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.