Coalescence ofFive-dimensional Black Holes( 5 次元ブラックホールの合

体 )

Ken Matsuno

( 松野 研 )

( H. Ishihara , S. Tomizawa , M. Kimura )

1. Introduction( なぜ高次元か , 次元低下 ,

コンパクトな余剰次元を持つブラックホール )

2. Coalescence of 5D Black Holes( 漸近構造の違いを調べる )

1. Introduction

我々は 4 次元時空 に住んでいる

量子論と矛盾なく , 4 種類の力を統一的に議論する

弦理論 超重力理論

余剰次元 の効果が顕著

高次元ブラックホール ( BH ) に注目

空間 3 次元

時間 1 次元

高次元時空 上の理論

高エネルギー現象

強重力場

次元低下高次元時空 ⇒ 有効的に 4 次元時空

a. Kaluza-Klein model “ とても小さく丸められていて見えない ”

b. Brane world model “ 行くことが出来ないため見えない”余剰次元方向

余剰次元方向

4 次元

Brane world model

Brane ( 4 次元時空 ) : 物質 と 重力以外の力 が束縛 Bulk ( 高次元時空 ) : 重力のみ伝播

重力の逆 2 乗則から制限 ⇒ ( 余剰次元 ) 0.1 mm≦

加速器内で ミニ・ブラックホール 生成 ?( 高次元時空の実験的検証 )

Brane

Bulk

5-dim. Black Objects

4 次元 : 軸対称 , 真空

⇒ Kerr BH with S2 horizon only

5 次元　 : 　軸対称 , 真空

⇒ Variety of Horizon Topologies

Black Holes

( S3 )

Black Rings

( S2×S1 )

[ 以降、 5 次元時空に注目 ]

4D Black Holes : Asymptically Flat

5D Black Holes : Variety of Asymptotic Structures

Asymptotically Flat :

Asymptotically Locally Flat :

: 5D Minkowski

: Lens Space

: 4D Minkowski + a compact dim.

Asymptotic Structures of Black Holes

( time ) ( radial ) ( angular )

Kaluza-Klein Black Holes

Kaluza-Klein Black Holes

4 次元 Minkowski

Compact S14 次元 Minkowski

[ 4 次元 Minkowski と Compact S1 の直積 ]

Squashed Kaluza-Klein Black Holes

4 次元 Minkowski

Twisted S1

[ 4 次元 Minkowski 上に Twisted S1 Fiber ]

異なる漸近構造を持つ 5 次元帯電ブラックホール解

5D Kaluza-Klein BH

( Ishihara - Matsuno )

r+

r-

4D Minkowski

+ a compact dim.

5D 漸近平坦 BH

( Tangherlini )r+r-

5D Minkowski

Two types of Kaluza-Klein BHs

Point Singularity Stretched Singularity

r+

r-

r+r-

同じ漸近構造

Five-dim. BHs : Variety of

S3 , S3 / Zn ( Lens Space ), S2×S1 , …

ex) Creation of Charged Rotating Multi-BHs in LHC( Coalescence of these BHs ? )

Change of Horizon Topologies ? ( S3 + S3 ? )⇒ Distinguishable of Asymptotic Structures ? ( From Behavior of Horizon Areas ? )

Study of Five-dimensional Black Holes

Horizon Topologies

Asymptotic Structures

2 種類の漸近構造

: 5D Minkowski

: Lens Space ここで

は

平坦空間上

Eguchi - Hanson 空間上 の 回転 BH の 合体( 本研究が初めて )

2. ブラックホールの合体

Multi-Black Holes

Multi-BHs : ( mass ) = ( charge )

重力場 ( 引力 ) とマックスウェル場 ( 斥力 ) のつりあい

Multi-Black Holes

Time

宇宙項

Time

時間反転

Time

BH の合体

Time

BH の合体

Time

5D Einstein-Maxwell system with Chern-Simons term and positive cosmological constant

System

Rotating Solution on Eguchi-Hanson space

Specified by ( m1 , m2 , j )

Three-sphere S3

( S2 base ) ( twisted S1 fiber )

S2

S1

S3

Three-sphere S3

S2×S1 S3

( S2 base ) ( twisted S1 fiber )

Lens space S3 / Zn

( S2 base ) ( S1 / Zn fiber )

S2

S1 S1 / Zn

( ex. Changing of Horizon Areas )

S3 S2 S3 / Zn

Eguchi-Hanson space 4D Ricci Flat ( Rij = 0 )

2 NUTs on S2 - bolt at ri = ( 0 , 0 , zi ) : 両極

( Fixed point of ∂/∂ζ )

Asymptotic Structure ( r ~ ∞) : R1×S3 / Z2

S2 - bolt

z

Rotating Solution on Eguchi-Hanson space

For Suitable ( m1 , m2 , j )

“ Mapping Rules ” of parameters ( mi , j )

S3

m2 , j

S3

m1 , j +

S3 / Z2

2(m1 + m2)

8 j

Early Time Late Time

S3

m2 , j

S3

m1 , j +

S3

m1 + m2

2 j

[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ] ( on EH space )

[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )

“ Mapping Rules ” of parameters ( m , j )

S3

m , j

S3

m , j +

S3 / Z2

4 m

8 j

Early Time Late Time

S3

m , j

S3

m , j +

S3

2 m

2 j

m = m1 = m2

[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ] ( on EH space )

[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )

“ Mapping Rule ” of ( m , j ) for coalescence on BHs on EH space

( mλ2 , j 2 / m 3 ) ( 4 mλ⇒ 2 , j 2 / m 3 ) ( we set m = m1 = m2 )

mλ2

j2 / m

3

ODEC : Two S3 BHs at Early time

OAFC : Single S3 / Z2 BH at Late time

OABC : Coalescence of 2 BHs ( S3 → S3 / Z2 )

“ Mapping Rules ” of parameters ( m , j )

S3

m , j

S3

m , j +

S3 / Z2

4 m

8 j

Early Time Late Time

S3

m , j

S3

m , j +

S3

2 m

2 j

m = m1 = m2

[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ] ( on EH space )

[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )

“ Mapping Rule ” of ( m , j ) for coalescence on BHs on Flat space

j2 / m

3

ODEC : Two S3 BHs at Early time

OGKL : Single S3 BH at Late time

OGHC : Coalescence of 2 BHs ( S3 → S3 )

mλ2

( mλ2 , j 2 / m 3 ) ( 2 mλ⇒ 2 , ( j 2 / m 3 ) / 2 ) ( we set m = m1 = m2 )

Comparison of Horizon Areas

S3

m , j

S3

m , j +

Early Time

Late Time

( Lens space S3 / Z2 )

S3 / Z2

4 m

8 j

S3

2 m

2 j

Horizon Area の変化

S3

m , j

S3

m , j +

S3 / Z2

4 m

8 j

Early Time Late Time

S3

m , j

S3

m , j +

S3

2 m

2 j

[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]

[ 漸近平坦 ( R1×R1×S3 ) な時空 ]

Comparison of Horizon Areas A(l) / A(e) > 1

漸近平坦な時空 漸近的に lens space な時空

j2 / m

3

j2 /

m3

mλ2 mλ2

j2 /

m3

mλ2

j2 /

m3

mλ2

j2 / m

3

mλ2

Horizon Area の変化

S3

m , j

S3

m , j +

S3 / Z2

4 m

8 j

Early Time Late Time

S3

m , j

S3

m , j +

S3

2 m

2 j

[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]

[ 漸近平坦 ( R1×R1×S3 ) な時空 ]

Horizon Area の変化

S3

m , j

S3

m , j +

S3 / Z2

4 m

8 j

Early Time Late Time

S3

m , j

S3

m , j +

S3

2 m

2 j

[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]

[ 漸近平坦 ( R1×R1×S3 ) な時空 ]

Comparison of Horizon Areas AEH(l) / AFlat

(l)

j2 / m3

mλ2

j → 0

Comparison of Horizon Areas AEH(l) / AFlat

(l) | j → 0

Comparison of Horizon Areas AEH(l) / AFlat

(l)

j2 / m3

mλ2

λ→ 0

Comparison of Horizon Areas AEH(l) / AFlat

(l) | λ→ 0

We construct5D new Rot. Multi-BH Sol.s on Eguchi-Hanson space

Coalescence of Rotating BHs

with Change of Horizon Topology : S3 S⇒ 3 / Z2

( Lens Space )

Comparing with that on Flat space

without change of Horizon Topology : S3 S⇒ 3

Horizon Areas の振る舞い

Conclusion

回転の影響

漸近構造を区別可能

Measurement of Extra Dimension by Kaluza-Klein Black Holes

( Gravity Probe B 実験結果から 余剰次元サイズ を見積もる )

Rotating Squashed Multi-Black Holes with Godel Parameter

( コンパクトな余剰次元を持つ 多体 BH の合体 )

Future Works

Large Scale Extra Dimension in Brane world model

D 次元時空 ( D 4 ) ≧ ( 余剰次元サイズ L )

: D 次元重力定数

: D 次元プランクエネルギー

When EP,D TeV , D = 6≒

ミニ・ブラックホールの形成条件

コンプトン波長

ブラックホール半径

[ 4 次元 ]

[ D 次元 ]

例 . LHC 加速器内 : EP,D TeV≒

⇒ mc2 TeV (proton mass)×10≧ ≒ 3 ミニ・ブラックホール !

≫ 1 GeV : 1 Proton

Kaluza-Klein model

余剰次元 : 小さくコンパクト化

⇒ 量子力学

[ 例 . 5 次元 ]

L

余剰次元

余剰次元を観測する為に必要な

励起エネルギー

加速器実験から制限 ⇔ L 10 ≒ -17 cm

2. 歪んだ Kaluza-Klein Black Holes

Creation of mini-black holes in the LHC

String Theory

Brane world scenario

Near horizon region : Higher-dim. spacetime

Far region from BHs : Effectively 4D spacetime

Background

Spacetime with large scale extra dim.

Higher-dim. Multi-BHs with compact extra dimensions( R.C. Myers (1987) )

5D Kaluza-Klein Black Holes

Near horizon region : ~ 5D black holeFar region : ~ 4D black hole × S1

Black Holes with a Compact Dimension

5D Einstein-Maxwell-Chern-Simons system

( Bosonic part of the ungauged SUSY 5-dim. N=1 SUGRA )

Solutions

角度成分

Squashed S3

( S2 base ) ( Twisted S1 fiber )

S2

S1 S1

( ex. Shape of Horizons )

S3 S2 Sq. S3

Solutions

Squashed S3

Squashed S3

Spatial cross section of r = const. surface Σr

S2 S1

Oblate

( k > 1 )

Round S3

( k = 1 )

Prolate

( k < 1 )

Near Horizon Region

Shapes of squashed S3 horizons r = r±

outer horizon r+ : Oblate

inner horizon r- : Prolate

( degenerate horizon r+ = r- : round S3 )

Far Region

Coord. Trans. : r ρ ( r = r⇒ ∞ ρ= ∞ )⇒

Far Region

Asymptotically Locally Flat

( a twisted constant S1 fiber bundle over 4D Minkowski )

ρ ∞⇒

4 次元 Minkowski

Twisted S1

Whole Structure

Singularity

r = 0

Spatial Infinity

r = r∞

0 < r < r∞

Outer Horizon

r = r+

Inner Horizon

r = r-

Two Regions of r coordinate

Here, we consider the region

Furthermore, we can consider the region for BH

Two types of Singularities

Point Singularity : shrink to a point as

Stretched Singularity : S2 → 0 and S1 → ∞ as

Two types of Black Holes

Point Stretched

Black Hole

Naked Singularity

2. の まとめ We construct charged static Kaluza-Klein black holes with squashed S3 horizons in 5D Einstein-Maxwell theory

These black holes asymptote to the effectively 4D Minkowski with a compact extra dimension at infinity

We obtain two types of Kaluza-Klein black holes related to the shapes of the curvature singularities

Point Singularity & Stretched Singularity

Asymptotic Behaviors

r ≒ ri　近傍r ∞ ( ≒ 遠方 )

Klemm – Sabra 解

Klemm-Sabra Solution

BH Horizon x+ in this coord.s is given by sol.s of

Specified by ( m , j )

Killing Vector Fields : ∂/∂ψ 　∂ /∂φ

: outgoing null expansion

( S3 )

x についての 3 次方程式　⇒　 ( m , j ) に制限

Region of ( m , j )

Black Hole

No Horizon

Absence of Closed Timelike Curves ( CTCs )

No CTC for x > x+ > 0

⇔ ( ψ , φ ) part of metric g2D has no negative eigenvalue

⇔ gψψ (x) > 0 and det g2D (x) > 0

In this case ,

gψψ (x+) > 0 and det g2D (x+) > 0

No CTC !

x の単調増加関数

Early Time

BH Horizon in this coord.s is given by sol.s of

[ Specified by ( mi , j ) ]

( outgoing null expansion )

Rot. 2 BHs at Early time

( outer trapped small S3 )S3 S3

For suitable ( mi , j )

( S3 )

Late Time

BH Horizon in this coord.s is given by sol.s of

[ Specified by ( 2( m1 + m2 ) , 8 j ) ]

Rot. 1 BH at Late time

( outer trapped large S3 )

S3 / Z2

( outgoing null expansion )

( Lens space S3 / Z2 )

For suitable ( mi , j )