Shin Nakamura Shin Nakamura (Center for Quantum Space (Center for Quantum Space time (CQUeST) , Sogang Un time (CQUeST) , Sogang Un iv.) iv.) Based on S. Kinoshita, S. Mukohyama, S.N. and K. Oda, arXiv:0807.3797 A Holographic Dual of A Holographic Dual of Bjorken Flow Bjorken Flow
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Shin Nakamura (Center for Quantum Spacetime (CQUeST), Sogang Univ.) Based on S. Kinoshita, S. Mukohyama, S.N. and K. Oda, arXiv:0807.3797 A Holographic.
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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)
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.