Shin NakamuraShin Nakamura(Center for Quantum Spacetime (C(Center for Quantum Spacetime (C

QUeST) , Sogang Univ.)QUeST) , Sogang Univ.)

Based on S. Kinoshita, S. Mukohyama, S.N. and K. Oda, arXi

v:0807.3797

A Holographic Dual of BjorkeA Holographic Dual of Bjorken Flown Flow

Motivation: quark-gluon plasmaRHIC: Relativistic Heavy Ion Collider (@ Brookhaven National Laboratory)

http://www.bnl.gov/RHIC/inside_1.htm

Heavy ion:e.g. 197Au ~200GeV.NNS

Quark-gluonPlasma (QGP)is observed.

Also at LHC

Similar exp. at• FAIR@ GSI• NICA@ JINR

http://aliceinfo.cern.ch/Public/en/Chapter4/Chapter4Gallery-en.html

ALICEATLASCMS

ALICE

QGP• Strongly coupled system• Time-dependent system

• Lattice QCD: a first-principle computation

However, it is technically difficult to analyzetime-dependent systems.

• (Relativistic) Hydrodynamics

• This is an effective theory for macroscopic physics. (entropy, temperature, pressure, energy density,….)• Information on microscopic physics has been lost. (correlation functions of operators, equation of state, transport coefficients such as viscosity,…)

Possible frameworks:

Alternative framework: AdS/CFTAn advantage:

Both macroscopic and microscopic physics can be analyzed within a single framework.

This feature may be useful in the analysis of non-equilibrium phenomena, like plasmainstability.

Plasma instability: seen only in time-dependent systems.

However,Construction of time-dependent AdS/CFT itself is a challenge.

We construct a holographic dual of Bjorken flow of large-Nc, strongly coupled N=4 SYM plasma.

Our work

A standard model of the expanding QGP

We deal with N=4 SYMinstead of QCD.

Quark-gluon plasma (QGP) as a one-dimensional expansion

http://www.bnl.gov/RHIC/heavy_ion.htm

Bjorken flow (Bjorken 1983)

• (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

“A standard model”of QGP expansion

Local rest frame(LRF)

τ=const.

y=const.

22222 dxdydds

yxyt sinh,cosh 1

Minkowski spacetime

x1

t

Rapidity

Proper-time

The fluid looks static on this frame

Boost invariance ： y-independence

Rindler wedge withMilne coordinates

Boost invariance

Taken from Fig. 5 in nucl-ex/0603003.

Stress tensor on the LRF

uugpuuT

Local rest frame: )0,0,0,1(u

The stress tensor is diagonal.

22222 dxdydds

Bjorken flow:

The stress tensor is diagonal on the Milne coordinates: ),,,( 32 xxy

Hydrodynamics

Hydrodynamic equation: 0 T

We can solve the hydrodynamic equationfor the Bjorken flow of conformal fluid, sincethe system has enough symmetry.

Hydrodynamics describes spacetime-evolutionof the stress tensor.

TTT

TTT

T

xx

yy

2

1

21

2

20

3/40

Solution important T~τ-1/3

Once the parameters (transport coefficients) are given, Tμν(τ) is completely determined.

expansion w.r.tτ-2/3

But, hydro cannot determine them.

AdS/CFT dictionary

Bulk on-shell action = Effective action of YM

The boundary metric(source)

T

4d stress tensor

Time-dependent geometry

Time-evolution of the stress tensor

How to obtain the geometry?

The bulk geometry is obtained by solvingthe equations of motion of super-gravitywith appropriate boundary data.

5d Einstein gravitywith Λ<0

• The boundary metric is that of the comoving frame: 22222

dxdydds

• The 4d stress tensor is diagonal on this frame.

Bjorken’s case:

We set (the 4d part of) the bulk metric diagonal. (ansatz)

This tells our fluid undergoes the Bjorken flow.

Time-dependent AdS/CFT

Earlier works

A time-dependent AdS/CFT

A time-dependent geometry that describes Bjorken flow of N=4 SYM fluid was first obtained within a late-timeapproximation by Janik-Peschanski.

Janik-Peschanski, hep-th/0512162

They have used Fefferman-Graham coordinates:

.............)(~)(~),(~

),(~

4)4()0(

2

22

zggzg

z

dzdxdxzgds

stress tensor of YM4d geometry (LRF)

boundary condition to 5d Einstein’s equation with Λ<0

geometry as a solution

Unfortunately, we cannot solve exactly

They employed the late-time approximation:

fixed with ,3/1

vz

x

xy

y ggg ~,~,~

........)()( 3/2)2()1( vfvf

have the structure of

We discard the higher-order terms.

Janik-Peschanski hep-th/0512162

Janik-Peschanski’s result at the leading order

2

2222

32

3

23

22 )1(

1

)1(1 4

4

4

z

dzxddyd

zds z

z

z

...)( 3/40 Hydrodynamics

The statement

,3/4 with ,)( 0 ppIf we start with unphysical assumption like

the obtained geometry is singular:

RR at the point gττ=0.

Regularity of the geometry tells us what the correct physics is.

fixed, with ,4/

vz

p

Many success

• 1st order: Introduction of the shear viscosity:

• 2nd order: Determination of from the regularity:

• 3rd order: Determination of the relaxation time from the absence of the power singularity:

4

1

S

S.N. and S-J.Sin, hep-th/0607123

Janik, hep-th/0610144

Heller and Janik, hep-th/0703243

For example:

same as Kovtun-Son-Starinets

But, a serious problem came out.

• An un-removable logarithmic singularity appears at the third order.

(Benincasa-Buchel-Heller-Janik, arXiv:0712.2025)

This suggests that the late-time expansion they are using is not consistent.

Our work:

Formulation without singularity.

What is wrong?The location of the horizon (where the problematic singularity appears) is the edge of the Fefferman-Graham (FG) coordinates.

Schwarzschild coordinates

2

1

4

40222222

4

4022 11 dr

r

rrdxdyrd

r

rrds

0

20

2220 //

z

zzzzr

0

0

2r

z

Only outside thehorizon!

2

22222

2

22 )1(

1

)1(140

4

40

4

40

4

z

dzdxdyd

zds

zz

z

z

z

z

FG coordinates

Static AdS-BH case:

This is also the case for the time-dependent solutions.

Better coordinates?

boundaryfu

ture

eve

nt h

orizo

n

past event horizon

singularity

dynamicalapparent horizon

Eddington-Finkelstein coordinates

trapped region

un-trapped region

Cf.Bhattacharyya-Hubeny-Minwalla-Rangamani (0712.2456)Bhattacharyya et. al. (0803.2526, 0806.0006)

Eddington-Finkelstein coordinates

Static AdS-BH:

2224

4022 21 xdrdtdrdt

r

rrds

• The trapped region and the un-trapped region are on the same coordinate patch.

At least for the static case,

Our proposal

Construct the dual geometry on the EF coordinates.

You may say, coordinate transformation doesnot remove the singularity.

where the potential singularity appears

The coordinate transformation from FG coordinatesto EF coordinates is singular at the “horizon”.

What we will do is not merely acoordinate transformation.

Our parametrization

Parametrization of the dual geometry:

We assume a, b, c depend only on τand r, because of the symmetry.

boundary metric: 22222 dxdydds

222212222222 )(12 xderdyrerdrddards ccb

Differential equations of a, b, cThe 5d Einstein’s eq.

,0,0,1 cbaThe boundary condition: at r= ∞.

Late-time approximation

It is very difficult to obtain the exact solution.

We introduce a late-time approximation by making an analogy with what Janik-Peschanski did on the FG coordinates.

Janik-Peschanski:τ-2/3 expansion with zτ-1/3 = v fixed.

Now, r ~ z-1 (around the boundary).

Let us employ τ-2/3 expansion with rτ1/3 = u fixed.

Our late-time approximation

More explicitly,

222212222222 )(12 xderdyrerdrddards ccb

We solve the differential equations for a(τ,u), b(τ,u), c(τ,u) order by order:

......)()()(),( 3/42

3/210 uauauaua

(similar for b and c)zeroth order first order second order

(u=rτ1/3)

The zeroth-order solution

)(2)(

1 2222243/1

422

xddyrdrdd

r

wrds

This is u

• This reproduces the correct zeroth-order stress tensor of the Bjorken flow.

3/40

1

T

• We have an apparent horizon.

4412

9wueF

trapped region if u<w.

The location of the apparent horizon: u=w+O(τ-2/3 )

2

2

04

8

3

cN

w

r = w τ-1/3

Location of the apparent horizonThe location of the apparent horizon is given by

,0Fe

,log ggggg yyzzz

“expansion”

Lie derivative along the null direction

volume element of the 3d surface

0

0 : un-trapped region

: trapped region

normalization

The (event) horizon is necessary

)(9

58 3/28

82

O

u

wR

We have a physical singularity at the origin.

However, this is hidden by the apparent horizonat u=w hence the event horizon (outside it).

OK, from the viewpoint of the cosmic censorshiphypothesis.

Not a naked singularity.

The first-order solution

222212222222 )(12 xderdyrerdrddards ccb

uuw

wuwuuww

c

ub

u

uwwua

3244

2

21

21

21

1

11

5

40

41

41

1

10 )1log(

)log()log()arctan(3

1

1

3)1(

3

2

gauge degree of freedom

c1 is regular at u=w, only when .3

10 w

Regularity of c1 is necessary.We can show

)(regular)(6

31

)(12

31

)(regular2

1

3/22

02

0

3/21

1

Owuw

w

wuw

w

Ocu

cNNR y

y

2

2,0,0,0,1

2

1 2arN : a regular space-like

unit vector

Riemann tensor projected onto a regular orthonormal basis

This has to be regular to make the geometry regular.

(projected onto a local Minkowski)

w3

10

What is this value?

422

8

3TNc 322

2

1TNs c

Gubser-Klebanov-Peet, hep-th/9602135

First law of thermodynamics

4/300 0

2

2

04

8

3

cN

w

Our definition and result:

ws 03

4

1

Combine all of them:

w3

10

4

1

sThe famous ratio by Kovtun-Son-Starinets (2004)

Second-order results:

• We have obtained the solution explicitly, but it is too much complicated to exhibit here.

• From the regularity of the geometry, “relaxation time” is uniquely determined.

consistent with Heller-Janik, Baier et. al., and Bhattacharyya et. al.

2nd-order transport coefficient

)(regular21

)(3

)19(4 23/422

02

Owuwu

wR•

Second-order results:

• We have obtained the solution explicitly, but it is too much complicated to exhibit here.

• From the regularity of the geometry, “relaxation time” is uniquely determined.

consistent with Heller-Janik, Baier et. al., and Bhattacharyya et. al.

2nd-order transport coefficient

)(regular21

)(3

)19(4 23/422

02

Owuwu

wR•

All-order results

n-th order Einstein’s equation:

Diff eq. for bn

= source which contains only k(<n)-th order metric

“Regular enough” to show the regularityof bn and its arbitrary-order derivatives(except at the origin).

We can show the regularity by induction.

All-order results

n-th order Einstein’s equation:

Diff eq. for an

= source which contains only k(<n)-th order metric and bn

“Regular enough” to show the regularityof an and its arbitrary-order derivatives(except at the origin).

All-order resultsn-th order Einstein’s equation:

Diff eq. for cn

= source which contains only k(<n)-th order metric and an , bn

Not always regular enough!

Unique choice of the transport coefficients.

)(3

lower43regular

44

24

wuu

bwaunc nn

n

n-th order transport coefficients comes here linearly

All-order results

• There is no un-removable logarithmic singularity found on the FG coordinates.

• We can make the geometry regular at all orders by choosing appropriate values of the transport coefficients.

Our model is totally consistent and healthy!

Area of the apparent horizon

....2log62

24

1

2

11

443/4

23/2

3

wwG

w

G

Aap

• This is consistent with the time evolution of the entropy density to the first order.

• There is some discrepancy at the second order. However, it does not mean inconsistency immediately.

....24

12log

2

11 3/4

23/2

wwSS

From Hydro.

greater!

What we have done:

• We constructed a consistent gravity dual of the Bjorken flow for the first time.

(cf. Heller-Loganayagam-Spalinski-Surowka-Vazquez, arXiv:0805.3774)

• Our model is a concrete well-defined example of time-dependent AdS/CFT

based on a well-controled approximation.

Hydrodynamics

Time evolution of the stress tensor

• hydrodynamic equation (energy-momentum conservation)• equation of state　 (conformal invariance)

• transport coefficients

Our model

5d Einstein’s eq. at the vicinity of the boundary

Reguarity aroundthe horizon

Related to local thermal equilibrium

Discussion

• The definition of the late-time approximation is a bit artificial.

(τ-2/3 expansion with rτ1/3 = u fixed.)

• Is this a unique choice? • Can we derive it purely from gravity?

• What is the physical meaning?• Is this related to the choice of the vacuum?

Discussion

• At this stage, the connection among our method and other methods are not clear.

• Kubo formula:

)0,0(),,(2

1lim 4

0xyxy

ti TxtTexd

• Quasi normal modes

In this case, we impose the “ingoing boundarycondition” at the horizon.

regularity?

Comparison with Veronicka’s work Veronica’s work Our work

Derivative expansion w.r.t4d coordinates by fixing r.

Derivative expansion w.r.t4d coordinates by fixing u= rτ1/3 .

20

3/40

21

T

The leading order is alreadytime-dependent

T const. + higher derivative(static)

Full Minkowski spacetime Rindler wedge

Expansion around a regular exact solution

The leading-order metricis not an exact solution