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 )

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 )

1. Introduction

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

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

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 )

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 )

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

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

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

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 ? )

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 )

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 S3 S2

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

( 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 )

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 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

“ 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

“ 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 )

“ 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

“ 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 )

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 )

[ 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

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

[ 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

[ 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

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

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 )

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

⇒ 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)×103 ミニ・ブラックホール !

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 ≒ 10 -17 cm

2. 歪んだ Kaluza-Klein Black Holes

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

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

Solutions Squashed S3

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 )

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 )

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

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

Whole Structure 0 < r < r∞ Inner Horizon r = r- Singularity 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 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

: 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

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 In this case , gψψ (x+) > 0 and det g2D (x+) > 0 No CTC ! x の単調増加関数

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

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