Docstoc

Charged Slowly Rotating Kaluza Klein Black Holes with dilaton field (PowerPoint)

Document Sample
Charged Slowly Rotating Kaluza Klein Black Holes with dilaton field (PowerPoint) Powered By Docstoc
					                                                1




Charged Rotating
Kaluza-Klein Black Holes
in Dilaton Gravity

Ken Matsuno ( Osaka City University )

collaboration with Masoud Allahverdizadeh
                    ( Universitat Oldenburg )
               2




Introduction
                              3


                     空間 3次元
• 我々は 4次元時空 に住んでいる
                     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
             高次元時空 上の理論
   超重力理論


                高エネルギー現象
• 余剰次元 の効果が顕著
                強重力場

      高次元ブラックホール ( BH ) に注目
                                                       4

     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
                                          5

“ 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 )
                                                           6


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
                                                        7

 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 !
                                                  8

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 )
                                                                                9

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
                                                   10


Kaluza-Klein Black Holes




              4次元 Minkowski           Compact S1

       [ 4次元 Minkowski と Compact S1 の直積 ]




            4次元 Minkowski
                                                   11


Squashed Kaluza-Klein Black Holes




                                     Twisted S1
           [ 4次元 Minkowski 上に Twisted S1 Fiber ]




                 4次元 Minkowski
                                                               12


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.
                                                        13

Two types of Kaluza-Klein BHs



                         同じ漸近構造

        r-
              r+



                                      r+
                                               r-


    Point Singularity           Stretched Singularity
                                                         14

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
                           15




Varieties of Black Holes
                                                     16


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


                      Static            Rotating


  Uncharged        Schwarzschild          Kerr
                       (M)               (M,J)


   Charged      Reissner-Nordstrom    Kerr-Newman
                      (M,Q)            (M,J,Q)
                                                            17


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


                   Static                Rotating


Uncharged       Tangherlini             Myers-Perry
                   (M)                  ( M , J1 , J2 )


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

                                  Kunz et al. (Numerical)
                                    ( M , J1 = J2 , Q )
                                                                  18


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


                     Static                 Rotating


Uncharged       Dobiash-Maison               Rasheed
                   ( M , r∞ )            ( M , J1 , J2 , r∞ )



  Charged       Ishihara-Matsuno                  ?
                   ( M , Q , r∞ )      ( M , J1 , J2 , Q , r∞ )
                                                                19


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


                     Static                  Rotating


Unsquashed   Horowitz-Strominger     Sheykhi-Allahverdizadeh
                 (M,Q,Φ)                     ( Slowly )
                                       ( M , Q , Φ , J1 , J2 )



 Squashed          Yazadjiev                      ?
                ( M , Q , Φ , r∞ )    (M , Q , Φ , J1 , J2 , r∞ )
                                                              20


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


                     Static                  Rotating


Unsquashed   Horowitz-Strominger     Sheykhi-Allahverdizadeh
                 (M,Q,Φ)                     ( Slowly )
                                       ( M , Q , Φ , J1 , J2 )


 Squashed          Yazadjiev         Allahverdizadeh-Matsuno
                ( M , Q , Φ , r∞ )            ( Slowly )
                                      (M , Q , Φ , J1 = J2 , r∞ )
                                21




Charged Rotating Kaluza-Klein
     Dilaton Black Holes
                                                  22


5D Einstein-Maxwell-Dilaton System
• Action




            ( α = 0 : Einstein-Maxwell system )

• Equations of motion
                                                      23

Anzats
• metric




       • Killing vector fields : ∂/∂t , ∂/∂φ , ∂/∂ψ
       • Black Holes with two equal angular momenta

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




• dilaton field
                                                              24


How to obtain slowly rotating solutions



 (1) Static part ( a = 0 ) is given by Yazadjiev’s solution
                                                                       25

Functions for static part



• Yazadjiev’s solution ( a = 0 )




        ( α → 0 : charged static Kaluza-Klein black hole solutions )
                                                              26


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)
                                                         27

Slowly Rotating Solution




new KK BH without closed timelike curve & naked singularity
                       r+ : Horizon
                       r∞ : Infinity
                                                       28

Three-sphere S3




                  ( S2 base )   ( twisted S1 fiber )




                         S1




                  S2                    S3
                                                       29

Three-sphere S3




                  ( S2 base )   ( twisted S1 fiber )




              S2×S1                           S3
                                                          30

  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 )
                                    31

   Asymptotic Structure
coordinate transformation




                            0<ρ<∞
• metric




• gauge potential




• dilaton field
                            32

Functions in ρ coordinate
                                    33

   Asymptotic Structure
coordinate transformation




                            0<ρ<∞
• metric




• gauge potential




• dilaton field
                                                           34

Asymptotic Structure




 Taking ρ → ∞ ( r → r∞ )
 with coordinate transformation :




                 Asymptotically Locally Flat
 ( twisted S1 fiber bundle over 4D Minkowski spacetime )
               35




Three Limits
                                                         36

No dilaton Limit: α → 0
 coordinate transformation




 Slowly rotating charged squashed Kaluza-Klein black holes
                                                             37


Asymptically Flat Limit: r∞ →     ∞




Asymptotically Flat slowly rotating charged dilaton black holes
                                                    38

Black String Limit: ρ0 → 0
coordinates transformation




             Charged static dilaton black strings
                      39




Physical Quantities
                                                       40

Mass and Angular Momenta




            consistent with asymptotically flat case



              Gyromagnetic ratio g
                                                                        41

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                                      B
                                                           μ



   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” )
                                                        42

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
                                                            43


Gyromagnetic ratio of Asymptotically Flat dilaton BHs
                              ( Sheykhi-Allahverdizadeh )
                                     r+ = 2 & r- = 1

                                                       6D



                                                       5D



                                                       4D
                                                             44


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 )
                                                               45


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
                                                          46

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)
                                                             47

  Future works (1)
• 5D charged (slowly) rotating Kaluza-Klein black holes
  in Einstein-Maxwell-Chern-Simons-Dilaton theory

Chern-Simons       naturally introduced by
Dilaton field      low energy limit of string theory ...


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

⇒ numerical solutions ... ?
                                              48

Future Works (2)
More Higher-dimensions

                 S3    : S1 bundle over CP1
                 ・・・
                 S2n+1 : S1 bundle over CPn

Ex) S7 : S1 bundle over CP3
                                          49

Future Works (2)

  Black Objects …




  Kasner spacetime ( Bianchi types ) …

  Dynamical ( Rotating ) BHs without Λ
                                                50

Future Works (3)
  Test Maxwell Fields
Ex) Wald Solutions ( vacuum background )




  Kerr BH in Uniform Magnetic Field

  “ Misner effect ” for extreme BH         BH
  最内部安定円軌道 ( ISCO )
                                     51

Future Works (3)

  Black Strings in …

  Black Rings in …




  ( Charged ) squashed KK BH in …

				
DOCUMENT INFO