ1 ;
HZ2 ; HZ3 . For example, it was proved that the bound is saturated by the
Sachdev-Ye-Kitaev model, which shows the existence of Einstein gravity dual
AK1 ; AK2 . On the other hand, Hashimoto and Tanahashi considered a particle
moving in the radial direction of a black hole by outward external forces and
studied its chaos near the horizon HT . They found that the exponent is
independent of the strength and species of the external forces and its upper
bound is determined by the surface gravity of the horizon,

$\displaystyle\lambda_{L}\leq\kappa.$ (1)

Considering the relation between the surface gravity and temperature of the
black hole, this result is fully consistent with that gotten in MSS .

Recently, the cases of the violation for the bound have been found ZLL ; KG1 ;
KG2 ; LG1 ; LG2 ; GCYW ; LTW . When a particle moves around a black hole, the
particle can be in equilibrium due to an external force such as an
electromagnetic force. The charge-to-mass ratio and angular momentum of the
particle can be adjusted to let it move close to the horizon. Based on this,
Zhao et al derived the exponent by calculating the effective potential ZLL .
To evaluate the violation for the bound in the near-horizon regions of the
spherically symmetric black holes, they expanded the exponent at the horizons.
They found that the bound is satisfied by the Reissner-Nordström (RN) and
Reissner-Nordstrröm anti-de Sitter (RNAdS) black holes, and is violated by a
large number of the black holes. In their work, the contribution of the
particle’s angular momentum was neglected. When this contribution was taken
into account, Kan and Gwak studied the exponent in the near-horizon regions
and at the certain distances from the horizons of the Kerr-Newman and Kerr-
Newman AdS black holes and found that the bound is violated by these black
holes KG1 ; KG2 . The exponent can be obtained by calculating the eigenvalue
of a Jacobian matrix in a phase space. Using this method, Lei and Ge found
that the bound is violated by the RN and RNAdS black holes when the
contribution of the particle’s angular momentum is taken into account LG2 . In
their work, to discuss the exponent in the near-horizon regions, the exponent
was also expanded at the horizons. To evaluate the value of the exponent at a
certain position from the event horizon, one of us adopted the matrix method
and fixed the particle’s charge to discuss the influence of the angular
momentum of the charged particle on the exponent GCYW . In our calculation,
the charge of the particle was fixed as a specific value, and the exponent did
not expand at the horizon. The violation for the bound was found outside the
charged Kiselev black hole. However, the range of the angular momentum and
spatial region for the violation were not found.

In this paper, we investigated the influences of the charge and angular
momentum of a particle around non-extremal and extremal Reissner-Nordstrröm
black holes with a scalar hair on the Lyapunov exponent, and find ranges of
the angular momentum and charge and spatial regions where the chaos bound is
violated. Hairy black holes are important solutions of Einstein’s theory of
gravity and modified gravity theories. They have been widely studied mainly in
connection with no-hair theorem. The hair is used to describe properties of
collapsed stars besides masses, electromagnetic charges and angular momenta.
People hope to find hairy black hole solutions with primary scalar hairs.
However, most of the solutions are characterized by secondary hairs, and
impossible to continuously connect a hair black hole’s solution to a black
hole’s solution with the same mass but no scalar field. On the other hand,
hairs might develop in the vicinity of charged black holes. The research of
hair black holes is conducive to our deep understanding of the AdS/CFT
correspondence SSG ; KAVM ; GPSV . The exponent is derived by calculating the
eigenvalue of a Jacobian matrix in a phase space $(r,\pi_{r})$. We first fix
the angular momentum at a specific value to discuss the influence of the
charge on the exponent and find the range of the charge and spatial region
where the bound is violated. Then we fix the charge to discuss the influence
of the angular momentum on the exponent and find the spatial region for the
violation. The influence of the hair parameter on the exponent is discussed.

The rest is organized as follows. In the next section, we obtain the exponent
by calculating the eigenvalue of the Jacobian matrix in the phase space
outside the event horizon of the Reissner-Nordström black hole with a scalar
hair. In Section 3, the influences of the charge and angular momentum of the
particle around non-extremal and extremal Reissner-Nordstrröm-scalar hair
black holes on the Lyapunov exponent are investigated, respectively. The
spatial regions where the bound is violated are found. The last section is
devoted to our conclusions and discussions.

##  2 Lyapunov exponent outside RNSH black holes

The action of general relativity coupled to a Maxwell field and conformally
coupled to a scalar field is

$\displaystyle S=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left[R-F_{\mu\nu}F^{\mu\nu}-8\pi
G\left(\nabla_{\mu}\Psi\nabla^{\mu}\Psi+\frac{R}{6}\Psi^{2}\right)\right],$
(2)

where $F_{\mu\nu}$ is the Maxwell field, and $\Psi$ is the scalar field. From
the action, Astorino got a static spherically symmetric solution with a scalar
hair MA . The solution is given by

$\displaystyle
ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (3)

with an electromagnetic potential $A_{t}=\frac{Q}{r}$, where

$\displaystyle
f(r)=1-\frac{2M}{r}+\frac{Q^{2}+s}{r^{2}},\quad\quad\Psi=\pm\sqrt{\frac{3}{4\pi
G}}\sqrt{\frac{s}{Q^{2}+s}},$ (4)

$s$ is the scalar hair, $M$ and $Q$ are the mass and charge, respectively.
This black hole is referred to as a Reissner-Nordström-scalar hair (RNSH)
black hole. The scalar field $\Psi$ exists even in the absence of the
electromagnetic field, therefore, the scalar hair is a primary hair. There are
two roots for $f(r)=0$,

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}-s},$ (5)

which describe the event horizon ($r_{+}$) and the inner horizon ($r_{-}$).
The surface gravity is

$\displaystyle\kappa=\frac{r_{+}-r_{-}}{2r_{+}^{2}}.$ (6)

When the inner and event horizons coincide with each other, there is
$M^{2}=Q^{2}+s$ and the surface gravity disappears.

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

$\displaystyle\frac{dX_{i}(t)}{dt}=F_{i}(X^{j}),$ (7)

we use a certain orbit and linearize the equations,

$\displaystyle\frac{d\delta X_{i}(t)}{dt}=K_{ij}(t)\delta X_{j}(t),$ (8)

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. (8) satisfies

$\displaystyle\delta X_{i}(t)=L_{ij}(t)\delta X_{j}(0),$ (9)

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

$\displaystyle\lambda=\lim_{t\to\infty}\frac{1}{t}ln\left(\frac{L_{jj}(t)}{L_{jj}(0)}\right).$
(10)

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

$\displaystyle\lambda=\sqrt{K_{12}K_{21}}.$ (11)

To derive the exponent via the matrix method, we first find the motion
equation of a particle with charge $q$ around the equatorial plane of the
black hole. The Lagrangian of the particle is

$\displaystyle\mathcal{L}=\frac{1}{2}\left(-f\dot{t}^{2}+\frac{\dot{r}^{2}}{f}+r^{2}\dot{\phi}^{2}\right)-qA_{t}\dot{t},$
(12)

where $f=f(r)$, $\dot{x^{\mu}}=\frac{dx^{\mu}}{d\tau}$ and $\tau$ is a proper
time. We use the definition of generalized momenta
$\pi_{\mu}=\frac{\partial\mathcal{L}}{\partial\dot{x}}$ and get

$\displaystyle\pi_{t}=-f\dot{t}-qA_{t}=-E,\quad\quad\pi_{r}=\frac{\dot{r}}{f},\quad\quad\pi_{\phi}=r^{2}\dot{\phi}=L.$
(13)

Due to Killing vectors in the black hole background, $E$ and $L$ are constants
in the above equation and denote an energy and angular momentum, respectively.
The particle’s Hamiltonian is

$\displaystyle
H=\frac{-(\pi_{t}+qA_{t})^{2}+\pi_{r}^{2}f^{2}+\pi^{2}_{\phi}r^{-2}f}{2f}.$
(14)

Equations of motion of the particle are obtained from the Hamiltonian, which
are

$\displaystyle\dot{t}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial\pi_{t}}=-\frac{\pi_{t}+qA_{t}}{f},\quad\dot{\pi_{t}}=-\frac{\partial
H}{\partial t}=0,\quad\dot{r}=\frac{\partial H}{\partial\pi_{r}}=\pi_{r}f,$
$\displaystyle\dot{\pi_{r}}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial
r}=-\frac{1}{2}\left[\pi^{2}_{r}f^{\prime}-\frac{2qA^{\prime}_{t}(\pi_{t}+qA_{t})}{f}+\frac{(\pi_{t}+qA_{t})^{2}f^{\prime}}{f^{2}}-\pi^{2}_{\phi}(r^{-2})^{\prime}\right],$
$\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial\pi_{\phi}}=\frac{\pi_{\phi}}{r^{2}},\quad\dot{\pi_{\phi}}=-\frac{\partial
H}{\partial\phi}=0,$ (15)

where "$\prime$" denotes a derivative with respect to $r$. We derive the
exponent in a phase space $(r,\pi_{r})$. Therefore, a radial equation at a
coordinate time $t$ should be first obtained. From the equations of motion, we
get

$\displaystyle\frac{dr}{dt}$ $\displaystyle=$
$\displaystyle\frac{\dot{r}}{\dot{t}}=-\frac{\pi_{r}f^{2}}{\pi_{t}+qA_{t}},$
$\displaystyle\frac{d\pi_{r}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\dot{\pi_{r}}}{\dot{t}}=-qA^{\prime}_{t}+\frac{1}{2}\left[\frac{\pi^{2}_{r}ff^{\prime}}{\pi_{t}+qA_{t}}+\frac{(\pi_{t}+qA_{t})f^{\prime}}{f}-\frac{\pi^{2}_{\phi}(r^{-2})^{\prime}f}{\pi_{t}+qA_{t}}\right].$
(16)

There is a normalization condition for the four-velocity of a particle,
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\eta$, where $\eta=0$ describes a case
of a massless particle and $\eta=-1$ denotes a case of a massive particle.
When a charged particle is considered, the normalization yields
$\pi_{t}+qA_{t}=-\sqrt{f(1+\pi_{r}^{2}f+\pi_{\phi}^{2}r^{-2})}$. We use this
constrain and define $F_{1}=\frac{dr}{dt}$, $F_{2}=\frac{d\pi_{r}}{dt}$. Then
Eq. (16) is rewritten as

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\frac{\pi_{r}f^{2}}{\sqrt{F(1+\pi_{r}^{2}f+\pi_{\phi}^{2}r^{-2})}},$
$\displaystyle F_{2}$ $\displaystyle=$ $\displaystyle-
qA^{\prime}_{t}-\frac{(2\pi^{2}_{r}f+1)f^{\prime}}{2\sqrt{f(1+\pi_{r}^{2}f+\pi_{\phi}^{2}r^{-2})}}-\frac{\pi^{2}_{\phi}(r^{-2}f)^{\prime}}{2\sqrt{f(1+\pi_{r}^{2}f+\pi_{\phi}^{2}r^{-2})}}.$
(17)

In the phase space $(r,\pi_{r})$, the elements of the matrix is defined by

$\displaystyle K_{11}=\frac{\partial F_{1}}{\partial r},\quad\quad
K_{12}=\frac{\partial F_{1}}{\partial\pi_{r}},\quad\quad K_{21}=\frac{\partial
F_{2}}{\partial r},\quad\quad K_{22}=\frac{\partial F_{2}}{\partial\pi_{r}}.$
(18)

When the particle is in equilibrium in a circular orbit, its trajectory
satisfies a constraint $\pi_{r}=\frac{d\pi_{r}}{dt}=0$. Then locations of
equilibrium orbits are derived by using the constraint and Eq. (17). The
exponent of the chaos of the charged particle in an equilibrium orbit is
obtained by calculating the eigenvalue of the matrix, which is identifies

$\displaystyle\lambda^{2}=\frac{1}{4}\left[\frac{f^{\prime}+\pi_{\phi}^{2}(r^{-2}f)^{\prime}}{1+\pi_{\phi}^{2}r^{-2}}\right]^{2}-\frac{1}{2}f\frac{f^{\prime\prime}+\pi_{\phi}^{2}(r^{-2}f)^{\prime\prime}}{1+\pi_{\phi}^{2}r^{-2}}-\frac{qA_{t}^{\prime\prime}f^{2}}{\sqrt{f(1+\pi_{\phi}^{2}r^{-2})}}.$
(19)

At the event horizon, we get $f(r_{+})=0$ and

$\displaystyle\lambda^{2}=\frac{1}{4}(f^{\prime})^{2}=\kappa^{2},$ (20)

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. (19), 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  can be easily recovered. It is clearly that the exponent is
affected by the angular momentum and charge of the particle. When the angular
momentum is neglected and the expression of the charge is adopted, the
exponent obtained in ZLL  is gotten.

##  3 Bound on Lyapunov exponent outside RNSH black holes

When a charged particle moves around a black hole, the charge-to-mass ratio
and angular momentum of the particle can be adjusted to make it in equilibrium
at a certain position from the event horizon. In this section, we fix the
particle’s charge (or angular momentum) to investigate the influences of the
angular momentum (or the charge) on the Lyapunov exponent, and to find spatial
regions where the chaos bound is violated. The values of the exponent and
surface gravity are gotten by numerically calculating Eqs. (6) and (19),
respectively. Without loss of generality, we order $M=1$ in this paper.

Figure 1: The influence of the charge of the particle around the non-extremal
RNSH black hole on the Lyapunov exponent, where $L=0$.  Figure 2: The
influence of the charge of the particle around the non-extremal RNSH black
hole on the Lyapunov exponent, where $L=5$ and $s=0.20$. The chaos bound is
violated in the range $0.01<q<20.67$ (the corresponding spatial region is
$1.00327r_{+}<r_{0}<2.28201r_{+}$, where $r_{+}=1.08888$) when $Q=0.89$, in
the range $0.1<q<14.35$ ($1.01477r_{+}<r_{0}<1.85038r_{+}$, where
$r_{+}=1.20761$) when $Q=0.87$, in the range $0.71<q<11.97$
($1.03233r_{+}<r_{0}<1.61754r_{+}$, where $r_{+}=1.27839$) when $Q=0.85$, and
in the range $2.39<q<7.66$ ($1.07181r_{+}<r_{0}<1.32367r_{+}$, where
$r_{+}=1.33332$) when $Q=0.83$. There is no violation when $Q=0.80$. Figure 3:
The influence of the charge of the particle around the non-extremal RNSH black
hole on the Lyapunov exponent, where $L=5$ and $Q=0.70$. The chaos bound is
violated in the range $0.01<q<24.86$ ($1.00041r_{+}<r_{0}<1.97703r_{+}$, where
$r_{+}=1.10000$) when $s=0.50$, in the range $0.14<q<15.51$
($1.02205r_{+}<r_{0}<1.81672r_{+}$, where $r_{+}=1.24495$) when $s=0.45$, and
in the range $2.74<q<9.25$ ($1.06960r_{+}<r_{0}<1.33271r_{+}$, where
$r_{+}=1.33166$) when $s=0.40$. There is no violation when $s=0.35$ and
$s=0.30$.

The influence of the charge of the particle around the non-extermal RNSH black
hole on the exponent is plotted in Figure 1 - Figure 3 when the angular
momentum is fixed at $L=0$ and $L=5$, respectively. In Figure 1, we calculate
the exponent when the black hole’s parameters are certain values and the
angular momentum is zero, and find there is no violation for these values.
When the particle’s charge is small, equilibrium orbits don’t exist. The
influence of the particle’s charge on the exponent is described in Figure 2
and 3 when the angular momentum is fixed at $L=5$. In Figure 2, the range of
the charge $q$ and spatial region where the bound is violated increase with
the increase of the black hole’s charge. There is no violation when $Q=0.80$.
When the particle’s charge is less than $1$, equilibrium orbits exist, and the
violation occurs for $Q=0.89$, $Q=0.87$ and $Q=0.85$. In Figure 3, the range
of the particle’s charge and spatial region for the violation increase with
the increase of the hair parameter’s value when the black hole’s charge is
fixed at $Q=0.70$. A large value of the hair parameter is more likely to cause
a violation. In the above figures, the angular momentum plays an important
role in the violation of the bound. When the angular momentum is very small
(for example, $L=0$), there is no violation regardless of how the particle’s
charge, black hole’s charge and hair parameter change. Only when the angular
momentum is relatively large, the bound can be violated for this black hole.
Therefore, we will discuss the case where the angular momentum is not zero in
the following.

Figure 4: The influence of the angular momentum of the particle around the
non-extremal RNSH black hole on the Lyapunov exponent, where $q=0.90$ and
$s=0.20$. The chaos bound is violated in the range $L>1.38$ (the corresponding
spatial region is $1.85103r_{+}<r_{0}<1.98131r_{+}$, where $r_{+}=1.08888$)
when $Q=0.89$, in the range $L>2.70$ ($1.61888r_{+}<r_{0}<1.72215r_{+}$, where
$r_{+}=1.20761$) when $Q=0.87$, in the range $L>4.58$
($1.57125r_{+}<r_{0}<1.67105r_{+}$, where $r_{+}=1.27839$) when $Q=0.85$, and
in the range $L>11.09$ ($1.57763r_{+}<r_{0}<1.64029r_{+}$, where
$r_{+}=1.33332$) when $Q=0.83$. There is no violation when $Q=0.80$. Figure 5:
The influence of the angular momentum of the particle around the non-extremal
RNSH black hole on the Lyapunov exponent, where $q=0.90$ and $Q=0.70$. The
chaos bound is violated in the range $L>1.85$
($1.83590r_{+}<r_{0}<2.13661r_{+}$, where $r_{+}=1.10000$) when $s=0.50$, in
the range $L>3.78$ ($1.64460r_{+}<r_{0}<1.69337r_{+}$, where $r_{+}=1.24495$)
when $s=0.45$, and in the range $L>11.56$ ($1.59273r_{+}<r_{0}<1.64115r_{+}$,
where $r_{+}=1.33166$) when $s=0.40$. There is no violation when $s=0.35$ and
$s=0.30$. Figure 6: The influence of the angular momentum of the particle
around the non-extremal RNSH black hole on the Lyapunov exponent, where $q=15$
and $s=0.20$. The chaos bound is violated in the range $L>4.25$
($1.00463r_{+}<r_{0}<1.84862r_{+}$, where $r_{+}=1.08888$) when $Q=0.89$, in
the range $L>5.79$ ($1.02301r_{+}<r_{0}<1.67098r_{+}$, where $r_{+}=1.27839$)
when $Q=0.85$, and in the range $8.30<L<41.86$
($1.05939r_{+}<r_{0}<1.34503r_{+}$, where $r_{+}=1.40000$) when $Q=0.80$.
There is no violation when $Q=0.75$ and $Q=0.70$. Figure 7: The influence of
the angular momentum of the particle around the non-extremal RNSH black hole
on the Lyapunov exponent, where $q=15$ and $Q=0.70$. The chaos bound is
violated in the range $L>3.84$ ($1.00698r_{+}<r_{0}<1.83414r_{+}$, where
$r_{+}=1.10000$) when $s=0.50$, in the range $L>4.92$
($1.02286r_{+}<r_{0}<1.69188r_{+}$, where $r_{+}=1.24495$) when $s=0.45$, in
the range $L>6.05$ ($1.04015r_{+}<r_{0}<1.63982r_{+}$, where $r_{+}=1.33166$)
when $s=0.4$, and in the range $7.97<L<36.21$
($1.06953r_{+}<r_{0}<1.34309r_{+}$, where $r_{+}=1.40000$) when $s=0.35$.
There is no violation when $s=0.30$.

The influence of the angular momentum of the particle around the non-extermal
RNSH black hole on the exponent is plotted in Figure 4 - Figure 7 by fixing
$q=0.9$ and $q=15$, respectively. Figure 4 shows that the angular momentum’s
range and spatial region increase with the increase of the black hole’s charge
when $q=0.9$ and $s=0.2$. There is no violation when the black hole’s charge
less than a certain value. For example, there is no violation when $Q=0.80$.
In Figure 5, we observe that the angular momentum’s range and spatial region
decrease with the decrease of the hair parameter’s value. There is no
violation when $s=0.35$ and $s=0.30$. Although the ranges of the angular
momentum in these two figures are very large, the corresponding spatial
regions are not large. In ZLL ; LG1 ; LG2 , the authors did not directly give
the charge-to-mass ratio of the particle, but gave a general expression about
the charge. In KG1 ; KG2 , the authors found the violation for the bound when
the charge-to-mass ratio is greater than 1. In above discussion, there is also
a violation when the ratio is less than 1. Therefore, the violation found here
does not require a weak gravity conjecture. We will discuss the case where the
ratio is greater than 1 in the following.

In Figure 6, the angular momentum’s range and spatial region increase with the
increase of the black hole’s charge. When the charge is less than a certain
value, there is no violation. When $Q=0.89$, $s=0.20$ and the angular momentum
approaches 4.25, there is $r_{0}\to 1.00463r_{+}$, which indicates that the
equilibrium orbit is in the near-horizon region. Therefore, the bound is
violated from the near-horizon region to a certain distance from the horizon.
The similar case also occurs in Figure 7, where $L\to 3.84$ and $r_{0}\to
1.00698r_{+}$ when $Q=0.70$ and $s=0.50$. In Figure 7, the angular momentum’s
range and spatial region increase with the increase of the hair parameter’s
value. When the hair parameter is less than a certain value, there is no
violation. Comparing Figure 4 and 5 with Figure 6 and 7, we find that for the
same black hole’s parameters, the spatial regions are relatively large when
the particle’s charge is large. The reason for this phenomenon is that the
electromagnetic force applied to the particle increases with the increase of
its charge. The particle with large charge is farther away from the horizon
than that with less charge.

Figure 8: The influence of the charge of the particle around the extremal RNSH
black hole on the Lyapunov exponent, where $L=5$. The chaos bound is violated
in the range $0<q<16.99$ ($1.000206r_{+}<r_{0}<2.192235r_{+}$) when $Q=0.3000$
and $s=0.9100$, in the range $0<q<20.39$ ($1.000155r_{+}<r_{0}<2.192235r_{+}$)
when $Q=0.2500$ and $s=0.9375$, in the range $0<q<25.49$
($1.000104r_{+}<r_{0}<2.192235r_{+}$) when $Q=0.2000$ and $s=0.9600$, in the
range $0<q<33.99$ ($1.000053r_{+}<r_{0}<2.192235r_{+}$) when $Q=0.1500$ and
$s=0.9775$, and in the range $0<q<50.98$ ($1.000002r_{+}<r_{0}<2.192235r_{+}$)
when $Q=0.1000$ and $s=0.9900$. Figure 9: The influence of the angular
momentum of the particle around the extremal RNSH black hole on the Lyapunov
exponent, where $q=0.9$. The chaos bound is violated in the range $L>2.34$
$(1.98144r_{+}<r_{0}<3.73791r_{+})$ when $Q=0.3000$ and $s=0.9100$, in the
range $L>2.43$ $(1.98718r_{+}<r_{0}<3.78975r_{+})$ when $Q=0.2500$ and
$s=0.9375$, in the range $L>2.51$ $(1.99183r_{+}<r_{0}<3.83103r_{+})$ when
$Q=0.2000$ and $s=0.9600$, in the range $L>2.59$
$(1.99542r_{+}<r_{0}<3.86573r_{+})$ when $Q=0.1500$ and $s=0.9775$, $L>2.67$
$(1.99797r_{+}<r_{0}<3.89825r_{+})$ when $Q=0.1000$ and $s=0.9900$.  Figure
10: The influence of the angular momentum of the particle around the extremal
RNSH black hole on the Lyapunov exponent, where $q=15$. The chaos bound is
violated in the range $L>4.39$ ($1.00029r_{+}<r_{0}<1.99910r_{+}$) when
$Q=0.3000$ and $s=0.9100$, in the range $L>3.62$
($1.00080r_{+}<r_{0}<1.99925r_{+}$) when $Q=0.2500$ and $s=0.9375$, in the
range $L>2.83$ ($1.00028r_{+}<r_{0}<1.99940r_{+}$) when $Q=0.2000$ and
$s=0.9600$, in the range $L>2.02$ ($1.00110r_{+}<r_{0}<1.99955r_{+}$) when
$Q=0.1500$ and $s=0.9775$, and in the range $L>1.12$
($1.00088r_{+}<r_{0}<1.99970r_{+}$) when $Q=0.1000$ and $s=0.9900$.

When the inner and event horizons coincide with each other, the black hole is
extremal and the surface gravity disappears. Then there is $Q^{2}+s=1$ and the
horizon is located at $r_{+}=1$. To investigate the influences of the charge
and angular momentum of the particle around the extremal RNSH black hole on
the exponent, we plot Figure 8 - Figure 10.

The influence of the charge on the exponent is plotted in Figure 8. We observe
that the range of the particle’s charge and spatial region increase with the
increase of the hair parameter’s value. When the value of the particle’s
charge decreases, the values of the exponent for different black hole’s
charges gradually approaches. Here we find that when the particle’s charge
takes the certain values, the equilibrium orbits are very close to the
horizon. They are closer to the horizon of the extremal black hole than that
of the non-extremal black hole. For example, when $Q=0.3000$, $s=0.9100$ and
$q\to 16.99$, we get $r_{0}\to 1.000206r_{+}$. The similar case also occurs in
Figure 10. In Figure 10, when the value of the hair parameter increases, the
range of angular momentum increases, while the spatial range decreases. In
Figure 9, when the value of the hair parameter decreases, the range of angular
momentum and spatial region increase. The value of the exponent gradually
approaches as the angular momentum increases. When $q=0.9$, the particle’s
angular momentum corresponding to the critical value is not much different for
different black hole’s charges. When the particle’s charge is very large, such
as $q=15$, the angular momentum changes significantly. The reason for this
phenomenon may be that the particles are subject to different electromagnetic
forces from the black hole and the black hole is extremal.

##  4 Conclusions and discussions

In this paper, we investigated the influences of the charge and angular
momentum of the particle around the non-extremal and extremal RNSH black holes
on the Lyapunov exponent, and found the spatial regions where the chaos bound
is violated. For the certain values of the black holes’ parameters, the bound
is violated in the regions that from the near-horizon regions to the certain
distances from the horizons of the non-extremal and extremal black holes.

The case of the non-extremal black hole was first investigated. For the fixed
angular momentum $L=5$, the spatial region for the violation increases with
the increase of the black hole’s charge when $s=0.20$, and with the increase
of the hair parameter’s value when $Q=0.70$. For the fixed particle’s charges,
the angular momentum’s ranges and spatial regions increase with the increase
of the black hole’s charge when $s=0.20$, and with the increase of the hair
parameter’s value when $Q=0.70$. For the extremal black hole, the range of the
particle’s charge and spatial region decrease when the angular momentum is
fixed at $L=5$ and the hair parameter’s value decreases. When the particle’s
charge is fixed at $q=0.9$ and the hair parameter’s value decreases, the
angular momentum’s range increases and spatial region decreases. For the
particle with charge $q=15$, there is no obvious trend in the spatial region
with the decrease of the hair parameter’s value. This may be caused by the
special properties of the extremal black hole.

Recently, the weak gravity conjecture has attracted much attention. It 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. The ratio of the extremal RNSH black hole is one when the hair
parameter is zero. In fact, the value of the hair parameter can be negative CB
. Therefore, it is feasible that the charge-to-mass ratio of the particle
discussed in this paper is greater than 1. In the calculation, we found that
the bound is violated in the near-horizon regions of both extremal and non-
extremal black holes when the particle’s charge is relatively large. For
example, the bound is violated at $r_{0}\to 1.00463r_{+}$ when $q=15$,
$Q=0.89$, $s=0.20$ and $L\to 4.25$, and at $r_{0}\to 1.000206r_{+}$ when $q\to
16.99$, $Q=0.3000$, $s=0.9100$ and $L=5$. This result is consistent with that
obtained in ZLL ; LG2 . In their work, they found that the bound is violated
by considering the contributions of the sub-leading terms in the near-horizon
expansion. In LG2 , the authors perceived that this violation should be
related to the dynamical stability of the black hole. The study on the
stability of the extremal black holes may help us to understand the violation.
In LTW , the authors taken into account the minimal length effects on the
chaotic motion of particles, and found the violation of the bound. They
believed that their result does not necessarily implies the violation for the
bound conjectured in MSS , and the bound should be corrected. In our work,
because the backreaction of the particle on the background spacetime was
neglected, we got the case that the bound is violated. If the backreaction is
taken into account LG2 , the result may change, which requires the further
study.

###### Acknowledgements.

This work is supported by the NSFC (Grant No. 12105031) and Tianfu talent
plan.

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