5D Black Holes with Compact Dimension

Coalescence of
Five-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
空間 3次元

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

時間 1次元
高次元時空上の理論
余剰次元の効果が顕著
高エネルギー現象
強重力場
高次元ブラックホール ( BH ) に注目
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 ”
余剰次元方向
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
4次元
Brane world model
Bulk
Brane
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 )
5-dim. Black Objects
[ 以降、5次元時空に注目 ]
 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
Black Rings
( S3 )
( S2×S1 )
Asymptotic Structures of Black Holes

4D Black Holes : Asymptically Flat
( time )

( radial )
( angular )
5D Black Holes : Variety of Asymptotic Structures
Asymptotically Flat :
: 5D Minkowski
: Lens Space
Asymptotically Locally Flat :
: 4D Minkowski
+ a compact dim.
Kaluza-Klein Black Holes
Kaluza-Klein Black Holes
4次元 Minkowski
Compact S1
[ 4次元 Minkowski と Compact S1 の直積 ]
4次元 Minkowski
Squashed Kaluza-Klein Black Holes
Twisted S1
[ 4次元 Minkowski 上に Twisted S1 Fiber ]
4次元 Minkowski
異なる漸近構造を持つ5次元帯電ブラックホール解
5D 漸近平坦 BH
5D Kaluza-Klein BH
( Tangherlini )
( Ishihara - Matsuno )
r-
r+
r-
r+
4D Minkowski
5D Minkowski
+ a compact dim.
Two types of Kaluza-Klein BHs
同じ漸近構造
rr+
r+
Point Singularity
r-
Stretched Singularity
Study of Five-dimensional Black Holes

Five-dim. BHs : Variety of
Horizon Topologies
Asymptotic Structures
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 ? )
2種類の漸近構造
 ここでは
: 5D Minkowski
: Lens Space
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
System
5D Einstein-Maxwell system with
Chern-Simons term and positive cosmological constant
Rotating Solution on Eguchi-Hanson space
Specified by ( m1 , m2 , j )
Three-sphere S3
( S2 base )
( twisted S1 fiber )
S1
S2
S3
Three-sphere S3
( S2 base )
S2×S1
( twisted S1 fiber )
S3
Lens space S3 / Zn
( S2 base )
S1 / Zn
S1
S2
( S1 / Zn fiber )
S3
S2
( ex. Changing of Horizon Areas )
S3 / Zn
Eguchi-Hanson space
4D Ricci Flat ( Rij = 0 )
z
S2 - bolt
 2 NUTs on S2 - bolt at ri = ( 0 , 0 , zi ) : 両極
( Fixed point of ∂/∂ζ )
 Asymptotic Structure ( r ~ ∞) : R1×S3 / Z2
Rotating Solution on Eguchi-Hanson space
For Suitable ( m1 , m2 , j )
“ Mapping Rules ” of parameters ( mi , j )
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ] ( on EH space )
Early Time
m1 , j
+
Late Time
m2 , j
S3
S3
2(m1 + m2)
8j
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )
m1 , j
S3
+
m2 , j
S3
m1 + m2
2j
S3
“ Mapping Rules ” of parameters ( m , j )
m = m1 = m2
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ] ( on EH space )
Early Time
m , j
+
Late Time
4m
m , j
8j
S3
S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )
m , j
+
m , j
2m
2j
S3
S3
S3
“ Mapping Rule ” of ( m , j ) for coalescence on BHs on EH space
( we set m = m1 = m2 )
j 2 / m3
( mλ2 , j 2 / m 3 ) ⇒ ( 4 mλ2 , j 2 / m 3 )
mλ2
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 )
m = m1 = m2
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ] ( on EH space )
Early Time
m , j
+
Late Time
4m
m , j
8j
S3
S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )
m , j
+
m , j
2m
2j
S3
S3
S3
“ Mapping Rule ” of ( m , j ) for coalescence on BHs on Flat space
( we set m = m1 = m2 )
j2 / m3
( mλ2 , j 2 / m 3 ) ⇒ ( 2 mλ2 , ( j 2 / m 3 ) / 2 )
mλ2
ODEC : Two S3 BHs
at Early time
OGKL : Single S3 BH at Late time
OGHC : Coalescence of 2 BHs ( S3 → S3 )

Comparison of Horizon Areas
Early Time
m , j
+
S3

m , j
S3
Late Time
2m
4m
2j
8j
S3
S3 / Z2
( Lens space S3 / Z2 )
Horizon Area の変化
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Early Time
m , j
+
Late Time
m , j
4m
8j
S3
S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ]
m , j
+
m , j
2m
2j
S3
S3
S3
Comparison of Horizon Areas A(l) / A(e) > 1
漸近平坦な時空
j 2 / m3
m33
j2 // m
漸近的に lens space な時空
mλ2
mλ
mλ22
Horizon Area の変化
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Early Time
m , j
+
Late Time
m , j
4m
8j
S3
S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ]
m , j
+
m , j
2m
2j
S3
S3
S3
Horizon Area の変化
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Early Time
m , j
+
Late Time
m , j
4m
8j
S3
S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ]
m , j
+
m , j
2m
2j
S3
S3
S3
Comparison of Horizon Areas AEH(l) / AFlat(l)
j2 / m3
j→0
mλ2
Comparison of Horizon Areas AEH(l) / AFlat(l) | j → 0
Comparison of Horizon Areas AEH(l) / AFlat(l)
λ→ 0
j2 / m3
mλ2
Comparison of Horizon Areas AEH(l) / AFlat(l) | λ→ 0
Conclusion
We construct
5D new Rot. Multi-BH Sol.s on Eguchi-Hanson space
 Coalescence of Rotating BHs
with Change of Horizon Topology : S3 ⇒ S3 / Z2
( Lens Space )
 Comparing with that on Flat space
without change of Horizon Topology : S3 ⇒ S3
 Horizon Areas の振る舞い
回転の影響
漸近構造を区別可能
Future Works
 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 )
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 ]
≫ 1 GeV : 1 Proton
[ D次元 ]
例. LHC 加速器内 : EP,D ≒ TeV
⇒ mc2 ≧ TeV ≒ (proton mass)×103
ミニ・ブラックホール !
Kaluza-Klein model
余剰次元 : 小さくコンパクト化
L
⇒ 量子力学
[ 例. 5次元 ]
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
Background
String Theory
Brane world scenario
Spacetime with large scale extra dim.
Creation of mini-black holes in the LHC
Near horizon region : Higher-dim. spacetime
Far region from BHs : Effectively 4D spacetime
Black Holes with a Compact Dimension

Higher-dim. Multi-BHs with compact extra dimensions
( R.C. Myers (1987) )

5D Kaluza-Klein Black Holes
Near horizon region : ~ 5D black hole
Far region
: ~ 4D black hole × S1
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 )
S1
S1
S2
( Twisted S1 fiber )
S3
S2
( ex. Shape of Horizons )
Sq. S3
Solutions
Squashed S3
Squashed S3

Spatial cross section of r = const. surface Σr
S2
S1
Oblate
Round S3
Prolate
(k >1)
(k=1)
(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
ρ⇒ ∞
4次元 Minkowski
Twisted S1
Asymptotically Locally Flat
( a twisted constant S1 fiber bundle over 4D Minkowski )
Whole Structure
Inner Horizon
Singularity
r=0
r = r-
Outer Horizon
r = r+
0 < r < r∞
Spatial Infinity
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
Black Hole
Naked
Singularity
Stretched
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
( S3 )
 Specified by ( m , j )
 Killing Vector Fields : ∂/∂ψ ∂/∂φ
 BH Horizon x+ in this coord.s is given by sol.s of
: outgoing null expansion
x についての3次方程式 ⇒ ( m , j ) に制限
Region of ( m , j )
No
Horizon
Black Hole
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
x の単調増加関数
In this case ,
gψψ (x+) > 0 and det g2D (x+) > 0
No CTC !
Early Time
( S3 )
[ Specified by ( mi , j ) ]

BH Horizon in this coord.s is given by sol.s of
( outgoing null expansion )
For suitable ( mi , j )
S3
S3
( outer trapped small S3 )
Rot. 2 BHs at Early time
Late Time
( Lens space S3 / Z2 )
[ Specified by ( 2( m1 + m2 ) , 8 j ) ]

BH Horizon in this coord.s is given by sol.s of
( outgoing null expansion )
For suitable ( mi , j )
S3 / Z2
( outer trapped large S3 )
Rot. 1 BH at Late time

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