Thermalization of Gauge Theory and Gravitational Collapse

Shu LinSUNY-Stony Brook

SL, E. Shuryak. arXiv:0808.0910 [hep-th]

Basic elements of AdS/CFTIn large Nc, strong coupling limit, string

theory in AdS5xS5 background is dual to N=4 SYM

pure AdS background AdS-Blackhole

2

2222

zdzxddtds

2

2222 )(/)(

zzfdzxddtzfds

4

4

1)(hzzzf

z=0 z=0

z= z=

horizon: z=zh

N=4 SYM at T=0 N=4 SYM at T=1/(zh)

thermalization

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)

hologram of the debris

QQbar

SL, E.Shuryak arXiv:0711.0736 [hep-th]

Gravitational Collapse Model

• Israel: spherical collapsing in Minkowski background.

Gravitational Collapse in AdS (backreaction included)

shell falling

boundary z=0

“horizon”: z=zh

AdS-Blackhole

pure AdS

z=

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.

Israel junction condition

• continuity of metric on the shell• matching of extrinsic curvature

where

Shell:gij: induced metric on the shell

ijijij

ijijij

KKK

SKgK

][

][][ 25

areapgdpS ij *det4

ijij pgS

Falling of shell-z0

-zh

Initial acceleration

Intermediate near constant fall

Final near horizon freezing

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)

z0~1/Qs~1/1GeV zh=1/(T)~1/1.5GeVQs: saturation scalezh: initial temperature of RHIC

The initial temperature of RHIC is determined from initial collision condition

6)

61(4

25

25

4

40 ppzz

h

Quasi-equilibriumaxial gauge where =z, t, xgraviton probe where m=t, xone-point function of stress energy tensor the same as thermal case

Two-point function deviates from thermal case

thermalmnshellmn xTxT )()(

thermalklmnshellklmn TxTTxT )0()()0()(

mnh0zh

infalling

infalling

outfalling

graviton probe h_mn:

horizon: zm=zh

AdS-BH (thermal) limit

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

ijijijij SKgKgK 25][][][

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(zm)^(1/2) inside

),(~),( 3 qhqeddxth mnxqiti

mn

inoutm dtdtzf )(

t_out

t_in

Asymptotic ratioStarinets and Kovtun hep-th/0506184

• scalar channel: hxy

• shear channel: htx, hxw

• sound channel: htt, hxx+hyy, htw, hww

where um=zm^2/zh^2as um1, f(um) 0. Infalling wave dominates the outfalling one.

6/8/11

)()1( 2

5 pufiu

OutfallingInfallingr

m

im

),( wthh mnmn

Retarded Correlator and Spectral Density

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

),(Im2),(

)]0(),([),(

)]0(),([)(),(

,,

4,

04,

qGq

TxTxedq

TxTxxediqG

klmnR

klmn

klmnikx

klmn

klmnikx

klmnR

spectral density mn,kl

deviation from thermal

scalar channel: q=1.5

black um=0.1, red um=0.3, blue um=0.5,

green um=0.7, brown um=0.9

thermalklmn

thermalklmn

shellklmn

klmnR,

,,,

Rxy,xy

shear channel: q=1.5black um=0.1, red um=0.3, blue um=0.5, green um=0.7, brown um=0.9

Rtx,tx

sound channel: q=1.5

black um=0.1, red um=0.3, blue um=0.5,

green um=0.7, brown um=0.9

Rtt,tt

• 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.

noscillatiothermalshell

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

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.