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					   Rotating Black Holes
        @Colliders

      Seong Chan Park
  Korea Institute for Advanced Study (KIAS)

       6th RESCEU International Symposium:
“Frontiers in Astroparticle Physics and Cosmology”
                    Nov. 6 , 2003

  S. Park ,H. S. Song J. Korea Phys. Soc. 43: 30-33 (2003)
   D. Ida, K. Oda, S. Park Phys. Rev. D67:064025 (2003)
 Q. Does higher energy lead us to
        smaller distance?
Uncertainty relation:
 to see smaller distance,
 we need higher energy.
We cannot see the smaller
distance than 1/Mp even in
trans-Planckian domain :
 Black Hole production
     (t’ Hooft 1982)

              Seong Chan Park, 6th RESCEU,
                         2003                2
            Distance VS Energy

        L


CG


                                            We don’t know much
QG                                           about this area:
     1/Mp                                      String/M ??!?


                                                    E
              Mp
                   Seong Chan Park, 6th RESCEU,
                              2003                               3
    Review: One minute for Black Hole

**Classical: Strong Gravity
-nothing goes out (Ve>c)
**Quantum: Hawking Radiation
-thermal radiation: T=Surface
   Gravity
-Anything goes out!
   Test? No, Not yet….
:only astronomical indirect
   observations have been found.




                      Seong Chan Park, 6th RESCEU,
                                 2003                4
                    Planck Energy
**Planck Energy: Energy
   scale where gravity
   becomes strong.
   Well measured only for:
   r>0.2 mm (Hoyle et. al. PRL03’)
   Enormous extrapolation
   assumed!!
  : from 1mm to 10-34 cm
    No example such a big
   extrapolation works in
   history.
                        Seong Chan Park, 6th RESCEU,
                                   2003                5
New understanding of Planck energy:
      TeV gravity scenarios
**Brane world (1996-)
   Talks by Langlois, Maartens, Tanaka, Soda,
    Khoury, Frolov, Ichiki in this symposium
    Horawa-Witten(1996)
    ADD(1998)
    RS(1999)
: What did we learn?
Newton’s constant is just a derived quantity which is
    depending on geometry of higher dimensional
    spacetime.
          Mp*=TeV possible!
              Seong Chan Park, 6th RESCEU,
                         2003                           6
      Black Holes at Collider?
Any colliders with ECM>> MPl have chances to produce BHs.
In TeV gravity models, we have Big chances in near future
experiments.
   LHC Ecm=14 TeV, VLHC Ecm=100 TeV or SLHC..
   UHECR E>> TeV

Such BHs are typically
   Rh << RCompact
   Higher Dimensional
   Rh >> RCompton
   Semi-Classical




                        Seong Chan Park, 6th RESCEU,
                                   2003                     7
       Fate of BHs at collider
1. Balding Phase
  •   Dynamical production phase
  •   BH loses its “hair”.
2. Spin Down Phase
  •   BH loses its mass and angular momentum.
3. Schwartzschild Phase
  •   Angular momentum is small.
  •   BH loses its mass.
4. Planck Phase                               BH radiates mainly
  •   Truly QG, highly unpredictable            on the brane.
  •   A few quanta would be emitted.


               Temperature gets higher and higher.
                   Seong Chan Park, 6th RESCEU,
                              2003                                 8
           Rotating BH formation
                 parton
                  M/2
                                    Rh
                                                       b
                                         M/2
                                        parton
•Condition for BH production (Hoop Conjecture) : BH forms if and only if
the energy is compacted into a region whose impact parameter is less
than the horizon size of the BH.
    Rh(M,J) > b/2
•BH must be produced with finite Angular Momentum.
  : Angular momentum conservation J= M b/2
            Don’t forget : we need higher dimensional
            generalization of Kerr Black Hole Solution!
                        Seong Chan Park, 6th RESCEU,
                                   2003                                9
   Metric for rotating black hole: D=4+n
                             Myers-Perry 1986

ds 2 = g (4) (r, ϑ ) + r 2 cos 2 ϑ dΩ2
                                     n    vanishes on the brane
g (4 ) (r, ϑ ) =
 ∆ − a 2 sin 2 ϑ              (∆ − r 2 − a 2 )asin 2 ϑ               
−                                                                 0 0
             Σ                           Σ                           
                       [(r 2 + a 2 )2 − ∆a 2 sin 2 ϑ ]sin 2 ϑ        
           *                                                      0 0
                                          Σ
                                                                  ∆  
           0                              0                         0
                                                                  Σ  
           0                              0                       0 Σ
                                                         Σ = r 2 + a 2 cos ϑ
                                                               2
                                                                      µ a2 
                            Seong Chan Park, 6th RESCEU,
                                                         ∆ = r 1− n+1 + 2
                                       2003
                                                                  r         r    10
::continued::   Rotating BHs at colliders
When we neglect balding phase,
                        phase
                                        There is maximum b allowed for
b < 2rh ( M ,J )                        BH formation.
gives condition for                                                     −1
BH formation.                                           n + 2       
                                                                      2 n+1

                                              bmax = 21+                  rS (M )
         rS ( M )                                     
                                                          2         
                                                                       
 rh =
      (1+ a* )1/(n+1)
            2
                                                 2.9 (n = 1)
rescaled angular momentum                       
                                                 4.5 (n = 2)
    a= (n+2)J/2M                        J max ~              for M / M p = 5
                                                 M
                                                11.5 (n = 7) (typical
                                                
                                                                LHC energy)
              How good is this estimation?
                        Seong Chan Park, 6th RESCEU,
                                   2003                                           11
            Our formula nicely fits
          numerical result with full GR
        n           1          2        3        4        5        6          7
       RNY         1.056   1.158       1.228   1.276    1.314    1.344    1.368
       Rours       1.110   1.170       1.218   1.262    1.300    1.334    1.364

                                         A few % for n =1 and ~1% for n >1.
         bmax
  R(n) =
          rS                       cf) Numerical result utilizes
                                   the Aichelburg-Sexl solution
               b                   (Eardley, Giddings 02) Yoshino, Nambu 02
                                       Setup: two particles (SBHs) with
                                           • boost      ,
                           t               • mass 0,
                                   z       • energy: fixed.


Closed trapped surface forms
                                                       There are two direct
when b < bmax.                                         consequences. 12
 1. Form factor becomes larger
       in higher dimensions.
    n      1       2          3          4            5    6           7
   FNY    1.084   1.341     1.515      1.642     1.741    1.819       1.883
   Four   1.231   1.368     1.486      1.592     1.690    1.780       1.863


   σ                            n + 2
                                                                 −2
                                                                  
F= 2                                                           2 n+1

  πRS             σ = πbmax = 41+ 
                        2
                                                                         πr  2

                                 2 
                                                                 
                                                                  
                                                                              S




This growing pattern can be understood when
one take angular momentum into account.
                       Seong Chan Park, 6th RESCEU,
                                  2003                                        13
        2. Most BHs are produced with
           large angular momenta
Initial angular momentum:   J = bM / 2
                   dσ  8 πJ / M 2 (J < J max )
                      =
 db                dJ  0
                                  (J > J max )
         dσ = 2πbdb                             (J max = bmax M /2)

                             Differential cross section
                             increases linearly with J.

                            Seong Chan Park, 6th RESCEU,
                                       2003                           14
      Production of Black Ring
Topological no-go                  Q. D=5 BR production?
theorem in D=4.                    A. Not so probable.
     horizon = S2
                                   See the maximum J in D=5:
     Simple generalization
      S2+n in D=4+n(??)
Black Ring solution in
D=5 (Emparan and Reall PRL 2002)
    horizon = S2 * S1
    theorem broken in D>4!!
                                    Q. How about D>5 ?
                                    We do not have the explicit
    Gravitational attraction VS         BR sol. In D>5, yet!!
    Centrifugal force
                                     Naïve estimation about the
                                     minimum value of angular
                                    momentum might be helpful.
                                        ::SEE NEXT PAGE::
                                                               15
   ::Continue from previous page:: Combine above relations
                                   gives the minimum angular
                                   momentum for rotating Black
  D=4+n, Black Ring                Ring:
  Horizon =



                                  Compare with Maximum
                                  value of Angular Momentum:
  Reduced Schwarzshild            D      6      7      8      9     10      11
radius
                                 Min/
                                        0.43   0.31   0.20   0.11   0.06   0.03
                                 Max

                                      This means that there
      L>R
                                      could be large
 Centrifugal force
                                      probability to produce
 > Gravitational attraction
                                      BRs in d>5.
                                                                           16
    Hawking Radiations from rotating BH
 BH radiates mainly into the brane fields via Hawking radiation.
                                                      radiation




  Greybody factors for Brane fields determine the evolution of BH.
                        d M  1                 Γs,l,m ω 
                       −  =        ∑ gs ∫ dω eω −mΩ m 1m
                        dt  J  2 π s ,l,m               
Greybody factor= absorption cross-section of Black Hole.
  can be obtained by solving the brane field equations.
                                             equations
                                                                  actually g.o.
   Conventionally people used g.o. approximation:
                                                     Γ ∝ω   2
                                                                approx. is not
                                                                so nice.

                            Seong Chan Park, 6th RESCEU,
                                       2003                                17
       Derivation of master equation:
                    Newman-Penrose formalism
We set null tetrad as follows:                                       Σ = r 2 + a 2 cos ϑ
                     ϕ    Σ r
 nµ = δµ − asin 2ϑδµ − δµ
        t

                         ∆                                                        µ a2 
        ∆ t                     1 r                                  ∆ = r 2 1− n+1 + 2
 n'µ =
       2Σ
           (δµ − asin 2 ϑδµ )+ δµ
                            ϕ
                                2
                                                                              r         r 

             i sin ϑ                                r − ia cos ϑ ϑ
 mµ =
          2(r + ia cos ϑ )
                           [aδµ − (r 2 + a 2 )δµ ]−
                              t                ϕ
                                                           2
                                                                δµ

 m'µ = mµ                                 (4 )µν
                                                   ∇ µ∂ν φ = 0
                 Scalar:              g

                 Spinor:

                Vector:
                                                                                      18
       Master equation for Brane fields
                                         −iω t + imϕ
Decomposition: φ, ψ0 , φ0 ~ R(r)S(ϑ )e                     Σ = r 2 + a 2 cos ϑ
   angular part:                                                 2
                                                                        µ a2 
                                                           ∆ = r 1− n+1 + 2
      1 d           dS                                            r         r 
              sin ϑ    
    sin ϑ dϑ        dϑ 
     + [(s − aω cos ϑ )2 − (s cot ϑ + m csc ϑ )2 − s(s −1) + A]S = 0
   radial part:
      −sd  s +1 dR  as 4-dim. Can be treated in a standard manner.
                  Same                           K = (r 2 + a 2 )ω − ma
    ∆      ∆       
        dr      dr 
       K 2            ∆,r K                                    
     +  + s  4iωr − i        + (∆,rr − 2)+ 2maω − (aω ) − A R = 0
                                                            2

       ∆               ∆                                       
           D=4 case was done by S. A. Teukolsky 1972-1973.
           Separable for any spin and in any dimensions!!
                            Seong Chan Park, 6th RESCEU,
                                       2003                                          19
    Solving the wave equation
     D=5 rotating Black Hole
D=4 case was done by D. N. Page in 1976



  NH limit:                                    FF limit:

                Matching here!



              Overlapping region




                Seong Chan Park, 6th RESCEU,
                           2003                            20
              Greybody factors
        for Rotating Black Hole (D=5)




Greybody factors in low
frequency expansion.
                                        21
                    Power Spectrums
                         S=0
              dE
        rh
             dtdω




S=1/2                     rhω         S=1




                                       22
                Angular dependence of emission: a=1.5

                                                       S=0




          dE
rh
     dt dω d cos ϑ


                                                             S=1
S=1/2




                        Seong Chan Park, 6th RESCEU,
                                   2003                        23
                   Summary
Lessons from New Paradigm of high energy scattering:
  High energy does not leads us to smaller distance.
  Possible TeV gravity scenarios in Brane world.
We have shown that
  Production of rotating Black Holes.
     Black holes are mostly produced with large angular momentum.
     Form factor of BH production becomes larger in higher dimensions.
  Production of Black Rings.
     Promising in D>5.(need for explicit solution)
  Evaporation of rotating BH on the brane.
     We derived the master equation for rotating Black Holes in D>4.
     Analytic expression of greybody factors for D=5 rotating black hole.
     Decay signals are significantly affected by the angular momentum of BHs.
We are working on
  to find out greybody factors for D>5 (numerically),
  to find out greybody factors for (bulk) graviton emission.

                         Seong Chan Park, 6th RESCEU,
                                    2003                                 24

				
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posted:12/29/2011
language:English
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