Chaotic dynamics of string s around the Bardeen -AdS black hole surrounded by quintessence dark energy Jiayu Xie Yaxuan Wang Bing Tang

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Chaotic dynamics of strings around the Bardeen-AdS black

hole surrounded by quintessence dark energy

Jiayu Xie, Yaxuan Wang, Bing Tang*

Department of Physics, Jishou University, Jishou 416000, China

We study the motion of a ring string in the background of the Bardeen-AdS black

hole surrounded by the quintessence dark energy. The effects of the magnetic

monopole charge



, the quintessence state parameter



, and the quintessence

normalization parameter

on the dynamical behavior of the ring string are

respectively analyzed. Our numerical results show that the chaotic behavior of string

generally becomes stronger with the increase of the quintessence normalization

parameter. In particular, the conditions for the existence of chaos are distinctly diverse

for two different quintessence state parameters. Furthermore, it is found that the

magnetic charge does not significantly affect the chaotic behavior of the string in a

specific range.

Ⅰ. INTRODUCTION

In recent years, due to an intrinsic non-linearity of the General Relativity, the

dynamic of chaos has already been one of the great focus issues in these relativistic

systems. On account of the susceptivity to the initial condition, chaos is a kind of

random motion and cannot be predicted in deterministically and nonlinearly dynamic

systems, leading in turn to appeal to many researchers spread deep exploration to

chaotic phenomena. However, the chaos always emerges when the researchers

expanded their work to some complicated space-time backgrounds, for example,

exploring the dynamic behavior of the charged particle around one black hole

including quasi-topological electromagnetism[1], the test particles around deformed

* Corresponding author.

E-mail addresses: bingtangphy@163.com; bingtangphy@jsu.edu.cn

electrically and magnetically charged Reissner-Nordström black holes[2], and a

particle among the Rindler space in a 4-Dimension black hole having spherically

symmetric construction[3]. These researches reveal that the main reason for the

occurrence of chaos is that the equation of motion is variable-inseparable and the

corresponding nonlinear dynamical system is not integrable, which is led by those

complicated couplings. In addition, although the dynamic of a particle is integrable, it

has also been proved that the chaotic phenomena can occur in a certain potential, for

example, a massless and chargeless particle with the addition of harmonic

perturbation in a very near horizon region[4-7].

Until now, the motion of a string in space-time has also attracted the interest of

many researchers, which is rapidly becoming a vigorous area of research in string

theory. The more complex and more intriguing behaviors of the motion of the string

can be exhibited via the inherent outstretched nature of string. Aspects of circular

strings dynamics in the Schwarzschild black hole with the foundational background

have been profoundly studied[8-10]. As pointed out in Refs.[11-13], the influence of

the parameters of one charge Reissner-Nordström black hole or one anti-de

Sitter(AdS) Gauss-Bonnet black hole causes that dynamical behaviors of one circular

string generate a special transformation. There is potential significance to discuss the

conversion from order to chaos and the condition of transformation. Moreover, some

researchers also pay close attention to chaos bound and its violation[14-17]. Čubrović

has performed a systematic study in AdS black hole geometries with multiple

horizons[14]. Gao et al. have inquired into influencing mechanism of the angular

momentum in the charged Kiselev black hole surrounded by an anisotropic fluid[15].

At a classical level, the motion of one string in miscellaneous curved space-time

backgrounds is non-integrable. As already shown in Refs.[18-26], the dynamics of the

string behave features of chaotic motion, even on the radially symmetric

backgrounds[27-29]. In contrast, the motion of a test particle is integrable in most

radially symmetric backgrounds. Understanding the difference of motion between a

string and a particle in the cosmological or gravitational backgrounds may give us

some enlightening views and help us figure out various problems which are associated

with classical gravitational singularities. Another motivation to investigate string

dynamics in black holes arises from growing interest in chaos near black holes, which

is aroused by the approach on studying more complex quantum chaos through some

analysis of the dynamic behavior of those relatively simple black holes. Chaos of one

string around a black hole is an important topic of contemporary research in physics.

As far as we are aware, the characteristic of the chaos in the vicinity of a black hole is

linked to its character. Hence, investigating as many various black holes as possible

may be available to help us comprehend those chaos phenomena more clearly.

However, in masses of curved time-space contexts, string theory is not capable of

being resolved analytically. It is widely advised that such kinds of integrability can be

stretched to the field theory via taking advantage of the AdS/CFT theory duality[30].

As is well-known, the current cosmological observations forecast the presence of

a special kind of energy that pervades the whole of space, called dark energy, which

could be one of the origins of the negative pressure. Modern cosmological

observations have revealed that about 68.3% dark energy of the energy density

constitutes our current universe, according to the standard model of

cosmology[31-33]. As one of the most fitting dark energy candidates, the

quintessence has attracted much attention, whose state equation

q q q





satisfied for

1 1/ 3



   

. With the help of the phenomenological approach

suggested by Kiselev[34], one can effortlessly analyze a black hole encircled by

quintessence directly. The regular Bardeen black hole meets the weak energy

condition and is regular everywhere, which was first proposed in Ref.[35]. In previous

studies, those physical analyses on the matter and thermodynamical quantities have

indicated that the Bardeen model is a decent candidate for understanding

astrophysical black holes[36-38], and is also considered as a natural particle

accelerator[39]. So far, some successive studies on quintessential Bardeen-AdS black

hole have also been carried out[40-43]. Unfortunately, very little attention has been

paid to chaotic dynamic of strings in the context of a black hole surrounded with dark

energy. We are of interest to know the magnetic monopole charge whether still holds a

vital role in the chaotic motion of strings around the Bardeen-AdS black hole

surrounded by the quintessence dark energy.

In this work, we focus on the dynamic behavior of string around the

Bardeen-AdS black hole encircled by the quintessence dark energy. The aim of the

present paper is to present one comprehensive work to study the chaotic behavior in

the corresponding phase space by considering the influence of the quintessence state

parameter



and the normalization constant

. Particularly, we attempt to find out

what effect the quintessence on the dynamics of strings near the quintessential

Bardeen-AdS black hole. In addition, the impact of the parameter



corresponding

to the different dark energy backgrounds is essential to the motion of the ring string,

which is not fully characterized so far. We shall accomplish this work by calculating

chaos indicators: Poincaré sections and the maximum Lyapunov exponents.

We organize the present article as follows. In Sect.2, we introduce the solution

for one Bardeen-AdS black hole encircled by the quintessence dark energy. And then,

the equations of motion of the ring string around the quintessential Bardeen-AdS

black holes are derived in Sect.3. The numerical methods and chaos indicators used in

this paper are shown in Sect.4. What is more, the solution for the present dynamical

system and the characteristics of the dynamic behavior of strings are explored. The

last section is used to summarize and discuss the present work.

Ⅱ. THE BARDEEN-ADS BLACK HOLE SURROUNDED BY

QUINTESSENCE

The Bardeen black hole is one regular spacetime solution of Einstein field

equation with the coupling of a nonlinear electromagnetic field. The four-dimensional

asymptotic AdS spacetime action can be expressed as[35]

1 1 3 1

[ ( )]

16 8 4

S d x g R+ l

  

  



(1)

where

and

are the Einstein scalar and the AdS radius, respectively. The first

term in brackets signifies the Einstein gravity, the second represents the negative

cosmological constant

3/l  

, and the third term

()

is a nonlinear

electrodynamics source Lagrangian, which reads

5/2

() 14















(2)

Here, is denoted as





, where



represents the electromagnetic

field intensity and



corresponds to the positive magnetic monopole charge related

to it. In the weak field limit

0

, the Lagrangian of the density of the nonlinear

electrodynamics is degraded to

1/2 5/4





, which is marginally strong compared

to Maxwell field.

The investigation for the effect of the quintessence on the dynamic behavior of a

black hole is inescapable, since the quintessence exists across our entire universe.

Taking into account the quintessence, the above mentioned action adopts the

following form

1 1 3 1

[ ( ) ]

16 8 4 Q

S d x g R+ l

  

   



(3)

Altering the action Eq. (3) for the electromagnetic field and the metric tensor



, respectively, one can write down the equations of motion, namely,

3 ( )

2 ( )GT



     





   







()









 





(4)

Under this theoretical model, the quintessence is derived from one fluid

possessing the energy-momentum tensor



. Through the use of the Kislev

phenomenological model, the regular-Bardeen black hole surrounded by the

quintessence can be constructed as[34]

t r q





1(3 1)

2qq



 

   

(5)

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ChaoticdynamicsofstringsaroundtheBardeen-AdSblackholesurroundedbyquintessencedarkenergyJiayuXie,YaxuanWang,BingTang*DepartmentofPhysics,JishouUniversity,Jishou416000,ChinaWestudythemotionofaringstringinthebackgroundoftheBardeen-AdSblackholesurroundedbythequintessencedarkenergy.Theeffectsofthemagneti...

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Chaotic dynamics of string s around the Bardeen -AdS black hole surrounded by quintessence dark energy Jiayu Xie Yaxuan Wang Bing Tang

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