Coalescence of Fivedimensional Black Holes 5 Ken Matsuno

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

5 -dim. Black Objects [ 以降、5次元時空に注目 ] l 4次元 : 軸対称 , 真空 ⇒ Kerr BH with S 2 horizon only l 5次元 : 軸対称 , 真空 ⇒ Variety of Horizon Topologies Black Holes Black Rings ( S 3 ) ( S 2×S 1 )

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

Kaluza-Klein Black Holes 4次元 Minkowski Compact S 1 [ 4次元 Minkowski と Compact S 1 の直積 ] 4次元 Minkowski

Squashed Kaluza-Klein Black Holes Twisted S 1 [ 4次元 Minkowski 上に Twisted S 1 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 5 D 漸近平坦 BH 5 D Kaluza-Klein BH ( Tangherlini ) ( Ishihara - Matsuno ) r- r+ rr+ 4 D Minkowski 5 D 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 l Five-dim. BHs : Variety of Horizon Topologies Asymptotic Structures S 3 , S 3 / Zn ( Lens Space ), S 2×S 1 , … ex) Creation of Charged Rotating Multi-BHs in LHC ( Coalescence of these BHs ? ) Change of Horizon Topologies ? ( S 3 + S 3 ⇒ ? ) Distinguishable of Asymptotic Structures ? ( From Behavior of Horizon Areas ? )

Multi-Black Holes Time

BHの合体 Time

BHの合体 Time

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

Rotating Solution on Eguchi-Hanson space Specified by ( m 1 , m 2 , j )

Three-sphere S 3 ( S 2 base ) ( twisted S 1 fiber ) S 1 S 2 S 3

Three-sphere S 3 ( S 2 base ) S 2×S 1 ( twisted S 1 fiber ) S 3

Lens space S 3 / Zn ( S 2 base ) S 1 / Z n S 1 S 2 ( S 1 / Zn fiber ) S 3 S 2 ( ex. Changing of Horizon Areas ) S 3 / Z n

Eguchi-Hanson space 4 D Ricci Flat ( Rij = 0 ) z S 2 - bolt l 2 NUTs on S 2 - bolt at ri = ( 0 , zi ) : 両極 ( Fixed point of ∂/∂ζ ) l Asymptotic Structure ( r ~ ∞) : R 1×S 3 / Z 2

Rotating Solution on Eguchi-Hanson space For Suitable ( m 1 , m 2 , j )

“ Mapping Rules ” of parameters ( mi , j ) [ 漸近的に Lens Space ( R 1×S 3 / Z 2 ) な時空 ] ( on EH space ) Early Time m 1 , j + Late Time m 2 , j 2(m 1 + m 2) S 3 8 j S 3 / Z 2 [ 漸近平坦 ( R 1×S 3 ) な時空 ] ( on Flat space ) m 1 , j S 3 + m 2 , j S 3 m 1 + m 2 2 j S 3

“ Mapping Rules ” of parameters ( m , j ) m = m 1 = m 2 [ 漸近的に Lens Space ( R 1×S 3 / Z 2 ) な時空 ] ( on EH space ) Early Time m , j + Late Time m , j S 3 4 m 8 j S 3 / Z 2 [ 漸近平坦 ( R 1×S 3 ) な時空 ] ( on Flat space ) m , j S 3 + m , j S 3 2 m 2 j S 3

“ Mapping Rule ” of ( m , j ) for coalescence on BHs on EH space ( we set m = m 1 = m 2 ) j 2 / m 3 ( mλ 2 , j 2 / m 3 ) ⇒ ( 4 mλ 2 , j 2 / m 3 ) mλ 2 ODEC : Two S 3 BHs at Early time OAFC : Single S 3 / Z 2 BH at Late time OABC : Coalescence of 2 BHs ( S 3 → S 3 / Z 2 )

“ Mapping Rules ” of parameters ( m , j ) m = m 1 = m 2 [ 漸近的に Lens Space ( R 1×S 3 / Z 2 ) な時空 ] ( on EH space ) Early Time m , j + Late Time m , j S 3 4 m 8 j S 3 / Z 2 [ 漸近平坦 ( R 1×S 3 ) な時空 ] ( on Flat space ) m , j S 3 + m , j S 3 2 m 2 j S 3

“ Mapping Rule ” of ( m , j ) for coalescence on BHs on Flat space ( we set m = m 1 = m 2 ) j 2 / m 3 ( mλ 2 , j 2 / m 3 ) ⇒ ( 2 mλ 2 , ( j 2 / m 3 ) / 2 ) mλ 2 ODEC : Two S 3 BHs at Early time OGKL : Single S 3 BH at Late time OGHC : Coalescence of 2 BHs ( S 3 → S 3 )

l Comparison of Horizon Areas Early Time m , j l + S 3 Late Time m , j S 3 2 m 4 m 2 j 8 j S 3 / Z 2 ( Lens space S 3 / Z 2 )

Horizon Area の変化 [ 漸近的に Lens Space ( R 1×S 3 / Z 2 ) な時空 ] Early Time m , j + m , j S 3 Late Time 4 m 8 j S 3 / Z 2 [ 漸近平坦 ( R 1×S 3 ) な時空 ] m , j S 3 + m , j S 3 2 m 2 j S 3

Comparison of Horizon Areas A(l) / A(e) > 1 漸近的に lens space な時空 j 2 / m 3 j 2 // m m 33 漸近平坦な時空 mλ 2 mλ mλ 22

Horizon Area の変化 [ 漸近的に Lens Space ( R 1×S 3 / Z 2 ) な時空 ] Early Time m , j + m , j S 3 Late Time 4 m 8 j S 3 / Z 2 [ 漸近平坦 ( R 1×S 3 ) な時空 ] m , j S 3 + m , j S 3 2 m 2 j S 3

Horizon Area の変化 [ 漸近的に Lens Space ( R 1×S 3 / Z 2 ) な時空 ] Early Time m , j + m , j S 3 Late Time 4 m 8 j S 3 / Z 2 [ 漸近平坦 ( R 1×S 3 ) な時空 ] m , j S 3 + m , j S 3 2 m 2 j S 3

Comparison of Horizon Areas AEH(l) / AFlat(l) j 2 / m 3 j→ 0 mλ 2

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

Comparison of Horizon Areas AEH(l) / AFlat(l) λ→ 0 j 2 / m 3 mλ 2

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

Conclusion We construct 5 D new Rot. Multi-BH Sol. s on Eguchi-Hanson space l Coalescence of Rotating BHs with Change of Horizon Topology : S 3 ⇒ S 3 / Z 2 ( Lens Space ) l Comparing with that on Flat space without change of Horizon Topology : S 3 ⇒ S 3 l Horizon Areas の振る舞い 回転の影響 漸近構造を区別可能

Future Works l 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 ) l 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 )

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 4 D spacetime

Black Holes with a Compact Dimension l Higher-dim. Multi-BHs with compact extra dimensions ( R. C. Myers (1987) ) l 5 D Kaluza-Klein Black Holes Near horizon region : ~ 5 D black hole Far region : ~ 4 D black hole × S 1

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

Solutions 角度成分

Squashed S 3 ( S 2 base ) S 1 S 2 ( Twisted S 1 fiber ) S 3 S 2 ( ex. Shape of Horizons ) Sq. S 3

Solutions Squashed S 3

Squashed S 3 l Spatial cross section of r = const. surface Σr S 2 S 1 Oblate Round S 3 Prolate (k >1) (k=1) (k<1)

Near Horizon Region l Shapes of squashed S 3 horizons r = r± outer horizon r+ : Oblate inner horizon r- : Prolate ( degenerate horizon r+ = r- : round S 3 )

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

Far Region ρ⇒ ∞ 4次元 Minkowski Twisted S 1 Asymptotically Locally Flat ( a twisted constant S 1 fiber bundle over 4 D Minkowski )

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

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

Two types of Singularities l Point Singularity : shrink to a point as l Stretched Singularity : S 2 → 0 and S 1 → ∞ 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 S 3 horizons in 5 D Einstein-Maxwell theory l These black holes asymptote to the effectively 4 D Minkowski with a compact extra dimension at infinity l 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 ( S 3 ) l Specified by ( m , j ) l Killing Vector Fields : ∂/∂ψ ∂/∂φ l 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 ) l No CTC for x > x+ > 0 ⇔ ( ψ , φ ) part of metric g 2 D has no negative eigenvalue ⇔ gψψ (x) > 0 and det g 2 D (x) > 0 x の単調増加関数 In this case , gψψ (x+) > 0 and det g 2 D (x+) > 0 No CTC !

Early Time ( S 3 ) [ Specified by ( mi , j ) ] l BH Horizon in this coord. s is given by sol. s of ( outgoing null expansion ) For suitable ( mi , j ) S 3 ( outer trapped small S 3 ) Rot. 2 BHs at Early time

Late Time ( Lens space S 3 / Z 2 ) [ Specified by ( 2( m 1 + m 2 ) , 8 j ) ] l BH Horizon in this coord. s is given by sol. s of ( outgoing null expansion ) For suitable ( mi , j ) S 3 / Z 2 ( outer trapped large S 3 ) Rot. 1 BH at Late time