shin nakamura (center for quantum spacetime (cquest), sogang univ.) based on s. kinoshita, s....
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![Page 1: 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](https://reader035.vdocuments.site/reader035/viewer/2022062314/56649f385503460f94c555bc/html5/thumbnails/1.jpg)
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
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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.
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Also at LHC
Similar exp. at• FAIR@ GSI• NICA@ JINR
http://aliceinfo.cern.ch/Public/en/Chapter4/Chapter4Gallery-en.html
ALICEATLASCMS
ALICE
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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:
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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.
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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.
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Quark-gluon plasma (QGP) as a one-dimensional expansion
http://www.bnl.gov/RHIC/heavy_ion.htm
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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
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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
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Boost invariance
Taken from Fig. 5 in nucl-ex/0603003.
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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
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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.
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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.
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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
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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.
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Time-dependent AdS/CFT
Earlier works
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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
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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
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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
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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
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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.
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Our work:
Formulation without singularity.
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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.
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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)
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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,
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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.
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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= ∞.
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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
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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)
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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
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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
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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.
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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
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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
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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)
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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•
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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•
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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.
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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).
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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
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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!
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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!
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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.
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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
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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?
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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?
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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