coalescence of five-dimensional black holes ( 5 次元ブラックホールの合体 )
DESCRIPTION
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 時間 1 次元. - PowerPoint PPT PresentationTRANSCRIPT
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 )