Miami 2006
AdS/CFT Correspondence with Heat Conduction
GEORGE SIOPSIS
Department of Physics and AstronomyThe University of Tennessee, Knoxville
December 14, 2006
⇒
J. Alsup, C. Middleton and G. S., hep-th/0607139
AdS/CFT Correspondence with Heat Conduction 1
STRONG FORCE: Quantum Chromodynamics (QCD)
u u u
p ndd
d
nucleons consist of quarks.
Quarks and gluons have colorsI cannot be free (directly detected)Only color neutral objects can be “seen”(nucleons, mesons, etc)Scale:
ΛQCD = 0.236 GeV
g
du
u d
interaction via gluons.F asymptotic freedom
At high energies: coupling constant decreasesI spin-1 charged bosons (gluons) make vacuum paramagnetic
αs(E) =2π
(11− 23Nf) ln(E/ΛQCD)
[Politzer; Gross and Wilczek - Nobel Prize 2004] Nf : number of quarks.
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AdS/CFT Correspondence with Heat Conduction 2
“A second unexpected connection comes from studies carried out using the Relativistic HeavyIon Collider, a particle accelerator at Brookhaven National Laboratory. This machine smashestogether nuclei at high energy to produce a hot, strongly interacting plasma. Physicists havefound that some of the properties of this plasma are better modeled (via duality) as a tiny blackhole in a space with extra dimensions than as the expected clump of elementary particles inthe usual four dimensions of spacetime. The prediction here is again not a sharp one, as thestring model works much better than expected. String-theory skeptics could take the point ofview that it is just a mathematical spinoff. However, one of the repeated lessons of physics isunity - nature uses a small number of principles in diverse ways. And so the quantum gravitythat is manifesting itself in dual form at Brookhaven is likely to be the same one that operateseverywhere else in the universe.”
– Joe Polchinski
AuAufireball
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AdS/CFT Correspondence with Heat Conduction 3
AdS/CFT correspondence and hydrodynamics[Policastro, Son and Starinets]
correspondence between N = 4 SYM in the large N limit and type-IIB stringtheory in AdS5 × S5.
I in strong coupling limit of field theory, string theory is reduced to classi-cal supergravity, which allows one to calculate all field-theory correlationfunctions.→ nontrivial prediction of gauge theory/gravity correspondence
entropy of N = 4 SYM theory in the limit of large ’t Hooft coupling is precisely3/4 the value in zero coupling limit.long-distance, low-frequency behavior of any interacting theory at finite tem-perature must be described by fluid mechanics (hydrodynamics).universality: hydrodynamics implies very precise constraints on correlation func-tions of conserved currents and stress-energy tensor:I correlators fixed once a few transport coefficients are known.
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AdS/CFT Correspondence with Heat Conduction 4
Hydrodynamicsconserved current: jµ
chemical potential µ = 0, so in thermal equilibrium
〈j0〉 = 0
retarded thermal Green function
GRµν(ω, q) = −i
∫d4x e−iq·x θ(t)〈[jµ(x), jν(0)]〉 ,
where q = (ω, q), x = (t, x)
I determines response to a small external source coupled to the current.ω and q small:
• external perturbation varies slowly in space and time
• macroscopic hydrodynamic description for its evolution is possible.
diffusion equation
∂0j0 = D∇2j0 ,
where D is a diffusion constant with dimension of length.
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AdS/CFT Correspondence with Heat Conduction 5
⇒ overdamped mode, dispersion relation
ω = −iDq2 ,
pole at ω = −iDq2 in the complex ω-plane, in the retarded correlation func-tions of j0
stress-energy tensor Tµν
∂0T00 + ∂iT0i = 0 ,
∂0T0i + ∂jTij = 0 ,
where
T00 = T00 − ρ, ρ = 〈T00〉 ,T ij = T ij − pδij = − 1
ρ + p
[η
(∂iT
0j + ∂jT0i − 2
3δij∂kT0k
)+ζδij∂kT0k
],
ρ (p): energy density (pressure)η (ζ): shear (bulk) viscosity.two types of eigenmodes:
• the shear modes - transverse fluctuations of momentum density T0i, with
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AdS/CFT Correspondence with Heat Conduction 6
a purely imaginary eigenvalue
ω = −iDq2 , D =η
ρ + p,
• sound wave - simultaneous fluctuation of energy density T00 and longitu-dinal component of momentum density T0i, with dispersion relation
ω = usq − i
2
1
ρ + p
(ζ +
4
3η
)q2 , u2
s =∂p
∂ρ.
conformal theory ⇒ stress-energy tensor is traceless, so
ρ = 3p , ζ = 0 , us =1√3
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AdS/CFT Correspondence with Heat Conduction 7
GravityThe non-extremal 3-brane background is a solution of type-IIB low energyequations of motion.In the near-horizon limit r ¿ R, the metric becomes
ds210 =(πTR)2
u
(−f(u)dt2 + dx2 + dy2 + dz2
)+
R2
4u2f(u)du2+R2dΩ2
5 ,
where T = r0πR2 is Hawking temperature, u =
r20r2
, f(u) = 1− u2.
The horizon corresponds to u = 1, spatial infinity to u = 0.gauge theory/gravity correspondence:
• background metric with non-extremality parameter r0 is dual to N = 4SU(N) SYM at finite temperature T in the limit of N →∞, g2
Y MN →∞.
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AdS/CFT Correspondence with Heat Conduction 8
Thermal R-current correlators5d Maxwell action
S = − 1
4g2SG
∫d5x
√−g F aµνFµν a , , g2
SG =16π2R
N2
Fix gauge
Au = 0
residual gauge transformations
Aµ → Aµ + ∂µΛ
where Λ is independent of u.Fourier decomposition (momentum in z-direction)
Aµ =∫
d4q
(2π)4e−iωt+iqzAµ(q, u) .
Dimensionless energy and momentum,
w =ω
2πT, q =
q
2πT,
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AdS/CFT Correspondence with Heat Conduction 9
5d Maxwell equations,
1√−g∂ν[√−ggµρgνσ(∂ρAσ − ∂σAρ)] = 0 ,
reduce to ordinary differential equations
wA′t + qf A′
z = 0 ,
A′′t −
1
uf
(q2At + wqAz
)= 0 ,
A′′z +
f ′
fA′
z +1
uf2
(w2Az + wqAt
)= 0 ,
A′′α +
f ′
fA′
α +1
uf
(w2
f− q2
)Aα = 0 , α = x, y
prime denotes derivative w.r.t. u.I invariant under residual gauge transformations
At → At − ωΛ , Az → Az + qΛ
Deduce
Az =uf
wqA′′
t −qw
At ,
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AdS/CFT Correspondence with Heat Conduction 10
A′′′t +
(uf)′
ufA′′
t +w2 − q2f(u)
uf2A′
t = 0 ,
2nd-order differential equation for A′t. Define
A′t = (1− u)νF (u) , ν = ±iw/2
“incoming wave” boundary condition at horizon singles out ν = −iw/2.
Equation for F (u):
F ′′ +(
1− 3u2
uf+
iw1− u
)F ′ +
iw(1 + 2u)
2ufF +
w2[4− u(1 + u)2]
4uf2F − q2
ufF = 0 .
In long-wavelength, low-frequency limit, w and q are small,
⇒ solution double series in w and q2,
F (u) = F0 + wF1 + q2G1 + w2F2 + wq2H1 + q4G2 + · · · .
F0 = C , F1 =iC
2ln
2u2
1 + u, G1 = C ln
1 + u
2u.
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AdS/CFT Correspondence with Heat Conduction 11
take limit u → 0 assuming boundary conditions
limu→0
At(u) = A0t , lim
u→0Az(u) = A0
z .
C =q2A0
t + wqA0z
iw− q2
C is invariant under the residual gauge transformations.Similarly, we find
Aα =8A0
a(1− u)−iw/2
8− 2iw ln 2 + π2q2
[1 +
iw2
ln1 + u
2
+q2
2
(π2
12+ Li2(−u) + lnu ln(1 + u) + Li2(1− u)
)]+ O
(w2, q4,wq2
).
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AdS/CFT Correspondence with Heat Conduction 12
Green functions
Gabxx = Gab
yy = −iN2Tω δab
16π+ · · · ,
Gabtt =
N2Tq2 δab
16π(iω −Dq2)+ · · · ,
Gabtz = Gab
zt = − N2Tωq δab
16π(iω −Dq2)+ · · · ,
Gabzz =
N2Tω2 δab
16π(iω −Dq2)+ · · · ,
where
D =1
2πT.
correlation functions of j0 and jz contain diffusion poleI expected from hydrodynamics.
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AdS/CFT Correspondence with Heat Conduction 13
compare with weak coupling:I diffusion constant ∝ mean free path
D ∼ 1
(g2YMN)2T ln 1
(g2YMN)
, g2YMN ¿ 1 .
D as a function of the ’t Hooft coupling g2YMN is
D = fD(g2YMN)
1
T,
where fD(x) ∼ 1−x2 lnx
for x ¿ 1 and fD(x) = 12π for x À 1.
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AdS/CFT Correspondence with Heat Conduction 14
Shear Modecompute two-point function of stress-energy tensor in the boundary theoryI consider small perturbation of metric
gµν = g(0)µν + hµν
where g(0)µν is given by
ds25 =π2T2R2
u
(−f(u)dt2 + dx2
)+
R2
4f(u)u2du2 .
Einstein equations:
Rµν = R(0)µν +R(1)
µν + · · · = 2Λ
3gµν ,Λ =
−6
R2
To first order in hµν,
R(1)µν = − 4
R2hµν .
assume perturbation ∼ e−iωt+iqz
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AdS/CFT Correspondence with Heat Conduction 15
fix gauge
huµ = 0
3 classes of perturbations:
• hxy 6= 0, or hxx = −hyy 6= 0;
• hxt and hxz 6= 0, or hyt and hyz 6= 0;
• htz and all diagonal elements of hµν are nonzero, and hxx = hyy
(sound wave).
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AdS/CFT Correspondence with Heat Conduction 16
Perturbation with hxy 6= 0
h′′xy +1− 3u2
ufh′xy +
1
(2πT )2f2u
(f
∂2hxy
∂z2− ∂2hxy
∂t2
)− 1 + u2
fu2hxy = 0
Introduce φ =uhxy
(πTR)2= hx
y
Fourier component φk(u) satisfies eq. for minimally coupled scalar
φ′′k −1 + u2
ufφ′k +
w2 − q2f
uf2φk = 0 .
For incoming wave at horizon,
φk(u) = (1− u)−iw/2Fk(u) ,
where Fk(u) is regular at u = 1 and can be written as a series
Fk(u) = 1− iw2
ln1 + u
2+
w2
8
[(ln
1 + u
2+ 8(1− q2
w2)
)ln
1 + u
2−4 Li2
1− u
2
]+O(w3) ,
retarded Green function
Gµν,λρ(ω, q) = −i∫
d4x e−iq·x θ(t)〈[Tµν(x), Tλρ(0)]〉 .
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AdS/CFT Correspondence with Heat Conduction 17
Deduce
Gxy,xy(ω, q) = −N2T2
16
(i2πTω + q2
).
shear viscosity of strongly coupled N = 4 SYM plasma (Kubo formula)
η = limω→0
1
2ω
∫dt dx eiωt 〈[Txy(x), Txy(0)]〉 =
π
8N2T3 .
NEXT: Calculation involving different components leads to same result!
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AdS/CFT Correspondence with Heat Conduction 18
Perturbation with htx 6= 0, hxz 6= 0
In this channel correlation functions have diffusion pole.
Einstein equations for Fourier components of Ht = uhtx(πTR)2
, Hz = uhzx(πTR)2
:
H ′t +
qfw
H ′z = 0 ,
H ′′t −
1
uH ′
t −wquf
Hz − q2
ufHt = 0 ,
H ′′z −
1 + u2
ufH ′
z +w2
uf2Hz +
wquf2
Ht = 0 .
Deduce
Hz =uf
wqH ′′
t −f
wqH ′
t −qw
Ht .
H ′′′t −
2u
fH ′′
t +2uf − q2f + w2
uf2H ′
t = 0 .
In the low-frequency, long-wavelength limit, solve as before...
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AdS/CFT Correspondence with Heat Conduction 19
H ′t(u) = (1− u)−iw/2G(u)
G′′ −(
2u
f− iw
1− u
)G′ +
1
f
(2 +
iw2− q2
u+
w2[4− u(1 + u)2]
4uf
)G = 0 .
G(u) = C
[u− iw
(1− u− u
2ln
1 + u
2
)+
q2(1− u)
2
]+ O
(w2,wq2, q4
).
C =q2H0
t + qwH0z
iw− q2
2
.
Deduce correlators
Gtx,tx(ω, q) =N2πT 3q2
8(iω −Dq2)+O(w2,wq2, q4) ,
Gtx,xz(ω, q) = − N2πT 3ωq
8(iω −Dq2)+O(w2,wq2, q4) ,
Gxz,xz(ω, q) =N2πT 3ω2
8(iω −Dq2)+O(w2,wq2, q4) ,
where D = 14πT
I half the diffusion constant for the R-charge.
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AdS/CFT Correspondence with Heat Conduction 20
Deduce η:I recall from hydrodynamics D = η
ρ+p.
Entropy:
s =3
4s0 =
π2
2N2T3 ,
where s0 is entropy at zero coupling.
From s = ∂P∂T , ρ = 3p, deduce ρ + p = π2
2 N2T4, ∴
η =π
8N2T3 ,
η
s=
1
4π
I agrees with Kubo formula.
I no agreement unless s = 34s0.
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AdS/CFT Correspondence with Heat Conduction 21
behavior of η as a function of the ’t Hooft coupling
η = fη(g2YMN)N2T3
where fη(x) ∼ 1−x2 lnx
for x ¿ 1 and fη(x) = π8 for x À 1.
I At weak coupling,
η
sÀ 1
4π
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AdS/CFT Correspondence with Heat Conduction 22
Chemical potential3-Charged black hole:
ds2 = −H−2/3f(r)dt2 + H1/3(
dr2
f(r)+ r2dx2
)
where
f(r) = −m
r2+ r2H , H = (1 + q21/r2)(1 + q22/r2)(1 + q23/r2)
[Behrnd, Cvetic and Sabra]
I interesting thermodynamic properties (Cardy-Verlinde entropy formula, etc)[Klemm, Petkou, G.S. and Zanon]
Entropy density:
s =N2r3H2π
√H(rH)
Chemical potentials (i = 1,2,3)
µi =
√2qi
1 + qi/r2H
√H(rH)
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AdS/CFT Correspondence with Heat Conduction 23
Kubo formula for shear viscosity
η =N2r3H8π2
√H(rH)
so η/s = 14π !
I also agrees with η from diffusion pole![Son and Starinets]
For single charge (q2 = q3 = 0), Kubo formula for thermal conductivity:
κT = πN2T2H
(1 + q1/r2H)2
(2 + q1/r2H)q1/r2H
Wiedemann-Franz law:
κTµ2
ηTH= 8π2
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AdS/CFT Correspondence with Heat Conduction 24
AdS/CFT Correspondence with Heat ConductionGOAL: Understand flowI static black holes not sufficientI try Bjorken flowPerfect Fluid [Janik and Peschanski]
introduce proper time τ and rapidity y on longitudinal plane,
x0 = τ cosh y , x1 = τ sinh y
transverse coordinates: x⊥ = (x2, x3).
Let z = R2/r denote the fifth dimension.Einstein equations in the bulk
RAB −1
2gABR− 6gAB = 0
where the cosmological constant is Λ = −6.I metric ansatz
ds2 =1
z2
[−ea(τ,z)dτ2 + τ2eb(τ,z)dy2 + ec(τ,z)dx⊥2 + dz2
]
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AdS/CFT Correspondence with Heat Conduction 25
most general bulk metric obeying
• boost invariance
• symmetry under reflection in the longitudinal direction (y → −y)
• translational and rotational invariance.
introduce coordinate
v =z
τη
leading behavior of solution in the τ →∞ limit
a(v) = A(v)− 2m(v)b(v) = A(v) + 2(4η − 1)m(v)c(v) = A(v) + 2(1− 2η)m(v)
where
A(v) =1
2
[ln(1 + ∆(η)v4) + ln(1−∆(η)v4)
]
m(v) =1
4∆(η)
[ln(1 + ∆(η)v4)− ln(1−∆(η)v4)
]
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AdS/CFT Correspondence with Heat Conduction 26
with
∆(η) =
√1
3(6η2 − 4η + 1)
Demand regularity in the bulk,
I R2 = RµναβRµναβ non-singular
I unique value
η =1
3Holographic Renormalizationrelate bulk metric to the vacuum expectation value of the stress-energy tensorof the corresponding gauge theory on the boundaryI metric is of the general asymptotically AdS form
ds2 =gµνdxµdxν + dz2
z2
Near the boundary at z = 0 expand
gµν = g(0)µν + z2g
(2)µν + z4g
(4)µν + . . .
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AdS/CFT Correspondence with Heat Conduction 27
where g(0)µν = ηµν, g
(2)µν = 0,
〈Tµν〉 ∝ g(4)µν
from explicit form of g(4)µν , for η = 1/3,
⇒ perfect fluid
Tαβ = (ρ + p)uαuβ + pηαβ
equation of state p = 13ρ
∵ tracelessness due to conformal invariance
mass density and temperature fall off, respectively, as
ρ ∼ 1
τ4/3, T ∼ 1
τ1/3
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AdS/CFT Correspondence with Heat Conduction 28
An extension of the bulk metric [Alsup, Middleton, G.S.]
I relax requirement of invariance under reflection in longitudinal direction.I add appropriate off-diagonal terms in the bulk metric
ds2 = ds2perfect fluid+2τλh||(v)dτdy + 2τσh⊥(v)dτdx⊥
• exponents λ, σ < 0 of τ
• 3 functions: 1 longitudinal h||(v), 2 transverse h⊥(v) = (h2(v), h3(v))
to be determined by the Einstein equations:
¦ mixed transverse and two yx⊥ components are, respectively,
(vh′2 + (2− vc′)h2)(vh′3 + (2− vc′)h3) = 0
(vh′|| + (2− vb′)h||)(vh′⊥ + (2− vc′)h⊥) = 0
exact (no higher-order corrections)! solution:
h⊥(v) =ec(v)
v2
up to overall arbitrary multiplicative constant→ singular function (h⊥(v) ∼ v−2 as v → 0)
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AdS/CFT Correspondence with Heat Conduction 29
→ discard
h⊥(v) = 0
¦ two x⊥τ components yield
(2vh′′2 + (2 + vb′ − vc′)h′2)h3 − (2 ↔ 3) = 0
→ trivially solved.→ unnecessary to invoke regularity; condition of isotropy (h2 = h3) suf-
fices.
¦ vy and yτ components are, respectively,
a0(v)h||(v) + a1(v)h′||(v) + h′′||(v) +O(τ−2+4η+2λ) = 0
b0(v)h||(v) + b1(v)h′||(v) + h′′||(v) +O(τ−2+4η+2λ) = 0
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AdS/CFT Correspondence with Heat Conduction 30
where
a0 = −2(1 + λ)
ηv2+−a′ + 1+λ
ηb′ + 2c′
v+
1
2(a′ − b′ − 2c′)b′ − b′′
a1 =3η − λ− 1
ηv− 1
2(a′ + b′ − 2c′)
b0 = − 4
v2+
2(a′ + b′ + 4c′)v
− 1
2(a′2 − a′b′ + b′2)− a′c′ − b′c′ − 3
2c′2 − a′′ − b′′ − 2c′′
b1 =1
v2− 1
2(a′ + b′ − 2c′)
For consistency (a1 = b1), choose
λ = 2η − 1
for consistency of the expansion, we need
η <1
2
[for which τ has negative exponent, −2 + 4η + 2λ = −4 + 8η < 0]We need to ensure a0 = b0, as well.I To this end, we need to determine a, b, c from:
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AdS/CFT Correspondence with Heat Conduction 31
¦ diagonal components
6b′ − vb′2 + 12c′ − 2vb′c′ − 3vc′2 − 2vb′′ − 4vc′′ +O(τ−2+2η, τ−4+8η) = 0−6a′ + va′2 − 12c′ + 2va′c′ + 3vc′2 + 2va′′ + 4vc′′ +O(τ−2+2η, τ−4+8η) = 0
−6a′ + va′2 − 6b′ + va′b′ + vb′2 − 6c′ + va′c′ + vb′c′ + vc′2 + 2va′′ + 2vb′′ + 2vc′′
+O(τ−2+2η, τ−4+8η) = 0−6a′ − 6b′ + va′b′ − 12c′ + 2va′c′ + 2vb′c′ + vc′2 +O(τ−2+2η, τ−4+8η) = 0
¦ and vτ component,
2a′−6ηa′−2b′−4ηb′+vηb′2−8ηc′+2vηb′c′+3vηc′2+2vηb′′+4vηc′′+O(τ−2+2η, τ−4+8η) = 0
I To leading order, identical to perfect fluid case⇒ same a, b, c (satisfy a0 = b0).
¦ rest of eqs. coalesce, yielding an ODE which uniquely determines h||(v) atleading order,
h||(v) =1
v2
[1−∆v4
1 + ∆v4
](1
2−η)/∆ (
1−∆2v8)1
2− η
∆
[(1−∆v4)2η/∆ − (1 + ∆v4)2η/∆
]
we discarded singular solution (behaving ∼ v−2 as v → 0).expand
h||(v) ∼ v2 + (2η − 1)v6 + . . .
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AdS/CFT Correspondence with Heat Conduction 32
up to an irrelevant overall constant factor.For η = 1/3, reduces to
h||(v) =v2
1 + 13v4
special value of η derived by demanding regularity of R2 = RµναβRµναβ
- unchangedFinally, our metric reads
ds2 =1
z2
[−
(1− e
3z4
τ 4/3
)2
1 + e3
z4
τ 4/3
dτ2 +(1 + e
3z4
τ 4/3
)(τ2dy2 + dx2
⊥) + dz2 +2Az4
1 + e3
z4
τ 4/3
dydτ
τ
]
A is an arbitrary constant characterizing the departure from the symmetricstate giving rise to perfect fluid hyrdodynamics on the boundary.
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AdS/CFT Correspondence with Heat Conduction 33
HydrodynamicsFrom holographic renormalization
〈Tyτ〉 ∼ g(4)yτ ∼ 1
τ
In order to understand how our solution relates to the gauge theory fluid,choose stress-energy tensor with arbitrary energy flux,
Tµν =
T ττ T τy T τ2 T τ3
T τy T yy 0 0T τ2 0 T22 0T τ3 0 0 T33
.
Minkowski space metric
ds24 = −dτ2 + τ2dy2 + (dx⊥)2
From local conservation law
∇αTαβ = ∂αTαβ + ΓααλTλβ + Γβ
αλTαλ = 0
where Γyyτ = 1
τ = Γyτy and Γτ
yy = τ ,
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AdS/CFT Correspondence with Heat Conduction 34
♥ for β = τ ,
∂τT ττ + ∂yT τy +1
τT ττ + τT yy + ∂⊥T τ⊥ = 0
♥ for β = y,
∂τT τy + ∂yT yy +3
τT τy = 0
♥ for transverse components, β = i (i = 2,3),
∂τT τi +1
τT τi + ∂jT
ij = 0
♥ from conformal invariance (tracelessness)
−T ττ + τ2T yy + T22 + T33 = 0
match with bulk metric through holographic renormalization:
→ stress-energy tensor to the order we are considering should not dependon rapidity (y) or transverse coordinates (x⊥).
→ Solution:
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AdS/CFT Correspondence with Heat Conduction 35
T τy =Cτ3
, Tτy =−Cτ
T τ⊥ =C⊥τ
, Tτ⊥ =−C⊥
τ
C, C⊥ = (C2, C3) arbitrary constants.I matches prediction of gravity dual for T τy.I stress-energy tensor in terms of ρ = T ττ (energy density) and the arbitraryconstants C, C⊥ (determining energy flux),
Tµν =
ρ Cτ3
C2τ
C3τC
τ3 −1τ ∂τρ− 1
τ2ρ 0 0C2τ 0 ρ + 1
2τ∂τρ 0C3τ 0 0 ρ + 1
2τ∂τρ
.
demanding isotropy, τ2T yy = T22 = T33,
ρ ∼ 1
τ4/3
matching perfect fluid behavior.
I equation of state also unchanged, p = 13ρ.
George Siopsis Miami - 12/14/06
AdS/CFT Correspondence with Heat Conduction 36
For the energy flux, we obtain velocity components
uy ∼ τT τy
T ττ∼ 1
τ2/3, u⊥ ∼ T τ⊥
T ττ∼ τ1/3
transverse components grow with time (unstable solution). ∴ ought to set
C⊥ = 0
matches expected behavior from the gravity dual
I corresponding off-diagonal components of the metric were set to zero bydemanding regularity of the solution to the Einstein equations.
Summarizing,
T ττ =B
τ4/3, Tττ =
Bτ4/3
T yy =B
3τ10/3, Tyy =
B3
τ2/3
T ii =B
3τ4/3, Tii =
B3τ4/3
(i = 2,3)
T τy =Cτ3
, Tτy =−Cτ
George Siopsis Miami - 12/14/06
AdS/CFT Correspondence with Heat Conduction 37
To gain further insight into the nature of this fluid, consider the general caseof a dissipative relativistic fluid with stress-energy tensor and particle current,respectively,
Tαβ = (ρ + p)uαuβ + pηαβ + tαβ , Jα = nuα + jα
n is the particle density (for a perfect fluid, Jα = nuα)dissipative pieces satisfy
uαtαβ = uαjα = 0
Standard thermodynamic arguments using the relations
ρ + p = Ts + µn , dρ = Tds + µdn
lead to the general expressions
tαβ = −η[∇αuβ + uαuγ∇γuβ + (α ↔ β)]−(ζ − 2
3η
)[ηαβ + uαuβ]∇γuγ
jα = −κ(∂α + uαuβ∂β)µ
T
in terms of the coefficient of thermal conductivity κ and shear and bulk viscosi-ties, η and ζ, respectively.
George Siopsis Miami - 12/14/06
AdS/CFT Correspondence with Heat Conduction 38
Assuming no particle transport, ~J = ~0 (vectors in 3d),
n~u = κ(~∇+ ~uuβ∂β)µ
T
This can be solved as an expansion in the derivatives of µ/T ,
~u =κn
~∇µ
T+ . . .
justified because local velocities are small (|~u| ¿ 1).To match the desired behavior of the stress-energy tensor, assume no viscosity,
η = ζ = 0
energy flow in the longitudinal direction, ∴
u⊥ = 0
energy flux in the longitudinal direction
T τy = (ρ + p)uτuy
to leading order,
T τy =4
3ρuy + · · · = 4κρ
3nτ2∂y
µ
T+ . . .
George Siopsis Miami - 12/14/06
AdS/CFT Correspondence with Heat Conduction 39
gradient of chemical potential to leading order,
∂yµ
T=
3nC4κBτ1/3 + . . .
from the Gibbs-Duhem relation
dp = sdT + ndµ
we obtain the temperature gradient
∂yµ
T=
ρ + p
n∂y
1
T=
4B3nτ4/3
∂y1
T
∴ the temperature gradient is
∂y1
T=
9n2C16κB2
τ5/3 + . . .
energy flux
Tτy = κT∂yT
where κT is the standard definition of thermal conductivity in non-relativisticmechanics.
George Siopsis Miami - 12/14/06
AdS/CFT Correspondence with Heat Conduction 40
We obtain
κT = κ(
ρ + p
nT
)2
To leading order,
κT =16κB2
9n2T2τ8/3
¦ ¦ ¦I to add viscosity consider next perturbative order.
[Nakamura and Sin; Janik]
I Elliptic flow is also possible through modified metric ansatz.[Sin, Nakamura and Kim]
George Siopsis Miami - 12/14/06
AdS/CFT Correspondence with Heat Conduction 41
CONCLUSIONS
• We are beginning to understand non-perturbative QCD effects using stringtheory
• RHIC’s fireball can be described by a dual black hole
• RHIC and LHC may probe black holes and provide information on stringtheory as well as non-perturbative QCD effects.
George Siopsis Miami - 12/14/06