shin nakamura (center for quantum spacetime (cquest) and hanyang univ.)
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Shin NakamuraShin Nakamura(Center for Quantum Spacetime (C(Center for Quantum Spacetime (C
QUeST) and Hanyang Univ.)QUeST) and Hanyang Univ.)
Refs. : SN, Sang-Jin Sin, JHEP09(2006)020, hep-th/0607123 (Sang-Jin Sin, SN, Sang Pyo Kim, JHEP0612 (2006) 075, hep-th/0610113)
A Holographic Dual A Holographic Dual of of
QGP HydrodynamicsQGP Hydrodynamics
MotivationMany interesting phenomena in QCD lie in the strongly-coupled region.
We need non-perturbative methods for analysis.
Lattice QCD: a first-principle computation.
However, there are technical difficulties in thecomputations if the system has
• Finite baryon chemical potential• Time dependence• Large size (e.g. deuterons….)
Let us consider time-dependent fluid of YM theories.
An example: QGP
Quark-gluon plasma (QGP) at RHIC is a strongly coupled, expanding system.
• Relativistic hydrodynamics is useful.
However,we may be able to obtain something more than hydrodynamics by applying AdS/CFT, in principle.
• How about AdS/CFT?
Construction of AdS/CFT for time-dependent systems is still a challenge.
Why can AdS/CFT be better than hydrodynamics?
Hydrodynamics
Time evolution of macroscopic quantities temperature, pressure, entropy, energy, transport coefficients (viscosity,…)
microscopic quantities correlation functions, inter-quark potential, counting of entropy, equation of state,…
AdS/CFT
AdS/CFT(Weak version)
Classical Supergravity on
4dim. Large-Nc SU(Nc) N=4 Super Yang-Mills at the large ‘t Hooft coupling
55 SAdS
conjecture
=
Maldacena ‘97
Strongly interacting quantum YM !!
AdS/CFT at finite temperature
Classical Supergravity on AdS-BH×S5
4dim. Large-Nc strongly coupledSU(Nc) N=4 SYM at finite temperature(in the deconfinement phase).
conjecture
=
Witten ‘98
In any case,The bulk geometry is obtained by solvingthe equations of motion of super-gravitywith appropriate boundary conditions.
• The 4d geometry where the YM theory lives.• The energy-momentum tensor of the YM fluid.
Described by hydrodynamics.
What we will do
Something more than Hydro.
Solving the SUGRA equation (5d Einstein’s equation with )0
5d Bulk GeometryTime dependent
Without viscosity: Janik-Peschanski, hep-th/0512162
)(T4d relativistic hydrodynamics
Haro-Skenderis-Solodukhin, hep-th/0002330
boundary conditions
In this talk: N=4 SYM fluid with shear viscosity.
The system we consider
• (Almost) one-dimensional expansion.
• We have boost symmetry in the CRR.
Relativistically accelerated heavy nuclei
After collision
Velocity of light
Central Rapidity Region (CRR)
Time dependence of the physical quantities are written by the proper time.
Velocity of light
(Bjorken ’83)
Relativistic Hydrodynamics• We take local rest frame (LRF).
Our case:
• The energy-momentum tensor on this frame:
),,,( 32 xxy
RapidityProper-time
32
32
342
000
000
00)(0
000
p
p
pT
energy density
shear viscositypressure
The bulk viscosity is set to be zero.
22222 dxdydds
3 independent components: 22,, TTT yy
2 independent constraints:
0
0
T
T “Conformal invariance” (or equation of state)
Energy-momentum conservation
Only 1 independent quantity: )( T
or ,, p
3
4
3/
2
d
d
p
Solution:
1
03/4
00 1
31
4)( ,)(
)3/1( In the static N=4 SYM system:
3T
.........
1
)31(
1)(
0
03/10
TT
4T Stefan-Boltzmann’s low
1)( ,)( 0
in the slowly varying (late time) region
(dim. analysis)
Entropy creation
)(23
4)( 3/43/2
0
0
O
TS
TdS
The entropy (per unit volume on the LRF) at the infinitely far future is not determined in this framework. (Integration constant)
AdS/CFT gives more information.
The dissipation creates the entropy.
Now, the energy-momentum tensor is:
dd
dd
dd
T
21
21
2
000
000
00)(0
000
with20
3/40 2
)(
in the slowly varying (late time) region.
0)(
Gravity dual• 5d metric: asymptotically AdS
Solution of 5d Einstein’s eq. with
• 4d part of the metric depends only on z,
Boost symmetry
0
.............)(
)()()(),(
),(
6)6(
4)4(2)2()0(
2
22
zg
zgzggzg
z
dzdxdxzgds
The Minkowski metric on the LRF
T0
The higher-order terms are determined iteratively.
Haro-Skenderis-Solodukhin hep-th/0002330
Boundary cond.
Boundary cond.
The input (hydro.) is valid only at the late time (slowly varying region).
We employ the late time approximation:
fixed with ,3/1
vz
Late time approximation
xx
yy ggg ,,
......../)(/)()( 3/4)3(3/2)2(0
)1( vfvfvf
have the structure of
We discard the higher-order terms.
Janik-Peschanski hep-th/0512162
Solving the Einstein’s equation
3/1
40 ,3
zv
va
.....75533
13
2
1
.....75533
13
21
2
1
.....7654323
41
2
....44444431
5432
3/2
40
5432
3/2
40
2
54323/2
40
765432
aaaaaav
ag
aaaaaav
ag
aaaaaav
aaaaaaag
xx
yy
Solving the Einstein’s equation
.....75533
13
2
1
.....75533
13
21
2
1
.....7654323
41
2
....44444431
5432
3/2
40
5432
3/2
40
2
54323/2
40
765432
aaaaaav
ag
aaaaaav
ag
aaaaaav
aaaaaaag
xx
yy
3/1
40 ,3
zv
va
43
43
43
43
3/2
404
32
43
43
43
43
3/2
404
3
243
43
43
3/2
40
43
243
0
0
0
0
0
0
0
0
0
0
0
00
0
0
1
1log
11
31
1
1log
1
2
11
31
)1(
)3)(1(
31
)1(
v
v
v
vvv
g
v
v
v
vvvg
v
vvv
v
vg
yy
xx
The metric can be re-summed to be:
The late time geometry
2
22
3
33
22
2
3
33
2
3
23
22
4
4
4
4
4
4
4
4
1
1)1(
1
1)1(
1
)1(1
z
dzdxdy
dz
ds
z
zz
z
zz
z
z
0
20
3/40
3/20
0 )(,2
)( ,
This is correct up to the order of )( 2O .
This looks to be a black hole with time-dependent horizon.
Cf. AdS-Schwarzschild BH
2
2222
32
3
23
22 )1(
1
)1(1 4
4
4
z
dzdxdyd
zds z
z
z
In a more standard form,
2
3
~4
2222
22
2
23
~42
)1(~1
~)1(
4
4
z
z dzdxdy
zd
zds
3/41~
4z
zz
Various quantities from the geometry
Stefan-Boltzmann: )(8
3 422 Hc TN
)(
2
3122
3243/43/2
0
04/30
4/34/12
O
N
G
AS c
Entropy creation:
From hydrodynamics:)(2 3/43/2
0
0
OT
SS
Not only consistent with hydro but alsomore information in the holographic dual.
Integration constant is given.
Numerical coefficient is given.
More about the consistency
)(
2
3122
3243/43/2
0
04/30
4/34/1
O
N
G
AS c
)(21 3/43/2
0
0 O
TSSS
From gravity:
From hydrodynamics:
TT
S3
4
3
4
0
0
Correct relationship from thermodynamics
........./)(
........./)(3/4
0
3/10
TT
Thermal equiv.
Conclusions• We considered a holographic dual of hydro
dynamics of N=4 large-Nc SYM plasma that undergoes Bjorken expansion.
• We obtained the bulk geometry in the late time regime.
• The holographic dual is not only consistent with the hydrodynamics but also contains more information.
Recent progress
• (Janik hep-th/0610144)
• Relaxation time in the second-order dissipative hydrodynamics (Heller-Janik hep-th/0703243)
Computed by using regularity of the time-dependent geometry.
S
Conclusion/Question• Holographic dual “reduces” the problem to
that of dynamical black holes.• How much does the gravity “know” the
physics of non-equilibrium systems?• Holographic dual may provide us an
interesting playground not only for QCD but also for non-equilibrium physics.
We should appreciate the Newton’s discovery of gravity.
More about the late time limit
fixed with ,3/1
vz
zH3/14/1/3
The position of the horizon:
On the ),( v coordinate, the position of the horizonis constant. We are keeping our eyes close to thehorizon along the time evolution.
The above limit is valid around the horizon.
If we take the limit with fixing z, we cannot see thehorizon.
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