Chaos bound and its violation in the torus-like black hole Rui YinabJing Liangabyand Benrong Muabz aCenter for Joint Quantum Studies College of Medical Technology

2025-04-24 0 0 337.4KB 12 页 10玖币

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Chaos bound and its violation in the torus-like black hole

Rui Yina,b,∗Jing Lianga,b,†and Benrong Mua,b‡

aCenter for Joint Quantum Studies, College of Medical Technology,

Chengdu University of Traditional Chinese Medicine, Chengdu, 611137, PR China and

bCenter for Theoretical Physics, College of Physics,

Sichuan University, Chengdu, 610064, PR China

Abstract

In this paper, we have studied the variation of the chaos bound in two regions of the torus-like

black hole, i.e., the region close to the black hole horizon and the region at a certain distance from

the black hole horizon. The angular momentum of the particle aﬀects the eﬀective potential and

inﬂuences the magnitude of the chaotic behavior of the particle. Therefore, the angular momentum

of particle is important in the study. The angular momentum of a particle not only aﬀects the

particle equilibrium orbital position, but also aﬀects the Lyapunov exponent. As the angular

momentum of the particle increases, the particle equilibrium position gradually moves away from

the black hole horizon. In the near black hole horizon region, the chaos bound is not violated,

however, at the far black hole horizon region, the chaos bound is violated. In addition unlike the

charged AdS black hole which has a spherical topology of the horizon, the torus-like black hole has

a toroidal topology of the horizon.

∗Electronic address: yrphysics@126.com

†Electronic address: ljphysics@163.com

‡Electronic address: benrongmu@cdutcm.edu.cn

arXiv:2210.07799v1 [gr-qc] 13 Oct 2022

Contents

I. Introduction 2

II. Lyapunov exponent in the torus-like black hole 4

III. Chaos bound and its violation in the torus-like black hole 6

IV. Conclusions 8

Acknowledgments 9

References 9

I. INTRODUCTION

Chaos is a seemingly random, chance and irregular motion that occurs only in non-linear

and non-accumulable dynamical systems, which are sensitive to initial conditions [1–3]. Since

the tiny errors in chaotic motion grow rapidly with time, the motion at this point can be

quite diﬀerent from what it would be without these errors. This means that long-term

prediction of chaotic motion in general is very diﬃcult. It also means that chaotic motion

has many new properties that the usual dynamical systems do not have. This has led to

widespread interest in the study of chaos in various areas of physics.

In general relativity, the geodesic motion of particles in a generic Kerr-Newman black

hole spacetime [4] is integrable and there is no chaos in this system. In order to study

chaotic motion in general relativity and to ensure that the dynamical system describing the

motion of the mass is integrable, it is necessary to resort to some spacetime with a complex

geometry or to introduce some additional interactions. Along this spirit, the chaotic mo-

tions of particles have been studied in the multi-black hole spacetimes [5, 6], the perturbed

Schwarzschild spacetime [7–9], and in the non-standard Kerr black hole spacetime described

by MankoNovikov metric [10–12]. The study of chaotic behavior of the geodesic motion of

particles has now involved several spacetime contexts, and the main interest of researchers

in these systems is to use and further develop coordinate invariant descriptions and metrics

of chaotic behavior to make them applicable to general relativity where space and time are

not absolute. It has been recently shown that using the Melnikov method to identify chaotic

behavior in geodesic motion perturbed by the minimum length eﬀect around a Schwarzschild

black hole, there is Smale horseshoes chaotic structure in the phase space [13]. Based on the

Melnikov method, the existence of a critical amplitude aﬀecting temporal chaos is demon-

strated by studying the thermodynamic chaos of RN-AdS black holes immersed in Perfect

Fluid Dark Matter, while spatial chaos is always present regardless of the perturbation in-

tensity [14]. It has also been tentatively proved that the chaotic behavior of particles near

the black hole has quantum gravitational eﬀects [15]. In addition, chaotic phenomenon was

also investigated for the pinning particles in Kerr spacetime [16]. More interestingly, chaos

in loop string dynamics has been found in the asymptotically ﬂat Schwarzschild black hole

spacetime [17], in the AdS-Schwarzschild black hole [18] and in the AdS-Gauss-Bonnet black

hole spacetime [19] after the introduction of loop strings instead of point particles.

In recent studies, a number of violations of the chaos bound have been discovered [20–24].

The static equilibrium of charged probe particles around a black hole can be provided by the

Lorentz force. In Ref. [20], Zhao et al. considered the contribution of the sub-leading terms

in the expansion of the near-horizon regions and investigated the chaotic bound in the near-

horizon regions using the eﬀective potential. It is found that the Reissner-Nordstrom and

Reissner-Nordstrom-AdS black holes satisfy this bound, which is violated in a large number

of charged black holes. In their study, they only considered the radial contribution. In fact,

since angular momentum aﬀects the eﬀective potential and increases the magnitude of the

chaotic behaviour of the particles, angular momentum also has an eﬀect on the Lyapunov

exponent. In consideration of the above, Lei et al. again studied the chaos bound in the

near-visible region of Reissner-Nordstrom and Reissner-Nordstrom-AdS black holes [24]. It

is found that the bound is violated in the near-visible region when the angular momentum

of the charge and particles of the black hole is large. In rotating charged black holes, the

existence of a violation of the bound was also found by calculations of the eﬀective potential

[21, 22].

In this paper, we study the eﬀect of the angular momenta of charged particles on the chaos

bounded by the circular motion of particles around a torus-like black hole. The concept of

a torus black hole was ﬁrst introduced in the literature [25]. Unlike other black holes, the

topology of this black hole spacetime is S×S×M2. This has inspired many studies of

torus-like black holes [26–33].

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Chaosboundanditsviolationinthetorus-likeblackholeRuiYina;b,JingLianga;b,yandBenrongMua;bzaCenterforJointQuantumStudies,CollegeofMedicalTechnology,ChengduUniversityofTraditionalChineseMedicine,Chengdu,611137,PRChinaandbCenterforTheoreticalPhysics,CollegeofPhysics,SichuanUniversity,Chengdu,610064,PRC...

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Chaos bound and its violation in the torus-like black hole Rui YinabJing Liangabyand Benrong Muabz aCenter for Joint Quantum Studies College of Medical Technology

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