1. Thermalization of Gauge Theory and Gravitational Collapse Shu Lin SUNY-Stony Brook SL, E. Shuryak. arXiv:0808.0910 [hep-th]

2. Basic elements of AdS/CFT In large N c, strong coupling limit, string theory in AdS 5 xS 5 background is dual to N=4 SYM pure AdS background AdS-Blackhole z=0 z=  horizon: z=z h N=4 SYM at T=0 N=4 SYM at T=1/(  z h ) thermalization

3. Gravity Dual of Heavy Ion Collision E.Shuryak, S.Sin, I.Zahed hep-th/0511199 RHIC collisions produce debris consisting of strings and particles, which fall under AdS gravity SL, E.Shuryak hep-ph/0610168 studied the falling of debris and proposed to model the debris by a shell(ignoring the backreaction of the debris to AdS background)

4. hologram of the debris Q Qbar SL, E.Shuryak arXiv:0711.0736 [hep-th]

5. Gravitational Collapse Model Israel: spherical collapsing in Minkowski background.

6. Gravitational Collapse in AdS (backreaction included) shell falling boundary z=0 “horizon”: z=z h AdS-Blackhole pure AdS z= 

7. Gauge Theory Dual gravitational collapse in AdS is dual to the evolution of N=4 SYM toward equilibrium Different from hydrodynamics (locally equilibrated): non-equilibrium is due to spatial gradient. Our model: no spatial gradient. The SYM is approaching local equilibrium.

8. Israel junction condition continuity of metric on the shell matching of extrinsic curvature where Shell: g ij: induced metric on the shell

9. Falling of shell -z 0 -z h Initial acceleration Intermediate near constant fall Final near horizon freezing

10. Physical interpretation of p, z 0 and z h : The parameter p should be estimated from the initial condition on the boundary (energy density and particle number) z 0 ~1/Q s ~1/1GeV z h =1/(  T)~1/1.5GeV Q s : saturation scale z h : initial temperature of RHIC The initial temperature of RHIC is determined from initial collision condition

11. Quasi-equilibrium axial gauge where  =z, t, x graviton probe where m=t, x one-point function of stress energy tensor the same as thermal case Two-point function deviates from thermal case

12. infalling outfalling graviton probe h_mn: horizon: z m =z h AdS-BH (thermal) limit

13. Graviton passing the shell matching condition given by the variation of Israel junction condition: h mn outside and inside are continuous on the shell h mn outside and inside should preserve the EOM of the shell

14. Quasi-static limit Although the shell keeps falling, it can be considered as static for Fourier mode:  >> dz/dt NOTE: the frequency  outside corresponding to frequency  /f(z m )^(1/2) inside t_out t_in

15. Asymptotic ratio Starinets and Kovtun hep-th/0506184 scalar channel: h xy shear channel: h tx, h xw sound channel: h tt, h xx +h yy, h tw, h ww where u m =z m ^2/z h ^2 as u m  1, f(u m )  0. Infalling wave dominates the outfalling one.

16. Retarded Correlator and Spectral Density boundary behavior of h mn  retarded correlator G mn,kl  spectral density  mn,kl

17. spectral density  mn,kl deviation from thermal scalar channel: q=1.5 black u m =0.1, red u m =0.3, blue u m =0.5, green u m =0.7, brown u m =0.9  R xy,xy

18. shear channel: q=1.5 black u m =0.1, red u m =0.3, blue u m =0.5, green u m =0.7, brown u m =0.9  R tx,tx

19. sound channel: q=1.5 black u m =0.1, red u m =0.3, blue u m =0.5, green u m =0.7, brown u m =0.9  R tt,tt

20. spectral density the oscillation damps in amplitude and grows in frequency (reciprocal of  ) as u m  1. Eventually the shell spectral density relaxes to thermal one.

21. The WKB solution shows the oscillation of the shell spectral density rises from the phase difference between the infalling and outfalling waves. Further more, the frequency of oscillation in spectral density (reciprocal of  ) corresponds to the time for the wave to travel in the WKB potential (Echo Time) Echo Time approaches infinity as u m  1

22. Conclusion The evolution of SYM to equilibrium is studied by a gravitational collapse model Prescription of matching condition on the shell is given by variation of Israel junction condition. AdS-BH (thermal) limit is correctly recovered Spectral density at different stages of equilibration is obtained and compared with thermal spectral density. The deviation is general oscillations. The oscillation is explained by echo effect: damps in amplitude and grows in frequency, eventually relaxes to thermal case.

23. WKB solution(  >>1) with a_0, b_0, c_0 are functions of q, , u_m r = outfalling/infalling

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