Chains of mini-boson stars Shi-Xian Sun1 Yong-Qiang Wang2 and Li Zhao3 aLanzhou Center for Theoretical Physics Key Laboratory of Theoretical Physics of Gansu

2025-04-30 0 0 1023.33KB 15 页 10玖币

侵权投诉

Chains of mini-boson stars

Shi-Xian Sun1, Yong-Qiang Wang2, and Li Zhao 3

aLanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu

Province, School of Physical Science and Technology, Lanzhou University, Lanzhou

730000, People’s Republic of China

bInstitute of Theoretical Physics &Research Center of Gravitation, Lanzhou University,

Lanzhou 730000, People’s Republic of China

Abstract

In this paper, we re-investigate the stationary, soliton-like solutions in the model of the

Einstein gravity coupled to a free and complex scalar ﬁeld, which have been known as

mini-boson stars. With numerical method, we ﬁnd that in addition to the usual single

mini-boson star solutions, there exist a novel family of solutions interpreted as chains

of boson stars, which is made of some boson stars along the symmetry axis. We show

the conﬁguration of two types of chains, including an even number of constituents and

an odd number of constituents. Furthermore, we also study the eﬀect of the frequency

of the complex scalar ﬁeld on the ADM mass Mand the U(1) scalar charge Q. It

is interesting to note that the existence of chains of boson stars does not require the

introduction of a complex scalar ﬁeld with self-interacting potential.

1sunshx20@lzu.edu.cn

2yqwang@lzu.edu.cn

3lizhao@lzu.edu.cn, corresponding author

arXiv:2210.09265v1 [gr-qc] 17 Oct 2022

1 Introduction

It is well known that the most general stationary solution of the vacuum Einstein equation in

a four-dimensional spacetime is described by a so-called Kerr geometry, which is a rotating

black hole only with two parameters, the black-hole mass and angular momentum. The

above uniqueness of the Kerr black hole is sometimes referred to as the “no-hair” theorem.

When considering the model of a free scalar ﬁeld minimally coupled to Einstein’s gravity,

one could ﬁnd that it is diﬃcult to obtain analytical or numerical black hole solutions for

a long time. Until the last few years, a novel family of solutions of Kerr black holes with

scalar hair was presented by Herdeiro and Radu in [1]. Furthermore, when restrictions on

the stationary solution of compact object with event horizon are lifted, there exist a family

of horizonless solutions, which have become well-known as boson stars (BSs).

The study of BSs has a long history. Firstly, in the 1960s, a family of spherically

symmetric BSs were discovered by Kaup in the four-dimensional Einstein gravity coupled

to a free, complex scalar ﬁeld with the constant angular frequency of the phase of the ﬁeld in

the complex plane [2, 3]. Due to little astrophysical interest in directed searches for a boson

star, BSs with rotational symmetry were not studied until the 1990s. The ﬁrst rotating

solutions of boson stars in the Einstein-Klein-Gordon theory were found in [4]. BSs with a

free scalar ﬁeld without self-interaction were well known as mini-boson stars (BSs), which

can then be extended to the self-interacting BSs case [5], the excited case of scalar ﬁeld

with nodes [6] and the multistate boson stars [7, 8]. There are also studies on the solutions

of the self-gravitating solitons in Einstein-Proca or Einstein-Dirac models [9, 10]. Another

interest of BSs in astronomy is to investigate their application to axions [11, 12], cosmic

dark matter [13, 14], and black hole mimickers [15, 16, 17]. In addition, the collisions of

binary BSs have been studied extensively [18, 19, 20, 21, 22, 23, 24], which oﬀers a possible

way to detect the BSs with the gravitational waves generated by the merger of binary stars.

Recently, by introducing the self-interaction potential with the type of quartic and sextic

terms, which was studied in literatures on the Q-balls in the ﬁeld theory, a novel family of

solutions interpreted as chains of boson stars, which is made of some boson stars along the

symmetry axis, was obtained in the Einstein-Klein-Gordon theory [25]. Such solutions can

be divided into two classes, including even chains and odd chains according to the parity

at the two sides of the equatorial plane. The obvious diﬀerence is that in the case of even-

numbered chains, the curve of the relationship between mass and frequency could form a

spiral pattern similar to the case of a single boson star, while odd-numbered star chains

can form a kind of loop pattern, which is very diﬀerent from the case of a single boson

star. Moreover, in the second branch of odd-numbered star chains with loop patterns, the

system of star chains turn into a radially excited spherically symmetric boson star. Rotating

generalizations of chains of boson stars with the sextic potential also been studied in [26].

It is worth noting that in ﬂat spacetime there does not exist the chain of multisoliton in

the free scalar ﬁeld model. So, it will be interesting to see whether there are solutions of the

chains of mini-boson stars without the self-interaction potential in the model of free scalar

ﬁeld coupled to gravity. In the present paper, we numerically solve the coupled system of

nonlinear partial diﬀerential equations of scalar ﬁeld and Einstein equations, and obtain a

family of chains of mini -BSs, which can be divided into two classes, including even chains

and odd chains. Moreover, we also study the eﬀect of the frequency of the complex scalar

ﬁeld on the ADM mass and the Noether charge.

This paper is organized as follows. In Sect. 2, we brieﬂy review the model of a free,

complex scalar ﬁeld coupled to Einstein’s gravity. The boundary conditions are analyzed

in Sect. 3. We show the numerical results of two types of BSs chains in Sect. 4. The

conclusion and discussion are given in the last section. Throughout this paper, Roman

letters a,b,c, . . . denote spacetime indices ranging from 0 to 3.

Note added: when we are ﬁnishing this project, we notice that there appears a paper

[27], which overlaps with our results of chains of BSs with two constituents.

2 The Model

Let us introduce the model of Einstein gravity coupled to a free, complex massive scalar

ﬁeld in four-dimensional spacetime. The Lagrangian density reads

S=Zd4x√−gR

16πG − ∇aψ∗∇aψ−µ2ψ∗ψ,(2.1)

where Gand µare the Newton’s constant and the mass of the scalar ﬁeld ψ, respectively.

Note that we consider mini-boson stars without self-interaction potential, and the term

proportional to µ2is known as a mass term. The above action is invariant under a global

U(1) transformation ψ→ψeiθ, where θis constant. Variation of the action (2.1) with

respect to the complex scalar ﬁeld ψand the metric could lead to the following Klein-

Gordon (KG) equation of the scalar ﬁeld

ψ=µ2ψ , (2.2)

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：

Chainsofmini-bosonstarsShi-XianSun1,Yong-QiangWang2,andLiZhao3aLanzhouCenterforTheoreticalPhysics,KeyLaboratoryofTheoreticalPhysicsofGansuProvince,SchoolofPhysicalScienceandTechnology,LanzhouUniversity,Lanzhou730000,People'sRepublicofChinabInstituteofTheoreticalPhysics&ResearchCenterofGravitation,La...

展开>> 收起<<

Chains of mini-boson stars Shi-Xian Sun1 Yong-Qiang Wang2 and Li Zhao3 aLanzhou Center for Theoretical Physics Key Laboratory of Theoretical Physics of Gansu.pdf

共15页,预览3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！

Chains of mini-boson stars Shi-Xian Sun1 Yong-Qiang Wang2 and Li Zhao3 aLanzhou Center for Theoretical Physics Key Laboratory of Theoretical Physics of Gansu

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: