Thermalization of Gauge Theory and Gravitational Collapse

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Thermalization of Gauge Theory and Gravitational
Collapse

• Shu Lin
• SUNY-Stony Brook

SL, E. Shuryak. arXiv0808.0910 hep-th
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Basic elements of AdS/CFT

• In large Nc, strong coupling ? limit, string
  theory in AdS5xS5 background is dual to N4 SYM
• pure AdS background
  AdS-Blackhole

z0
z0
thermalization
horizon zzh
z?
z?
N4 SYM at T1/(?zh)
N4 SYM at T0
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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)

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hologram of the debris
Q
Qbar
SL, E.Shuryak arXiv0711.0736 hep-th
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Gravitational Collapse Model

• Israel spherical collapsing in Minkowski
  background.

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• Gravitational Collapse in AdS (backreaction
  included)

boundary z0
AdS-Blackhole
shell falling
horizon zzh
pure AdS
z?
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Gauge Theory Dual

• gravitational collapse in AdS is dual to the
  evolution of N4 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.

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Israel junction condition

• continuity of metric on the shell
• matching of extrinsic curvature
• where
• Shell
• gij induced metric on the shell

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Falling of shell
-z0
Initial acceleration
Intermediate near constant fall
Final near horizon freezing
-zh
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• Physical interpretation of p, z0 and zh
• The parameter p should be estimated from the
  initial condition on the boundary (energy density
  and particle number)
• z01/Qs1/1GeV zh1/(?T)1/1.5GeV
• Qs saturation scale
• zh initial temperature of RHIC
• The initial temperature of RHIC is determined
  from initial collision condition

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Quasi-equilibrium

• axial gauge where ?z, t, x
• graviton probe where mt, x
• one-point function of stress energy tensor
• the same as thermal case
• Two-point function deviates from thermal case

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graviton probe h_mn
infalling
outfalling
horizon zmzh
infalling
AdS-BH (thermal) limit
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Graviton passing the shell

• matching condition given by the variation of
  Israel junction condition
• hmn outside and inside are continuous on the
  shell
• hmn outside and inside should preserve the EOM of
  the shell

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Quasi-static limit

• Although the shell keeps falling, it can be
  considered as static for Fourier mode
• ?gtgt dz/dt
• NOTE
• the frequency ? outside corresponding to
  frequency ?/f(zm)(1/2) inside

t_out
t_in
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Asymptotic ratio

• Starinets and Kovtun hep-th/0506184
• scalar channel hxy
• shear channel htx, hxw
• sound channel htt, hxxhyy, htw, hww
• where umzm2/zh2
• as um?1, f(um) ?0. Infalling wave dominates the
• outfalling one.

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Retarded Correlator and Spectral Density

• boundary behavior of hmn ? retarded correlator
  Gmn,kl ? spectral density ?mn,kl

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spectral density ?mn,kldeviation from thermal
Rxy,xy
?
scalar channel q1.5 black um0.1, red um0.3,
blue um0.5, green um0.7, brown um0.9
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Rtx,tx
?
shear channel q1.5 black um0.1, red um0.3,
blue um0.5, green um0.7, brown um0.9
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Rtt,tt
?
sound channel q1.5 black um0.1, red um0.3,
blue um0.5, green um0.7, brown um0.9
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• spectral density
• the oscillation damps in amplitude and grows in
  frequency (reciprocal of ?) as um ? 1. Eventually
  the shell spectral density relaxes to thermal
  one.

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• 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 um ? 1

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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.

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WKB solution(?gtgt1)
with
a_0, b_0, c_0 are functions of q, ?, u_m r
outfalling/infalling
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