Ringing and echoes from black bounces surrounded by the string cloud Yi Yang1Dong Liu1Zhaoyi Xu1and Zheng-Wen Long1 1College of Physics Guizhou University Guiyang 550025 China

2025-05-03 0 0 950.62KB 13 页 10玖币

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Ringing and echoes from black bounces surrounded by the string cloud

Yi Yang,1, ∗Dong Liu,1, †Zhaoyi Xu,1, ‡and Zheng-Wen Long1, §

1College of Physics, Guizhou University, Guiyang, 550025, China

In the string theory, the fundamental blocks of nature are not particles but one-dimensional

strings. Therefore, a generalization of this idea is to think of it as a cloud of strings. Rodrigues

et al. embedded the black bounces spacetime into the string cloud, which demonstrates that the

existence of the string cloud makes the Bardeen black hole singular, while the black bounces space-

time remains regular. On the other hand, the echoes are the correction to the late stage of the

quasinormal ringing for a black hole, which is caused by the deviation of the spacetime relative to

the initial black hole spacetime geometry in the near-horizon region. In this work, we study the

gravitational wave echoes of black bounces spacetime surrounded by a cloud of strings under scalar

ﬁeld and electromagnetic ﬁeld perturbation to explore the eﬀects caused by a string cloud in the

near-horizon region. The ringing of the regular black hole and traversable wormhole with string

cloud are presented. Our results demonstrate that the black bounce spacetime with strings cloud is

characterized by gravitational wave echoes as it transitions from regular black holes to wormholes,

i.e. the echoes signal will facilitate us to distinguish between black holes and the wormholes in black

bounces surrounded by the string cloud.

I. INTRODUCTION

Recently, the LIGO and Virgo interferometers have

made signiﬁcant progress in the observation of grav-

itational waves (GWs) [1–6]. In addition, the Event

Horizon Telescope has also made a breakthrough in

the imaging of black hole shadows [7,8]. These

results validate the predictions of general relativity

(GR) about black holes (BH). It also allows physi-

cists to test new physical features beyond GR [9–14],

such as the existence of event horizons in compact

objects. Gravitational wave spectroscopy plays a cru-

cial role in the examination of new physical features

beyond general relativity [15,16]. For the gravita-

tional wave signal generated by the binary merger,

its late stage always decays in the form of the ring-

down. It can usually be described using a superposi-

tion of complex frequency damping exponents, which

are called quasinormal modes (QNMs) [17–19]. The

detection of QNMs can serve as a tool to test GR

predictions. Therefore, this makes gravitational wave

detectors (LIGO/Virgo and LISA, etc.) expected to

detect some new physical features in the future, such

as gravitational wave echoes and so on. Gravita-

tional wave echoes are an important observable for

probing the spacetime near the event horizon of the

black hole. In addition, gravitational wave echoes are

closely related to the unique characteristics of com-

pact objects.

Under the framework of general relativity, with the

perturbation of black hole spacetime, it must be ac-

companied by the emergence of quasinormal modes.

Because as long as a black hole is perturbed, it re-

∗yangyigz@yeah.net

†dongliuvv@yeah.net

‡zyxu@gzu.edu.cn

§zwlong@gzu.edu.cn (corresponding author)

sponds to the perturbation by emitting gravitational

waves, and the evolution of gravitational waves can

be divided into three stages [20,21]: ﬁrst, a relatively

short initial burst of radiation; then a longer damped

oscillation, which depends entirely on the parame-

ters of the black hole; and ﬁnally the exponentially

decays over a longer period of time. Note that the

three stages refer to the postmerger gravitational-

wave signal. Among these three stages, people are

generally most concerned about the middle quasinor-

mal ringing stage. The QNMs of black holes have at-

tracted extensive attention [22–41]. Although there

are many indirect ways to identify black holes in the

universe, gravitational waves emitted by perturbed

black holes will carry unique “ﬁngerprints” that allow

physicists to directly identify the existence of black

holes. In particular, Ref. [42] proposes that grav-

itational wave echoes can be used as a new feature

of exotic compact objects. Later, when people stud-

ied QNM in various spacetime backgrounds, gravi-

tational wave echoes were analysed in the late stage

of quasinormal ringing [43–71]. These works make

GWs echoes very important in studying the prop-

erties of compact objects. In Ref. [72], the author

found a new mechanism to produce the gravitational

wave echoes in the black hole spacetime. Bronnikov

and Konoplya [73] found that the echoes appeared in

the black hole-wormhole transition when studying the

quasinormal ringing of black hole mimickers in brane

worlds. In Ref. [74], the authors studied the time

evolutions of external ﬁeld perturbation in the asym-

metric wormhole and black bounce spacetime back-

ground, they observed echoes signals from the space-

time of asymmetric wormholes and black bounce. Es-

pecially, Churilova and Stuchlik in Ref. [75] studied

the quasinormal ringing of black bounce, and they

found the gravitational wave echoes signal during the

regular black-hole/wormhole transition. We need to

pay attention that not all compact objects can show

arXiv:2210.12641v3 [gr-qc] 16 Mar 2023

echoes signals in the late stage of quasinormal ring-

ing. Cardoso et al. [76] pointed out that the pre-

cise observation of the late stage of quasinormal ring-

ing allows us to distinguish diﬀerent compact objects.

Therefore, in our work, we plan to explore whether

the string cloud will destroy the gravitational wave

echoes signal in the black bounce spacetime. We hope

to provide some direction for probing black bounces

with strings cloud experimentally after obtaining its

relevant basic properties.

String theory points out that the fundamental

blocks of nature are not particles but one-dimensional

strings. Therefore, a generalization of this basic idea

is to think of it as a cloud of strings. On the other

hand, the black hole in general relativity usually has

singularities, which forces theoretical physicists to

constantly try to avoid the occurrence of singularities.

A black hole without singularities is called a regular

black hole (RBH). Bardeen was the ﬁrst theoretical

physicist to propose regular black hole [77]. Ayon-

Beato et al. interpret it as a black hole solution for

the Einstein equations under the presence of nonlin-

ear electrodynamics [78]. Letelier proposed a black

hole solution in 1979, which is surrounded by the

string cloud [79]. The string cloud is a closed system,

therefore its stress-energy tensor is conserved. Subse-

quently, black holes with strings have attracted a lot

of attention [80–82]. Sood et al. proposed an RBH

surrounded by the string cloud, but the string cloud

makes this black hole solution no longer regular [83].

It would be very fascinating if string cloud would not

insert singularities in the RBH. Simpson and Visser

proposed a type of regular black hole known as black

bounces [84]. The diﬀerence between this solution

and the standard RBH is that it is achieved by mod-

ifying the black hole area, and it allows a nonzero

radius throat in r= 0. Many studies have been done

on black bounces including analysis of their proper-

ties [85–92]. Recently, Rodrigues et al. embedded

the Simpson-Visser spacetime into a string cloud [93].

They demonstrate that the Simpson-Visser spacetime

is still regular even if the string cloud exists. In this

work, our goal is to study the eﬀect of the presence

of string cloud on the GW echoes of black bounces

spacetime and explore what gravitational eﬀects are

caused by string cloud.

Our work is organized as follows. In Sec. II, we

brieﬂy review the black bounces in a cloud of strings.

In Sec. III, we discuss the scalar ﬁeld and electromag-

netic ﬁeld perturbations for black bounces in a cloud

of strings. In Sec. IV, we outline the time-domain

integration method as well as the WKB method. In

Sec. V, we present the quasinormal ringing and grav-

itational wave echoes of the scalar ﬁeld and elec-

tromagnetic ﬁeld perturbations to black bounces in

a cloud of strings. Sec. VI is our main conclu-

sion of the full text. In this work, we use the units

G=~=c= 1.

II. A BRIEF REVIEW OF THE BLACK

BOUNCES IN STRINGS SLOUD

To gain black bounces in a cloud of string, Ro-

drigues et al. [93] considers the following Einstein

equations

Rµν −1

2Rgµν =κ2Tµν =κ2TM

µν +κ2TCS

µν ,(1)

where

µν =TSV

µν +TNMC

µν ,(2)

where TSV

µν denotes the stress-energy tensor related

to the Simpson-Visser spacetime, and the information

about the non-minimum coupling between the string

cloud and the Simpson-Visser spacetime is included

in the stress-energy tensor TNMC

µν . Furthermore, TCS

µν

in Eq. (1) represents the stress-energy tensor of the

string cloud, which can be written as

TCS

µν =ρΣα

µΣαν

8π√−γ,(3)

where ρrepresents the density of the string cloud.

TCS

µν must satisfy the following conservation laws

∇µTCSµν =∇µρΣµaΣαν

8π√−γ

=∇µ(ρΣµα)Σαν

8π√−γ+ρΣµα∇µΣαν

8π√−γ= 0.

(4)

By solving the above Einstein ﬁeld equations, Ro-

drigues et al. obtain the following black bounces with

the string cloud [93]

ds2=f(r)dt2−f(r)−1dr2− R2dθ2+ sin2θdφ2,

(5)

where

f(r)=1−L−2M

√a2+r2,R=pa2+r2.(6)

If a= 0, this spacetime can be reduced to the Letelier

spacetime, and this spacetime can be reduced to the

Simpson-Visser spacetime when L= 0. If L= 1, this

spacetime will have no event horizon, so the range of

the string parameter Lis 0 <L<1. In addition, the

value of the parameter ahas a critical value

ac=2M

√1−2L+L2.(7)

The black bounce with the string cloud will corre-

spond to a diﬀerent spacetime for diﬀerent a: i) reg-

ular black hole with string cloud for 0 < a < ac; ii)

one-way wormhole with string cloud for a=ac; iii)

traversable wormhole with string cloud for a > ac.

III. MASTER WAVE EQUATION

The covariant equations of scalar ﬁeld perturbation

can be written as

√−g∂µ√−ggµν ∂νΨ= 0,(8)

considering the black bounces surrounded by the

string cloud we studied, we can get

−1

f(r)

d2Ψ

d2t+1

(r2+a2)2rf(r)d

dr Ψ

+r2+a2df(r)

dr Ψ + r2+a2f(r)d2

d2rΨ

(r2+a2)1

sin θ∂θsin θ∂θΨ + 1

sin2θ∂2

φΨ= 0.

(9)

Since the spacetime we are studying is spherically

symmetric, we can achieve separation of variables

through the following ansatz

Ψ(t, r, θ, φ) = X

l,m

ψ(t, r)Ylm(θ, φ)/R,(10)

where Ris the function of radial coordinate rand the

parameter a, which has been deﬁned in equation (6),

and Ylm(θ, φ) are the spherical harmonic function.

After separating the variables and using the proper-

ties of spherical harmonics, we can simplify equations

(9) to the following form

d2ψ

dt2−d2ψ

dr2

∗

+V(r)ψ= 0,(11)

where tortoise coordinate r∗can be deﬁned by

dr∗=1

f(r)dr =1

1−L−2M

√r2+a2

dr. (12)

Moreover, the eﬀective potentials for scalar ﬁeld per-

turbation can be written as

V(r) = 1−L−2M

√r2+a2"`(`+ 1)

r2+a2+2Mr2+a2−2M−(−1 + L)√a2+r2

(a2+r2)5/2#.(13)

The motion equation of the electromagnetic ﬁeld

in the curved spacetime background can be written

√−g∂µ√−gFγσgγµgσν = 0 (14)

where Aµbeing the four vector potential, and Fγσ =

∂γAσ−∂σAγ. Since spacetime has spherical symme-

try, we have

Aµ(t, r, θ, φ) = X

l,m











plm(t,r)

sin θ∂φYlm

−plm(t, r) sin θ∂θYlm







+





flm(t, r)Ylm

hlm(t, r)Ylm

klm(t, r)∂θYlm

klm(t, r)∂φYlm











,(15)

where the term on the left has odd parity (−1)l+1,

and the term on the right has even parity (−1)l. Sub-

stituting the above equation into (14), we can get

∂2ψelec

∂t2−∂2ψelec

∂r2

∗

+Velec(r)ψelec = 0,(16)

where Velec(r) denotes the eﬀective potential of the

electromagnetic ﬁeld perturbation,

V(r) = 1−L−2M

√r2+a2`(`+ 1)

r2+a2.(17)

In Fig. 1, we present the eﬀective potential of

the scalar ﬁeld perturbation for diﬀerent awith

M= 0.5, l = 1, L = 0.1 and for diﬀerent Lwith

M= 0.5, l = 1, a = 0.1 as the function of the tor-

toise coordinate r∗. Here we are studying l= 1 mode

of scalar ﬁeld perturbation mainly because the peak

value of l= 0 mode is too small. From Fig. 1, we

can see that the eﬀective potential is the single peak,

which indicates that the black bounce in a cloud of

strings at this time is the black hole spacetime with

the string cloud. These results show that the eﬀec-

tive potential is very sensitive to changes in L, but

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RingingandechoesfromblackbouncessurroundedbythestringcloudYiYang,1,*DongLiu,1,ZhaoyiXu,1,andZheng-WenLong1,§1CollegeofPhysics,GuizhouUniversity,Guiyang,550025,ChinaInthestringtheory,thefundamentalblocksofnaturearenotparticlesbutone-dimensionalstrings.Therefore,ageneralizationofthisideaistothinkofi...

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Ringing and echoes from black bounces surrounded by the string cloud Yi Yang1Dong Liu1Zhaoyi Xu1and Zheng-Wen Long1 1College of Physics Guizhou University Guiyang 550025 China

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