charged rotating kaluza-klein black holes in dilaton gravity ken matsuno ( osaka city university )...
TRANSCRIPT
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Charged Rotating Kaluza-Klein Black Holesin 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
• 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 )
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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
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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
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( α → 0 : charged static Kaluza-Klein black hole solutions )
• Yazadjiev’s solution ( a = 0 )
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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
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r+ : Horizon
r ∞ : Infinity
new KK BH without closed timelike curve & naked singularity
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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 )
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Asymptotic Structure
• metric
• gauge potential
• dilaton field
coordinate transformation
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0 < ρ < ∞
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Functions in ρ coordinate
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Asymptotic Structure
• metric
• gauge potential
• dilaton field
coordinate transformation
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0 < ρ < ∞
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Asymptotic Structure
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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 → 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
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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
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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
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Gyromagnetic ratio of Asymptotically Flat dilaton BHs
4D
5D
6D
r+ = 2 & r- = 1
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( Sheykhi-Allahverdizadeh )
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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 ...
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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
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Black Strings in …
Black Rings in …
( Charged ) squashed KK BH in …
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