Charged Slowly Rotating Kaluza

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Charged Rotating
Kaluza-Klein Black Holes
in Dilaton Gravity
Ken Matsuno ( Osaka City University )
collaboration with Masoud Allahverdizadeh
( Universitat Oldenburg )
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Introduction
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• 我々は 4次元時空に住んでいる
空間 3次元
時間 1次元
• 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
高次元時空上の理論
• 余剰次元の効果が顕著
高エネルギー現象
強重力場
高次元ブラックホール ( BH ) に注目
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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)
余剰次元方向
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
4次元
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“ Hybrid ” Brane world model
Bulk
Brane
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 )
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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 ]
≫ 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 !
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5-dim. Black Objects
[ 以降、5次元時空に注目 ]
 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
Black Rings
( S3 )
( S2×S1 )
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Asymptotic Structures of Black Holes
• 4D Black Holes : Asymptically Flat
( time )
( radial )
( angular )
• 5D Black Holes : Variety of Asymptotic Structures
Asymptotically Flat :
: 5D Minkowski
: Lens Space
Asymptotically Locally Flat :
: 4D Minkowski
+ a compact dim.
Kaluza-Klein Black Holes
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Kaluza-Klein Black Holes
4次元 Minkowski
Compact S1
[ 4次元 Minkowski と Compact S1 の直積 ]
4次元 Minkowski
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Squashed Kaluza-Klein Black Holes
Twisted S1
[ 4次元 Minkowski 上に Twisted S1 Fiber ]
4次元 Minkowski
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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 漸近平坦 BH
5D Kaluza-Klein BH
( Tangherlini )
( Ishihara - Matsuno )
r-
r+
r-
r+
4D Minkowski
5D Minkowski
+ a compact dim.
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Two types of Kaluza-Klein BHs
同じ漸近構造
rr+
r+
Point Singularity
r-
Stretched Singularity
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Geodesics of massive particles
5D Sch. BH
Squashed KK BH
Stable circular orbit
⇒ 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
( α = 0 : Einstein-Maxwell system )
• Equations of motion
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Anzats
• metric
• Killing vector fields : ∂/∂t , ∂/∂φ , ∂/∂ψ
• Black Holes with two equal angular momenta
• gauge potential
• dilaton field
( r+ , r∞ : constants )
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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
• Yazadjiev’s solution ( a = 0 )
( α → 0 : charged static Kaluza-Klein black hole solutions )
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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
new KK BH without closed timelike curve & naked singularity
r+ : Horizon
r∞ : Infinity
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Three-sphere S3
( S2 base )
( twisted S1 fiber )
S1
S2
S3
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Three-sphere S3
( S2 base )
S2×S1
( twisted S1 fiber )
S3
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Shape of Horizon r+
• induced metric
Squashed S3 Horizon
• k(r+) > 1 ⇔ (S2 base) > (S1 fiber)
• No contribution of rotation parameter a
( cf. vacuum rotating Kaluza-Klein black holes )
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Asymptotic Structure
coordinate transformation
0<ρ<∞
• metric
• gauge potential
• dilaton field
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Functions in ρ coordinate
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Asymptotic Structure
coordinate transformation
0<ρ<∞
• metric
• gauge potential
• dilaton field
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Asymptotic Structure
Taking ρ → ∞ ( r → r∞ )
with coordinate transformation :
Asymptotically Locally Flat
( twisted S1 fiber bundle over 4D Minkowski spacetime )
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Three Limits
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No dilaton Limit: α → 0
coordinate 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
coordinates transformation
Charged static dilaton black strings
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Physical Quantities
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Mass and Angular Momenta
consistent with asymptotically flat case
Gyromagnetic ratio g
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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
μ
B
Dirac eq. と比較 ⇒
• g因子：磁気回転比 μ/S とボーア磁子μB の比
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)
: nD asymptotically flat dilaton BH
5D asymptotically Kaluza-Klein dilaton BH
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Gyromagnetic ratio of Asymptotically Flat dilaton BHs
( Sheykhi-Allahverdizadeh )
r+ = 2 & r- = 1
6D
5D
4D
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Gyromagnetic ratio of 5D Kaluza-Klein dilaton BHs
r+ = 2 & r- = 1
r∞ = 2.2
r∞ = 2.7
r∞ = rC
r∞ = 4.8
r∞ = ∞
( Asymptotically Flat )
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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
2 equal angular momenta
axisymmetric 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-Simons
Dilaton field
naturally introduced by
low energy limit of string theory ...
• 5D charged (slowly) rotating Kaluza-Klein dilaton black boles
with Cosmological Constant
⇒ numerical solutions ... ?
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Future Works (2)
More Higher-dimensions
S3
: S1 bundle over CP1
・・・
S2n+1 : S1 bundle over CPn
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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Future Works (3)
Test Maxwell Fields
Ex) Wald Solutions ( vacuum background )
Kerr BH in Uniform Magnetic Field
“ Misner effect ” for extreme BH
最内部安定円軌道 ( ISCO )
BH
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Future Works (3)
 Black Strings in …
 Black Rings in …
 ( Charged ) squashed KK BH in …

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