Charged Rotating Kaluza-Klein Black Holesin Dilaton Gravity

Ken Matsuno ( Osaka City University )

collaboration with Masoud Allahverdizadeh

( Universitat Oldenburg )

1

Introduction

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• 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

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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 ” ( 針金 )

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

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“ Hybrid ” 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

BulkBrane

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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≒

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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

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5-dim. Black Objects

4 次元 : 漸近平坦 , 真空 , 定常 , 地平線の上と外に特異点なし

⇒ Kerr BH with S2 horizon only

5 次元　 : For above conditions

⇒ Variety of Horizon Topologies

Black Holes

( S3 )

Black Rings

( S2×S1 )

[ 以降、 5 次元時空に注目 ]

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• 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

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Kaluza-Klein Black Holes

4 次元 Minkowski

Compact S14 次元 Minkowski

[ 4 次元 Minkowski と Compact S1 の直積 ]

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Squashed Kaluza-Klein Black Holes

4 次元 Minkowski

Twisted S1

[ 4 次元 Minkowski 上に Twisted S1 Fiber ]

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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

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Two types of Kaluza-Klein BHs

Point Singularity Stretched Singularity

r+

r-

r+r-

同じ漸近構造

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Geodesics of massive particles

Stable circular orbit

5D Sch. BH Squashed KK BH

⇒ 重力源周りの物理現象 ( 近日点移動等 ) に現れる高次元補正

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Varieties of Black Holes

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Varieties of Black Holes•4D Einstein-Maxwell Black Holes with S2 horizons

Static Rotating

Uncharged Schwarzschild( M )

Kerr( M , J )

Charged Reissner-Nordstrom( M , Q )

Kerr-Newman( M , J , Q )

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• 5D Einstein-Maxwell Asymptically Flat ( Unsquashed ) Black Holes with S3 horizons ( No Chern-Simons term )

Static Rotating

Uncharged Tangherlini( M )

Myers-Perry( M , J1 , J2 )

Charged Tangherlini( M , Q )

Aliev ( Slowly )( M , J1 , J2 , Q )

Kunz et al. (Numerical)( M , J1 = J2 , Q )

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• 5D Einstein-Maxwell Asymptically Locally Flat ( Squashed ) Black Holes with S3 horizons ( No Chern-Simons term )

Static Rotating

Uncharged Dobiash-Maison( M , r ∞ )

Rasheed( M , J1 , J2 , r ∞ )

Charged Ishihara-Matsuno( M , Q , r ∞ )

?( M , J1 , J2 , Q , r ∞ )

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• 5D Einstein-Maxwell-Dilaton Black Holes with S3 horizons ( general dilaton coupling constant α )

Static Rotating

Unsquashed

Horowitz-Strominger( M , Q , Φ )

Sheykhi-Allahverdizadeh

( Slowly )( M , Q , Φ , J1 , J2 )

Squashed Yazadjiev( M , Q , Φ , r ∞ )

?(M , Q , Φ , J1 , J2 ,

r ∞ )

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• 5D Einstein-Maxwell-Dilaton Black Holes with S3 horizons ( general dilaton coupling constant α )

Static Rotating

Unsquashed

Horowitz-Strominger( M , Q , Φ )

Sheykhi-Allahverdizadeh

( Slowly )( M , Q , Φ , J1 , J2 )

Squashed Yazadjiev( M , Q , Φ , r ∞ )

Allahverdizadeh-Matsuno( Slowly )

(M , Q , Φ , J1 = J2 , r ∞ )

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Charged Rotating Kaluza-Klein Dilaton Black Holes

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5D Einstein-Maxwell-Dilaton System• Action

• Equations of motion

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( α = 0 : Einstein-Maxwell system )

Anzats• metric

• gauge potential ( r+ , r ∞ : constants )

• dilaton field

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• Killing vector fields : ∂/∂t , ∂/∂φ , ∂/∂ψ

• Black Holes with two equal angular momenta

How to obtain slowly rotating solutions

(1) Static part ( a = 0 ) is given by Yazadjiev’s solution

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Functions for static part

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( α → 0 : charged static Kaluza-Klein black hole solutions )

• Yazadjiev’s solution ( a = 0 )

How to obtain slowly rotating solutions

(1) Static part ( a = 0 ) is given by Yazadjiev’s solution

(2) Substituting the anzats into equations of motion

(3) Discarding any terms involving O(a2) ⇔ Slow Rotation

(4) Solving ordinary differential equation of f(r)

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Slowly Rotating Solution

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r+ : Horizon

r ∞ : Infinity

new KK BH without closed timelike curve & naked singularity

Three-sphere S3

( S2 base ) ( twisted S1 fiber )

S2

S1

S3

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Three-sphere S3

S2×S1 S3

( S2 base ) ( twisted S1 fiber )

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Shape of Horizon r+

• induced metric

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Squashed S3 Horizon

• k(r+) > 1 ⇔ (S2 base) > (S1 fiber)

• No contribution of rotation parameter a

( cf. vacuum rotating Kaluza-Klein black holes )

Asymptotic Structure

• metric

• gauge potential

• dilaton field

coordinate transformation

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0 < ρ < ∞

Functions in ρ coordinate

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Asymptotic Structure

• metric

• gauge potential

• dilaton field

coordinate transformation

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0 < ρ < ∞

Asymptotic Structure

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Taking ρ → ∞ ( r → r ∞ )

with coordinate transformation :

Asymptotically Locally Flat( twisted S1 fiber bundle over 4D Minkowski

spacetime )

Three Limits

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No dilaton Limit: α → 0coordinate

transformation

Slowly rotating charged squashed Kaluza-Klein black holes

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Asymptically Flat Limit: r∞ →

∞

Asymptotically Flat slowly rotating charged dilaton black holes

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Black String Limit: ρ0 → 0

Charged static dilaton black strings

coordinates transformation

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Physical Quantities

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Mass and Angular Momenta

Gyromagnetic ratio g

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consistent with asymptotically flat case

Gyromagnetic ratio (g 因子 ) g

• 電子の磁気モーメント μ と外部磁場 B の相互作用

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Dirac eq. と比較 ⇒

• g 因子：磁気回転比 μ/S とボーア磁子 μB の比

μ B

Analogy : 電子 ⇔ charged rotating black holes (Carter, 1968)

( μ = Q a : “magnetic dipole moment” )

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Gyromagnetic ratios of (slowly) rotating black holes

g = 2 : 4D Kerr-Newman BH (Carter, 1968)

g = n-2 : nD asymptically flat BH (Aliev, 2006)

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: nD asymptotically flat dilaton BH

5D asymptotically Kaluza-Klein dilaton BH

Gyromagnetic ratio of Asymptotically Flat dilaton BHs

4D

5D

6D

r+ = 2 & r- = 1

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( Sheykhi-Allahverdizadeh )

Gyromagnetic ratio of 5D Kaluza-Klein dilaton BHs

r ∞ = rC

r ∞ = ∞( Asymptotically

Flat )

r ∞ = 4.8

r ∞ = 2.7

r ∞ = 2.2

r+ = 2 & r- = 1

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Conclusion

• We obtain a class of slowly rotating charged Kaluza-Klein black hole solutions of 5D Einstein-Maxwell-dilaton theory with arbitrary dilaton coupling constant α

( restricted to black holes with two equal angular momenta )

• At infinity, metric asymptotically approaches a twisted S1 bundle over 4D Minkowski spacetime

• Behaviors of gyromagnetic ratio g crucially depend on the size of extra dimension

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Future works (1)今回：

5D charged slowly rotating Kaluza-Klein dilaton black holes

with

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2 equal angular momentaaxisymmetric horizon

5D charged slowly rotating Kaluza-Klein dilaton black holes

with

2 independent angular momenta

3 軸不等な horizon (Bianchi IX)

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Future works (1)

• 5D charged (slowly) rotating Kaluza-Klein black holes in Einstein-Maxwell-Chern-Simons-Dilaton theory

Chern-SimonsDilaton field

• 5D charged (slowly) rotating Kaluza-Klein dilaton black boles with Cosmological Constant

⇒ numerical solutions ... ?

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naturally introduced by low energy limit of string theory ...

Future Works (2)

S3 : S1 bundle over CP1

・・・S2n+1 : S1 bundle over CPn

More Higher-dimensions

Ex) S7 : S1 bundle over CP3

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Future Works (2)

Black Objects …

Kasner spacetime ( Bianchi types ) …

Dynamical ( Rotating ) BHs without Λ

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Test Maxwell Fields

Kerr BH in Uniform Magnetic Field

“ Misner effect ” for extreme BH

最内部安定円軌道 ( ISCO )

Ex) Wald Solutions ( vacuum background )

BH

Future Works (3)

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Black Strings in …

Black Rings in …

( Charged ) squashed KK BH in …

Future Works (3)

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