Charged Rotating Kaluza-Klein Black Holes in Dilaton Gravity Ken Matsuno ( Osaka City University ) collaboration with Masoud Allahverdizadeh ( Universitat Oldenburg ) 1

Introduction 2

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 3

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 4 次元 4

“ 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 Bulk Brane 5

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 E P,D ≒ TeV, D = 6 6

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 : E P,D ≒ TeV ⇒ mc 2 ≧ TeV ≒ (proton mass)×10 3 ミニ・ブラックホール ! ≫ 1 GeV : 1 Proton 7

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 S 2 horizon only 5 次元 : For above conditions ⇒ Variety of Horizon Topologies Black Holes ( S 3 ) Black Rings ( S 2 ×S 1 ) [ 以降、 5 次元時空に注目 ] 8

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 9

4 次元 Minkowski Compact S 1 4 次元 Minkowski [ 4 次元 Minkowski と Compact S 1 の直積 ] 10

Squashed Kaluza-Klein Black Holes 4 次元 Minkowski Twisted S 1 [ 4 次元 Minkowski 上に Twisted S 1 Fiber ] 11

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+ r-r- 4D Minkowski + a compact dim. 5D 漸近平坦 BH ( Tangherlini ) r+r+ r-r- 5D Minkowski 12

Two types of Kaluza-Klein BHs Point SingularityStretched Singularity r+r+ r-r- r+r+ r-r- 同じ漸近構造 13

Geodesics of massive particles Stable circular orbit 5D Sch. BHSquashed 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 14

Varieties of Black Holes 15

Varieties of Black Holes 4D Einstein-Maxwell Black Holes with S 2 horizons StaticRotating UnchargedSchwarzschild ( M ) Kerr ( M, J ) ChargedReissner-Nordstrom ( M, Q ) Kerr-Newman ( M, J, Q ) 16

5D Einstein-Maxwell Asymptically Flat ( Unsquashed ) Black Holes with S 3 horizons ( No Chern-Simons term ) StaticRotating UnchargedTangherlini ( M ) Myers-Perry ( M, J1, J2 ) ChargedTangherlini ( M, Q ) Aliev ( Slowly ) ( M, J1, J2, Q ) Kunz et al. (Numerical) ( M, J1 = J2, Q ) 17

5D Einstein-Maxwell Asymptically Locally Flat ( Squashed ) Black Holes with S 3 horizons ( No Chern-Simons term ) StaticRotating UnchargedDobiash-Maison ( M, r ∞ ) Rasheed ( M, J1, J2, r ∞ ) ChargedIshihara-Matsuno ( M, Q, r ∞ ) ? ( M, J1, J2, Q, r ∞ ) 18

5D Einstein-Maxwell-Dilaton Black Holes with S 3 horizons ( general dilaton coupling constant α ) StaticRotating UnsquashedHorowitz-Strominger ( M, Q, Φ ) Sheykhi-Allahverdizadeh ( Slowly ) ( M, Q, Φ, J1, J2 ) SquashedYazadjiev ( M, Q, Φ, r ∞ ) ? (M, Q, Φ, J1, J2, r ∞ ) 19

5D Einstein-Maxwell-Dilaton Black Holes with S 3 horizons ( general dilaton coupling constant α ) StaticRotating UnsquashedHorowitz-Strominger ( M, Q, Φ ) Sheykhi-Allahverdizadeh ( Slowly ) ( M, Q, Φ, J1, J2 ) SquashedYazadjiev ( M, Q, Φ, r ∞ ) Allahverdizadeh-Matsuno ( Slowly ) (M, Q, Φ, J1 = J2, r ∞ ) 20

Charged Rotating Kaluza-Klein Dilaton Black Holes 21

5D Einstein-Maxwell-Dilaton System Action Equations of motion 22 ( α = 0 : Einstein-Maxwell system )

Anzats metric gauge potential ( r +, r ∞ : constants ) dilaton field 23 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 24

Functions for static part 25 ( 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(a 2 ) ⇔ Slow Rotation (4) Solving ordinary differential equation of f(r) 26

Slowly Rotating Solution 27 r + : Horizon r ∞ : Infinity new KK BH without closed timelike curve & naked singularity

Three-sphere S 3 ( S 2 base )( twisted S 1 fiber ) S2S2 S1S1 S3S3 28

Three-sphere S 3 S 2 ×S 1 S3S3 ( S 2 base )( twisted S 1 fiber ) 29

Shape of Horizon r + induced metric 30 Squashed S 3 Horizon k(r + ) > 1 ⇔ (S 2 base) > (S 1 fiber) No contribution of rotation parameter a ( cf. vacuum rotating Kaluza-Klein black holes )

Asymptotic Structure metric gauge potential dilaton field coordinate transformation 31 0 < ρ < ∞

Functions in ρ coordinate 32

Asymptotic Structure metric gauge potential dilaton field coordinate transformation 33 0 < ρ < ∞

Asymptotic Structure 34 Taking ρ → ∞ ( r → r ∞ ) with coordinate transformation : Asymptotically Locally Flat ( twisted S 1 fiber bundle over 4D Minkowski spacetime )

Three Limits 35

No dilaton Limit: α → 0 coordinate transformation Slowly rotating charged squashed Kaluza-Klein black holes 36

Asymptically Flat Limit: r ∞ → ∞ Asymptotically Flat slowly rotating charged dilaton black holes 37

Black String Limit: ρ 0 → 0 Charged static dilaton black strings coordinates transformation 38

Physical Quantities 39

Mass and Angular Momenta Gyromagnetic ratio g 40 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 41 Dirac eq. と比較 ⇒ g 因子：磁気回転比 μ/S とボーア磁子 μ B の比 μ B Analogy : 電子 ⇔ charged rotating black holes (Carter, 1968) ( μ = Q a : “magnetic dipole moment” ) 41

Gyromagnetic ratios of (slowly) rotating black holes g = 2 : 4D Kerr-Newman BH (Carter, 1968) g = n-2 : nD asymptically flat BH (Aliev, 2006) 42 : 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 43 ( Sheykhi-Allahverdizadeh )

Gyromagnetic ratio of 5D Kaluza-Klein dilaton BHs r ∞ = r C r ∞ = ∞ ( Asymptotically Flat ) r ∞ = 4.8 r ∞ = 2.7 r ∞ = 2.2 r + = 2 & r - = 1 44

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 S 1 bundle over 4D Minkowski spacetime Behaviors of gyromagnetic ratio g crucially depend on the size of extra dimension 45

Future works (1) 今回： 5D charged slowly rotating Kaluza-Klein dilaton black holes with 46 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) 46

Future works (1) 5D charged (slowly) rotating Kaluza-Klein black holes in Einstein-Maxwell-Chern-Simons-Dilaton theory Chern-Simons Dilaton field 5D charged (slowly) rotating Kaluza-Klein dilaton black boles with Cosmological Constant ⇒ numerical solutions... ? 47 naturally introduced by low energy limit of string theory...

Future Works (2) S 3 : S 1 bundle over CP 1 ・・・ S 2n+1 : S 1 bundle over CP n More Higher-dimensions Ex) S 7 : S 1 bundle over CP 3 48

Future Works (2) Black Objects … Kasner spacetime ( Bianchi types ) … Dynamical ( Rotating ) BHs without Λ 49

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) 50

Black Strings in … Black Rings in … ( Charged ) squashed KK BH in … Future Works (3) 51