Stable regular solution of Einstein-Yang-Mills equation Yuewen Chen1and Shing-Tung Yau1 2 3y 1Yau Mathematical Sciences Center Tsinghua University Beijing 100084 China

2025-05-03 0 0 385.76KB 4 页 10玖币

侵权投诉

Stable regular solution of Einstein-Yang-Mills equation

Yuewen Chen1, ∗and Shing-Tung Yau1, 2, 3, †

1Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

2Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China

3Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

(Dated: November 1, 2022)

In this letter, we ﬁnd the ﬁrst dynamically stable non-singular solution of spherically symmetric

SU(2) Einstein-Yang-Mills equation. This solution is regular at r= 0 and asymptotically ﬂat. Since

the Yang-Mills ﬁeld strength decay exponentially, the Einstein-Yang-Mills particle perhaps can be

used as a candidate for dark matter.

INTRODUCTION

In 1988, a global nontrivial static nonsingular particle-

like solution of coupled Einstein-Yang-Mills (EYM) equa-

tion was found by Bartnik and McKinnon numerically [1].

It aroused a great of interest in GR community. It was

established rigorously by Smoller-Wasserman-Yau et.al

[2–4]. It was also found there is an inﬁnite black solu-

tions diﬀerent coupling constants. Since it violated the

familiar no hair theorem, much excitement was gener-

ated. However, it was soon demonstrated by Strauumen-

Zhou [5] that these solutions are not dynamically stable

and therefore not physical. In this letter, we ﬁnd the

ﬁrst dynamically stable non singular solution of Einstein-

Yang-Mills. Since the ﬁeld strength of Yang Mills of our

solution decay exponentially, we believe such particle so-

lution can be a potential candidate for dark matter.

After Bartnik-McKinnon’s pioneering work, a large

number of soliton and black hole solutions of spherically

symmetric EYM equations were found [6–8]. The crit-

ical behavior of spherically symmetric collapse of EYM

equations were studied in [9–12]. Smoller and Yau et.al

proved rigorously that the SU (2) EYM equations admit

an inﬁnite family of black-hole solutions with a regular

event horizon [13]. However, the colored black hole found

numerically by Bizon is also unstable[14,15]. The non-

linear stability was studied in [16] and the work provided

numerical evidence for the instability of the colored black

hole solutions.

The new idea in this letter, is to study the evolution

of the time dependent EYM equation rather than only

studying the static equation [1]. Our main new result is

the observation that for suitable initial data, the EYM

equations would have a non trivial steady state solution.

Furthermore, we show this solution is stable under linear

perturbation. We write the spherically symmetric metric

ds2=−Ae2Qdt2+1

Adr2+r2dΩ2.

The Yang-Mills curvature tensor is given by

F=Wtτ1dt ∧dθ +Wtτ2dt ∧sin θdφ +W0τ1dr ∧dθ

+W0τ2dr ∧sin θdφ −(1 −W2)τ3dθ ∧sin θdφ,

where τiare the Pauli matrices. The evolution version of

Einstein-Yang-Mills equation are given by

AeQWt)t= (AeQWr)r

+eQ

r2W(1 −W2),

(1)

At=−4

rWtWrA, (2)

rAr+A(1 + 2(W2

r+1

A2e2QW2

t)) = 1 −1

r2(1 −W2)2,

(3)

rQr= 2(W2

r+1

A2e2QW2

t).

(4)

We need a new coordinate transformation

x= ln(r).

We deﬁne the auxiliary functions , S, α, θ, q and c, re-

spectively, as follows

= 2(W2

r+1

A2e2QW2

t),(5)

S= 1 −1

r2(1 −W2)2,(6)

α=eQ,(7)

θx=−,(8)

qx= 1 + ,(9)

c=Aα. (10)

To evolution Einstein-Yang-Mills system, we choose

equations (1), (3) and (4). Then, we rewrite the EYM

system as

c(1

cWt)t=c

r(c

rWx)x+cα

r2W(1 −W2),(11)

(Aeq)x=Seq,(12)

(αeθ)x= 0.(13)

arXiv:2210.09861v3 [gr-qc] 30 Oct 2022

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：

StableregularsolutionofEinstein-Yang-MillsequationYuewenChen1,andShing-TungYau1,2,3,y1YauMathematicalSciencesCenter,TsinghuaUniversity,Beijing100084,China2YanqiLakeBeijingInstituteofMathematicalSciencesandApplications,Beijing101408,China3DepartmentofMathematics,HarvardUniversity,Cambridge,MA02138,U...

展开>> 收起<<

Stable regular solution of Einstein-Yang-Mills equation Yuewen Chen1and Shing-Tung Yau1 2 3y 1Yau Mathematical Sciences Center Tsinghua University Beijing 100084 China.pdf

共4页,预览1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！

Stable regular solution of Einstein-Yang-Mills equation Yuewen Chen1and Shing-Tung Yau1 2 3y 1Yau Mathematical Sciences Center Tsinghua University Beijing 100084 China

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: