Sebastiano Sonego et al- Optical geometry for gravitational collapse and Hawking radiation
Thermalization of Gauge Theory and Gravitational Collapse
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Transcript of Thermalization of Gauge Theory and Gravitational Collapse
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.