CTP-SCU2022013 Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell Wormhole

2025-05-06 0 0 3.18MB 26 页 10玖币

侵权投诉

CTP-SCU/2022013

Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell

Wormhole

Yiqian Chena,∗Peng Wanga,†Houwen Wua,b,‡and Haitang Yanga§

aCenter for Theoretical Physics, College of Physics,

Sichuan University, Chengdu, 610064, China and

bDepartment of Applied Mathematics and Theoretical Physics,

University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

It has been recently reported that, at late times, the total luminosity of a star freely falling

in black holes decays exponentially with time, and one or two series of ﬂashes with decreasing

intensity are seen by a speciﬁc observer, depending on the number of photon spheres. In this

paper, we examine observational appearances of an infalling star in a reﬂection-asymmetric

wormhole, which has two photon spheres, one on each side of the wormhole. We ﬁnd that

the late-time total luminosity measured by distant observers gradually decays with time

or remains roughly constant due to the absence of the event horizon. Moreover, a speciﬁc

observer would detect a couple of light ﬂashes in a bright background at late times. These

observations would oﬀer a new tool to distinguish wormholes from black holes, even those

with multiple photon spheres.

∗chenyiqian@stu.scu.edu.cn

†pengw@scu.edu.cn

‡hw598@damtp.cam.ac.uk

§hyanga@scu.edu.cn

arXiv:2210.10948v1 [gr-qc] 20 Oct 2022

CONTENTS

I. Introduction 2

II. Setup 4

III. Numerical Results 7

A. Scenario I 10

B. Scenario II 14

IV. Conclusions 18

Acknowledgments 20

References 20

I. INTRODUCTION

The Event Horizon Telescope (EHT) collaboration released images of the supermassive black

holes M87* [1–8] and Sgr A* [9–14], which provides a new method to test general relativity in the

strong ﬁeld regime. The main feature displayed in these images is a central brightness depression,

namely black hole shadow, surrounded by a bright ring. The edge of black hole shadow involves

a critical curve in the sky of observers, which is closely related to some unstable bound photon

orbits. For static spherically symmetric black holes, unstable photon orbits form photon spheres

outside the event horizon. Since light rays undergo strong gravitational lensing near photon spheres,

black hole images encode valuable information of the geometry in the vicinity of photon spheres.

Therefore, black hole images have been widely studied in the context of diﬀerent theories of gravity,

e.g., nonlinear electrodynamics [15–21], the Gauss-Bonnet theory [22–25], the Chern-Simons type

theory [26,27], f(R) gravity [28–30], string inspired black holes [31–34] and other theories [35–46].

On the other hand, testing the nature of compact objects in the universe has been an important

question in astrophysics for decades. Although the black hole images captured by EHT are in

good agreement with the predictions of Kerr black holes, the black hole mass/distance and EHT

systematic uncertainties still leave some room within observational uncertainty bounds for black

hole mimickers. Among all black hole mimickers, ultra compact objects (UCOs), e.g., boson

stars, gravastars and wormholes, which are horizonless and possess light rings (or photon spheres

in the spherically symmetric case), are of particular interest since their observational signatures

can be quite similar to those of black holes [47–50]. Nevertheless, it is of great importance to

seek observational signals to distinguish UCOs from black holes. For example, due to a reﬂective

surface or an extra photon sphere, echo signals associated with the post-merger ringdown phase

in the binary black hole waveforms can be found in various ECO models [51–61]. In addition,

asymmetric thin-shell wormholes with two photon spheres were found to have double shadows and

an additional photon ring in their images [62–66]. For black holes with one photon sphere, there is

one shadow and one photon ring in black hole images, and no echo signal in late-time waveforms.

These observational features would allow us to distinguish wormholes from black holes with one

photon sphere.

Intriguingly, more than one photon sphere has been reported to exist outside the event horizon

for a class of hairy black holes in certain parameter regions [67–71]. Multiple photon spheres can

introduce distinctive features in black hole images, e.g., double shadows [71], extra photon rings

[72] and tripling higher-order images [73]. Furthermore, late-time echo signals were also observed

since the eﬀective potential of a scalar perturbation possesses a multiple-peak structure [74,75].

Can we distinguish black holes with multiple photon spheres from UCOs? To answer this ques-

tion, we investigate dynamic observations of a luminous object freely falling in an asymmetric

thin-shell wormhole in this paper. Lately, observational appearances of a star freely falling onto

black holes with a single or double photon spheres have been numerically simulated [76,77]. Partic-

ularly, the total observed luminosity fades out exponentially with a declining tail, which is caused

by photons orbiting around the photon sphere, in the single-photon-sphere case. In contrast, when

there exist two photon spheres, the total luminosity exhibits two exponential decays and a sharp

peak between them. In addition, due to photons trapped between two photon spheres, a speciﬁc

observer can detect one more cascade of ﬂashes in the double-photon-sphere case.

Recently, luminous matter falling onto a black hole has been reported to occur periodically near

the Cyg X-1 [78] and the Sgr A* source [79,80]. Moreover, a new way to measure the spin of Sgr

A* was proposed by simulating an infalling gas cloud [81]. In practice, detecting photons circling

around photon spheres several times at late times could be a challenging task due to the scarcity

of these photons. Interestingly, it showed that precise measurements of photon rings, which are

formed of photons circling around photon spheres more than once, may be feasible with a very

long baseline interferometry [82–84]. Therefore, it is timely to study observational appearances of

a freely-falling star in the wormhole background, which provides a new way to detect wormholes.

The rest of the paper is organized as follows. In Section II, we brieﬂy review the asymmetric

thin-shell wormhole and introduce our observational settings. Numerical results are presented in

Section III. Finally, we conclude with a brief discussion in Section IV. We set G=c= 1 throughout

this paper.

II. SETUP

As introduced in [62,65,85], an asymmetric thin-shell wormhole has two distinct spacetimes,

M1and M2, which are glued together by a thin shell at its throat. The metric of the wormhole

is described as

ds2

i=−fi(ri)dt2

i+dr2

fi(ri)+r2

idΩ2,(1)

where i= 1 and 2 indicate quantities in M1and M2, respectively. Focusing on the Schwarzschild

spacetime, we have

fi(ri) = 1 −2Mi

for ri≥R, (2)

where Miare the mass parameters, and Ris throat radius. Without loss of generality, we set

M1= 1 and M2=kin the rest of this paper. For more details of the asymmetric thin-shell

wormhole, refer to [62]. In M1and M2, the local tetrads are

eti=f−1

i(ri)∂

∂ti

,eri=f

i(ri)∂

∂ri

,eθi=1

∂

∂θi

,eφi=1

risin(θ)

∂

∂φi

.(3)

At the throat, one has et1=et2,er1=−er2,eθ1=eθ2and eφ1=eφ2, which yields the relations

between the bases of the tangent space of M1and M2,

∂

∂t1

=Z−1∂

∂t2

,∂

∂r1

=−Z∂

∂r2

,∂

∂θ1

=∂

∂θ2

,∂

∂φ1

=∂

∂φ2

, (4)

where Z≡pf2(R)/f1(R). Therefore, the components of a vector at the throat in M1and M2

are related by

Vt1=ZV t2,Vr1=−Z−1Vr2,Vθ1=Vθ2,Vφ1=Vφ2.(5)

In this paper, we study a point-like star freely falling along the radial direction at θi=π/2

and ϕi= 0, which emits photons isotropically in its rest frame. With spherical symmetry, we can

conﬁne ourselves to emissions on the equatorial plane. The geodesics on the equatorial plane are

described by the Lagrangian

L=−1

2fi(ri)˙

i+1

fi(ri)˙r2

i+r2

i˙ϕi2,(6)

where dots stand for derivative with respect to an aﬃne parameter τ. Since the Lagrangian Ldoes

not depend on coordinates tiand ϕi, the geodesics can be characterized by their conserved energy

Eiand angular momentum liin Mi,

Ei=−pti=fi(ri)˙

ti,li=pϕi=r2

i˙ϕi.(7)

Note that, according to eqn. (5), one has E1=E2/Z and l1=l2.

The Lagrangian of the freely-falling star obeys the constancy L=−1/2 when the aﬃne param-

eter τis chosen as the proper time. Since the star falls radially, its angular momentum li= 0. Due

to the traversability of the wormhole, we consider two scenarios with distinct trajectories of the

star. In the scenario I, the star with energy E1= 1/Z (E2= 1) has a nonzero initial velocity at

spatial inﬁnity of M1. So, the star can pass through the throat and travel towards spatial inﬁnity

of M2. With the relation (7), the four-velocities of the star in M1and M2are given by

vµ1

e(r1) = 1

1−2r−1

1rR−2

R−2k,−r2k−2

R−2k+2

,0,0!,

vµ2

e(r2) = 1

1−2kr−1

,r2k

,0,0!.(8)

In the scenario II, the star with energy E1= 1 is initially at rest at spatial inﬁnity of M1. At ﬁrst,

the star falls freely in M1, passes through the throat and reaches a turning point in M2. Then,

it moves towards the throat in M2, returns to M1and comes to rest at spatial inﬁnity of M1.

Similarly, the four-velocities of the star in M1and M2are

vµ1

e(r1) = 1

1−2r−1

,∓r2M

,0,0!,

vµ2

e(r2) = 1

1−2kr−1

2rR−2k

R−2,±r−2k+ 2

R−2+2k

,0,0!,(9)

where plus and minus signs represent outward and inward moving, respectively.

Moreover, null geodesics on the equatorial plane are also governed by the Lagrangian (6) with

L= 0, which rewrites the radial component of the null geodesic equations as

˙ri2

i−Vi,eﬀ (ri),(10)

where bi≡li/Eiis the impact parameter, and Vi,eﬀ (ri) = fi(ri)r−2

iis the eﬀective potential. Note

that the impact parameters of a null geodesic in M1and M2, namely b1and b2, are related by

b1=Zb2. A photon sphere in Miis constituted of unstable circular null geodesics, whose radius

rph

iis determined by

Vi,eﬀ(rph

i) = 1

(bph

i)2,V0

i,eﬀ(rph

i) = 0, V00

i,eﬀ(rph

i)<0,(11)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

CTP-SCU/2022013ObservationalAppearanceofaFreely-fallingStarinanAsymmetricThin-shellWormholeYiqianChena,PengWanga,yHouwenWua;b,zandHaitangYangaxaCenterforTheoreticalPhysics,CollegeofPhysics,SichuanUniversity,Chengdu,610064,ChinaandbDepartmentofAppliedMathematicsandTheoreticalPhysics,UniversityofCamb...

展开>> 收起<<

CTP-SCU2022013 Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell Wormhole.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

CTP-SCU2022013 Observational Appearance of a Freely-falling Star in an Asymmetric Thin-shell Wormhole

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: