Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qfthep.sinp.msu.ru/talks2011/PozdeevaQFTHEP11.pdf Äàòà èçìåíåíèÿ: Thu Oct 6 18:35:46 2011 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:50:01 2012 Êîäèðîâêà:

Charged wall-wall collision in AdS5 Ekaterina Pozdeeva SINP MSU, Moscow, Russia
based on work with I. Ya. Aref 'eva and A. Bagrov

QFTHEP 2011


· In a series of papers Amati, Ciafaloni, Veneziano and t'Hooft1 conjectured that black hole occur in the collision of two light particles at planking energies: E > Mpl · I.Ya. Arefeva, K.S. Viswanathan, I.V. Volovich mad a conjecture that ultra-relativistic particle generates a gravitational wave2 Then these plane gravitational waves collide and produce a black hole.
D.Amati, M. Ciafaloni and G. Veneziano, Phys.Lett B 197(1987)81, G.t Hooft, Phys.Lett.B 198(1987); I.Ya. Arefeva, K.S. Viswanathan, I.V. Volovich, Nucl.Phys.B 452:346368,1995.
1 2

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· The higher-dimensional, fundamental theory has a new scale for gravity, M, that is related to the effective 4-dimensional one through the equation 3:
2 2+ Mp RnM n,

D = 4 + n.

R is size of extra spacelike dimensions. · New development has been started, when people realized that Mpl can be about 1 TeV.

N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998); Phys. Rev. D 59, 086004 (1999). I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436, 257 (1998).
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· After this time the problem of black hole production has been considered as accessible at about Tev and estimation of BH production for b < RS ch(D), b is impact parameter.4 · The technic of entropy finding developed by S. B. Giddings and S. Thomas 5 were applied in AdS/dS space for without charge case 6 and for charged case 7
T. Banks and W. Fischler, A Model for High Energy Scattering in Quantum Gravity hep-th/9906038. I.Ya.Aref 'eva, High Energy Scattering in the Brane-World and Black Hole Production hep-th/9910269 Part.Nucl. 31, 169-180 (2000) S. B. Giddings and S. Thomas Phys. Rev. D 65, 056010 (2002) S.S. Gubser, S.S. Pufu and A. Yarom Phys.Rev. D 78:066014(2008) S.S. Gubser, S.S. Pufu and A. Yarom JHEP 0911:050(2009) L. Alvarez-Gaume, C. Gomez, A. Sabio Vera, A. Tavanfar, M. A. VazquezMozo JHEP 0902:009 (2009) I.Ya. Aref 'eva, A.A. Bagrov, E.A. Guseva JHEP 0912:009(2009) I.Ya. Aref 'eva, A.A. Bagrov, L.V. Joukovskaya JHEP 1003:002 (2010)
4 5 6 7

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· The metric of arbitrary gravitational shock wave in AdS5 spacetime in Poincare coordinates 8:
2 -dx+dx- + dx2 + (z ) (x+)dx+ + dz 2 ds2 = L2 , (1) 2 z where x+ = t + x1, x- = t - x1 are light-cone coordinates, x are coordinates transversal to the direction of moving.

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S.S. Gubser, S.S. Pufu and A. Yarom Phys.Rev. D 78:066014(2008)
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equation for the membrane wall without charge 9 ordinates 3 z0 z )(x+, z ) = -16 G5µ 3 (x+) (z - z0), L ) = (z ) (x+). That is equivalent to 3 z0 3 2 (z - z )(z ) = -16 G5µ 3 (x+). z L · We obtain the charged membrane wall Einstein equation of the form: 3 z0 3 2 ¯ (z - z )(z ) = -16 G5µ 3 (z - z0) - 16 G5Tx+x+ (Q, z ) z L (2)
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·The Einstein in Poincare co 2-3 (z z where (x+, z

S. Lin, E. Shuryak, Phys.Rew. D, 83, 045025 (2011)
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· In AdS5 charged wave shape in Poincare coordinates has the form 10 ¯ ¯ 3 M 5 Q2 F, F= F1 + 22 a 64a 2 + 8q + 1) - 4(2q + 1) q (1 + q ) (8q F1 = , 2 q (1 + q ) 144q 2 + 16q - 1 + 128q 4 + 256q 3 - 64(2q + 1)(q (q + 1))3/2 F2 = q (1 + q ) q (1 + q ) (x)2 + (z - z0)2 . where q = 4z z 0
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I. Ya. Aref 'eva, A. Bagrov and L. Joukovskaya, JHEP 1003:002 (2010).
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· Now we apply the charged wave shape 11 to the expression 12 related the shap e wave with corresp onding plane stress-energy tensor x+x+ component : q (q + 1)Fqq + (3/2)(1 + 2q )Fq - L Jx+x+ = , z Jx+x+ is the bulk stress-energy tensor x After that we make renormalization and tensor Jx+x+ all over surface. 3F = -16 G5L2, (3) (4)
+x+ for charged case Q.

spread the stress-energy

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I. Ya. Aref 'eva, A. Bagrov and L. Joukovskaya, JHEP 1003:002 (2010). S.S. Gubser, S.S. Pufu and A. Yarom JHEP 0911:050 (2009).
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· Let us return to the part of Einstein equation for the charged membrane wall 3 z0 3 2 (z - z )(z ) = -16 G5µ 3 (z - z0) - 16 G5Tx+x+ . (5) z L The charged part of x+x+ stress-energy tensor component corresponding to plane homogenous source has form 40 G5Q2E â (6) -16 G5Tx+x+ = - 4M 3L ) ( 5 2 2 z0 (-3z 2 + z0 ) z 4z0(-z 2 + 3z0 ) (z0 - z ) + (z - z0) . â 2 + z 2 )3 2 + z 2 )3 ( -z ( -z 0 0 Later we will consider the cases z < z0, z > z0 separately.
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· Let us consider the following two equations separately:
2 3 40 G5Q2E z 4z0(-z 2 + 3z0 ) 2 (z - z )qz0>z = - , z0 > z ; 4M 2 + z 2 )3 z 3L ( -z 0 5 2 3 40 G5Q2E z0 (-3z 2 + Z0 ) 2 (z - z )qz >z0 = - , z > z0 . 4M 2 + z 2 )3 z 3L ( -z 0

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· The function (z ) = z L (z ) is solution to the equation. Here the function (z ) can be represented in the form: C1 z 3C2 10 z0Q2 E G5z 3 , z0 > z , q z0>z = + 4- 4 5M (| - z 2 + z 2|) 3L zL L 0 5 z0 G5Q2E z 3C2 C1 10 q z >z0 = , z > z0 + 4- 4 5M (| - z 2 + z 2|)z 3L L zL 0 or C1 z 3C2 10 z0z 3Q2 E G5 ) , z0 > z , ( qz0>z = + 4- 4 5 M -z 2 + Z 2 3L zL L 0 5 z0 Q2 E G5 C1 10 z 3 C2 ( ) , z > z0 qz >z0 = + 4+ 4 5 M z -z 2 + z 2 3L L zL 0

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· A sufficient condition for a black hole formation is the existence of trapped surface. · At the trapped surface the solution to the Einstein equation for the membrane wall must satisfy to conditions 13: 1. (za) = (zb) = 0, ( za ) 2 ( zb ) 2 2. (za) = (zb) = 4, L L where za and zb are the boundaries of trapped surface.

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S. Lin, E. Shuryak, Phys.Rew. D, 83, 045025 (2011).
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· In 14 solution to non-charged membrane wall Einstein equation constructed by such way that conditions 1 satisfied automatically ) ( 4 z0 ) ( - 1 zb 4 4 G5µ zb z 3 za a = C ,C=- - , z < z0 3 4 4 - z4 z za L zb a 33 zazb = ( ) 4 z0 ( ) - 1 za 4 3 z = D z - zb , D = - 4 G5µ a b , z0 < z 3 4 4 - z4 z L zb zb a z 3z 3
ab
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S. Lin, E. Shuryak, Phys.Rew. D, 83, 045025 (2011).
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· We want that the solution to charged membrane wall Einstein equation satisfied automatically to the property 1 too. Thus, the solution part corresponding to charge must satisfy to the properties 1 and can be represented in the form: 2 10 µG5Q2 3 -za + z 2 aq = - z0 z , z < z0 5M 2 + z 2)(-z 2 + z 2) 3L ( -z a 0 0 (7) 2 5 -zb + z 2 10 µG5Q2 z0 = , z0 < z bq 5M z (-z 2 + z 2)(-z 2 + z 2) 3L 0 0 b

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· Let us consider the property 2 ( )2 ( (Z ) Za a = L for the complete solution: { a = b

(Z ) Zb b

)2 =4

L

+ +

q a q b

(8)

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· The first order derivatives of the solution to the Einstein equation at points za, zb are the following: ( ) 4 z0 zb -1 4 4 za 20 E G5Q2 16 G5E zb z0 za - (za) = - , 4 4 - z4 5M (-z 2 + z 2)2 3L L zb a a 0
4 z0 -1 4 16 G5E za (zb) = - 4 L4 zb - z

(

z

3 a

3 zb )

za zb

4 a

5 Z0 20 E G5Q2 + . 5 M ( -z 2 + z 2 ) 2 3L 0 b

z

3z 3 ab
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· The condition 2 gives us the following equations: (4 ) 4 z3 5 8 G5E z0 - zb a 10 E G5Q2 z0 za Fa = - - = 1, 6M (-z 2 + z 2)2 5(z 4 - z 4) 3L Lb a) a 0 (4 43 5 8 G5E z0 - za zb 10 E G5Q2 z0 zb Fb = - + = -1. 6 M ( -z 2 + z 2 ) 2 5(z 4 - z 4) 3L Lb a 0 b These equations have not analytical solutions. We look for numerical solutions and graphical analyse.

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Numerical calculations and graphical analyse Example of parameters choosing. Now let us use the following initial paraments: E = 19.7 T eV L = Z0 = 4.3f m, L3 = 1.9, G5 f m = 10-15m, T eV = 1012eV . (9)

(10)

Using 1T eV = 103GeV = (1.2)-1 · 103f m-1 0.8(3) · 103f m-1 and direct calculations we obtain: E 16416.7f m-1 Analogical numerical calculations we apply for another E .
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(11)


The function Fa for E = 19.7 TeV is in 1.
F
a

Q = 5 · 10

3

F

a

Q=0 Q = 20 · 10

3

Q = 10 · 10

3

Q = 5 · 10

3

z Q = 20 · 10
3

a

Q = 40 · 10

3

Q = 40 · 10

3

z

a

Figure 1:

The function Fa(za) and the function Fa(za) near the root

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The function Fb for E = 19.7 TeV is in 2.
F
b

F

b

l n ( zb )

Q = 40 · 10 Q = 20 · 10

3

z

b

3

Q = 10 · 10

3

Q = 5 · 10

3

Figure 2:

The function Fb(zb) and the function Fb(zb) near the root

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The square trapp ed surface calculation g dz d2x 2A S= = , 4G 5 2( 5 G ) S L3 1 1 s 2 = -2 . 2 d x 4G5 za zb (12) (13)

The trapped surface decreases with growth of a charge. The corresponding graphical representations are in the picture (3):

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s

s

Q

Q

Figure 3:

The trapped surface area s(Q) for E = 1 T eV E = 19.7 T eV

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l n( E )

E

l n( Q)

Q

Figure 4:

Phase diagram in the logarithmic scale for large Q and large E . The phase diagram for small E and small Q.

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Summarizing

At the present report we have considered the charged membrane walls collision in AdS5 using Einstein equation and shock waves approach. We have studied the influence of the charges to the trapped surface formation in the charged walls collision and obtain the diagram with allowed and forbidden zones for black hole formation.

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