1. 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 )

2. コンパクトな余剰次元を持つブラックホール )
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 )

3. 1. Introduction

4. 空間 3次元
時間 1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
余剰次元 の効果が顕著
高次元ブラックホール ( BH ) に注目
高次元時空 上の理論
高エネルギー現象
強重力場

5. 次元低下 高次元時空 ⇒ 有効的に 4次元時空 Kaluza-Klein model “ とても小さく丸められていて見えない ”
高次元時空 ⇒ 有効的に 4次元時空
Kaluza-Klein model “ とても小さく丸められていて見えない ”
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

6. 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
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 )

7. 5-dim. Black Objects [ 以降、5次元時空に注目 ] 4次元 : 軸対称 , 真空
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 Rings
( S2×S1 )
Black Holes
( S3 )

8. Asymptotic Structures of Black Holes
4D Black Holes : Asymptically Flat
5D Black Holes : Variety of Asymptotic Structures
( time )
( radial )
( angular )
Asymptotically Flat :
Asymptotically Locally Flat :
: 5D Minkowski
: Lens Space
: 4D Minkowski + a compact dim.
Kaluza-Klein Black Holes

9. Kaluza-Klein Black Holes
4次元 Minkowski
Compact S1
[ 4次元 Minkowski と Compact S1 の直積 ]
4次元 Minkowski

10. Squashed Kaluza-Klein Black Holes
Twisted S1
[ 4次元 Minkowski 上に Twisted S1 Fiber ]
4次元 Minkowski

11. 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
( Tangherlini )
5D Kaluza-Klein BH
( Ishihara - Matsuno ) r+ r- r- r+
4D Minkowski
+ a compact dim.
5D Minkowski

12. Two types of Kaluza-Klein BHs
同じ漸近構造 r- r+ r+ r-
Point Singularity
Stretched Singularity

13. Study of Five-dimensional Black Holes
Horizon Topologies
Asymptotic Structures
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 ? )

14. 2種類の漸近構造 ここでは 平坦空間上 Eguchi - Hanson 空間上 の 回転BH の 合体 : 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 )

15. 2. ブラックホールの合体

16. 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

17. Multi-Black Holes
Time

18. 宇宙項
Time

19. 時間反転
Time

20. BHの合体
Time

21. BHの合体
Time

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

23. Rotating Solution on Eguchi-Hanson space
Specified by ( m1 , m2 , j )

24. Three-sphere S3
( S2 base )
( twisted S1 fiber ) S1 S3 S2

25. Three-sphere S3
( S2 base )
( twisted S1 fiber )
S2×S1 S3

26. ( ex. Changing of Horizon Areas )
Lens space S3 / Zn
( S2 base )
( S1 / Zn fiber )
S1 / Zn S1 S2 S3 S2
S3 / Zn
( ex. Changing of Horizon Areas )

27. 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

28. Rotating Solution on Eguchi-Hanson space
For Suitable ( m1 , m2 , j )

29. “ Mapping Rules ” of parameters ( mi , j )
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ] ( on EH space )
Early Time
Late Time
2(m1 + m2)
8 j
m1 , j
m2 , j + S3 S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )
m1 + m2
2 j
m1 , j
m2 , j + S3 S3 S3

30. “ 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
Late Time
4 m
8 j
m , j
m , j + S3 S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )
2 m
2 j
m , j
m , j + S3 S3 S3

31. “ Mapping Rule ” of ( m , j ) for coalescence on BHs on EH space
( mλ2 , j 2 / m 3 ) ⇒ ( 4 mλ2 , j 2 / m 3 )
( we set m = m1 = m2 )
j2 / m3
mλ2
ODEC : Two S3 BHs at Early time
OAFC : Single S3 / Z2 BH at Late time
OABC : Coalescence of 2 BHs ( S3 → S3 / Z2 )

32. “ 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
Late Time
4 m
8 j
m , j
m , j + S3 S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ] ( on Flat space )
2 m
2 j
m , j
m , j + S3 S3 S3

33. “ Mapping Rule ” of ( m , j ) for coalescence on BHs on Flat space
( mλ2 , j 2 / m 3 ) ⇒ ( 2 mλ2 , ( j 2 / m 3 ) / 2 )
( we set m = m1 = m2 )
j2 / m3
mλ2
ODEC : Two S3 BHs at Early time
OGKL : Single S3 BH at Late time
OGHC : Coalescence of 2 BHs ( S3 → S3 )

34. Comparison of Horizon Areas
Early Time
m , j
m , j + S3 S3
Late Time
4 m
8 j
2 m
2 j S3
S3 / Z2
( Lens space S3 / Z2 )

35. [ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Horizon Area の変化
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Early Time
Late Time
4 m
8 j
m , j
m , j + S3 S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ]
2 m
2 j
m , j
m , j + S3 S3 S3

36. 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 / m3
j2 / m3
j2 / m3
j2 / m3
j2 / m3
mλ2
mλ2
mλ2
mλ2
mλ2

37. [ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Horizon Area の変化
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Early Time
Late Time
4 m
8 j
m , j
m , j + S3 S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ]
2 m
2 j
m , j
m , j + S3 S3 S3

38. [ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Horizon Area の変化
[ 漸近的に Lens Space ( R1×R1×S3 / Z2 ) な時空 ]
Early Time
Late Time
4 m
8 j
m , j
m , j + S3 S3
S3 / Z2
[ 漸近平坦 ( R1×R1×S3 ) な時空 ]
2 m
2 j
m , j
m , j + S3 S3 S3

39. Comparison of Horizon Areas AEH(l) / AFlat(l)
j2 / m3
j → 0
mλ2

40. Comparison of Horizon Areas AEH(l) / AFlat(l) | j → 0

41. Comparison of Horizon Areas AEH(l) / AFlat(l)
λ→ 0
j2 / m3
mλ2

42. Comparison of Horizon Areas AEH(l) / AFlat(l) | λ→ 0

43. 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

44. Measurement of Extra Dimension by Kaluza-Klein Black Holes
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 )

45. 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

46. ⇒ mc2 ≧ TeV ≒ (proton mass)×103 ミニ・ブラックホール !
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)× ミニ・ブラックホール !

47. Kaluza-Klein model L 加速器実験から制限 ⇔ L ≒ 10 -17 cm 余剰次元 : 小さくコンパクト化 ⇒ 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 ≒ cm

48. 2. 歪んだ Kaluza-Klein Black Holes

49. Far region from BHs : Effectively 4D spacetime
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

50. 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

51. 5D Einstein-Maxwell-Chern-Simons system
( Bosonic part of the ungauged SUSY 5-dim. N=1 SUGRA )

52. Solutions
角度成分

53. Squashed S3 ( S2 base ) ( Twisted S1 fiber ) S1 S1 S2 S3 S2 Sq. S3
( ex. Shape of Horizons )

54. Solutions
Squashed S3

55. Spatial cross section of r = const. surface Σr
Squashed S3
Spatial cross section of r = const. surface Σr S2 S1
Oblate
( k > 1 )
Round S3
( k = 1 )
Prolate
( k < 1 )

56. inner horizon r- : Prolate
Near Horizon Region
Shapes of squashed S3 horizons r = r±
outer horizon r+ : Oblate
inner horizon r- : Prolate
( degenerate horizon r+ = r- : round S3 )

57. Coord. Trans. : r ⇒ ρ ( r = r∞ ⇒ ρ= ∞ )
Far Region
Coord. Trans. : r ⇒ ρ ( r = r∞ ⇒ ρ= ∞ )

58. Asymptotically Locally Flat
Far Region
ρ⇒ ∞
4次元 Minkowski
Twisted S1
Asymptotically Locally Flat
( a twisted constant S1 fiber bundle over 4D Minkowski )

59. Whole Structure 0 < r < r∞ Inner Horizon r = r- Singularity
Outer Horizon
r = r+
0 < r < r∞
Spatial Infinity
r = r∞

60. Two Regions of r coordinate
Here, we consider the region
Furthermore, we can consider the region for BH

61. Two types of Singularities
Point Singularity : shrink to a point as
Stretched Singularity : S2 → 0 and S1 → ∞ as

62. Two types of Black Holes
Point
Stretched
Black Hole
Naked Singularity

63. 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

64. Asymptotic Behaviors
r ≒ ri　近傍
r ≒ ∞ ( 遠方 )
Klemm – Sabra 解

65. : outgoing null expansion
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

66. Region of ( m , j )
No Horizon
Black Hole

67. 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

68. Early Time S3 S3 Rot. 2 BHs at Early time [ 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

69. Late Time S3 / Z2 Rot. 1 BH at 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