Viscosity, Black Holes, and Quantum Field Theory
INT PUB 07-02
Viscosity, Black Holes, and Quantum Field Theory
Dam T. Son
Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA
Andrei O. Starinets
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
Key Words AdS/CFT correspondence, hydrodynamics
Abstract We review recent progress in applying the AdS/CFT correspondence to finite-temperature field theory.
In particular, we show how the hydrodynamic behavior of field theory is reflected in the low-momentum limit of
correlation functions computed through a real-time AdS/CFT prescription, which we formulate. We also show how
the hydrodynamic modes in field theory correspond to the low-lying quasinormal modes of the AdS black p-brane
metric. We provide a proof of the universality of the viscosity/entropy ratio within a class of theories with gravity
duals and formulate a viscosity bound conjecture. Possible implications for real systems are mentioned.
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
HYDRODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Kubo’s Formula For Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Hydrodynamic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Viscosity In Weakly Coupled Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
AdS/CFT CORRESPONDENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Review Of AdS/CFT Correspondence At Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Black Three-Brane Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
REAL-TIME AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Prescription For Retarded Two-Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Calculating Hydrodynamic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
THE MEMBRANE PARADIGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
THE VISCOSITY/ENTROPY RATIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
The Viscosity Bound Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
http://arXiv.org/abs/0704.0240v2
2 Son, Starinets
1 INTRODUCTION
This review is about the recently emerging connection, through the gauge/gravity correspondence,
between hydrodynamics and black hole physics.
The study of quantum field theory at high temperature has a long history. It was first motivated
by the Big Bang cosmology when it was hoped that early phase transitions might leave some
imprints on the Universe [1]. One of those phase transitions is the QCD phase transitions (which
could actually be a crossover) which happened at a temperature around Tc ∼ 200 MeV, when
matter turned from a gas of quarks and gluons (the quark-gluon plasma, or QGP) into a gas of
hadrons.
An experimental program was designed to create and study the QGP by colliding two heavy
atomic nuclei. Most recent experiments are conducted at the Relativistic Heavy Ion Collider (RHIC)
at Brookhaven National Laboratory. Although significant circumstantial evidence for the QGP
was accumulated [2], a theoretical interpretation of most of the experimental data proved difficult,
because the QGP created at RHIC is far from being a weakly coupled gas of quarks and gluons.
Indeed, the temperature of the plasma, as inferred from the spectrum of final particles, is only
approximately 170 MeV, near the confinement scale of QCD. This is deep in the nonperturbative
regime of QCD, where reliable theoretical tools are lacking. Most notably, the kinetic coefficients
of the QGP, which enter the hydrodynamic equations (reviewed in Sec. 2), are not theoretically
computable at these temperatures.
The paucity of information about the kinetic coefficients of the QGP in particular and of strongly
coupled thermal quantum field theories in general is one of the main reasons for our interest
in their computation in a class of strongly coupled field theories, even though this class does
not include QCD. The necessary technological tool is the anti–de Sitter–conformal field theory
(AdS/CFT) correspondence [3, 4, 5], discovered in the investigation of D-branes in string theory.
This correspondence allows one to describe the thermal plasmas in these theories in terms of black
holes in AdS space. The AdS/CFT correspondence is reviewed in Sec. 3.
The first calculation of this type, that of the shear viscosity in N = 4 supersymmetric Yang-
Mills (SYM) theory [6], is followed by the theoretical work to establish the rules of real-time
finite-temperature AdS/CFT correspondence [7, 8]. Applications of these rules to various special
cases [9, 10, 11, 12] clearly show that even very exotic field theories, when heated up to finite
temperature, behave hydrodynamically at large distances and time scales (provided that the number
of space-time dimensions is 2+1 or higher). This development is reviewed in Sec. 4. Moreover, the
way AdS/CFT works reveals deep connections to properties of black holes in classical gravity. For
example, the hydrodynamic modes of a thermal medium are mapped, through the correspondence,
to the low-lying quasi-normal modes of a black-brane metric. It seems that our understanding of the
connection between hydrodynamics and black hole physics is still incomplete; we may understand
more about gravity by studying thermal field theories. One idea along this direction is reviewed in
Sec. 5.
From the point of view of heavy-ion (QGP) physics, a particularly interesting finding has been
the formulation of a conjecture on the lowest possible value of the ratio of viscosity and volume
density of entropy. This conjecture was motivated by the universality of this ratio in theories with
gravity duals. This is reviewed in Sec. 6.
Viscosity, Black Holes, and QFT 3
This review is written primarily for readers with a background in QCD and QGP physics who are
interested in learning about AdS/CFT correspondence and its applications to finite-temperature
field theory. Some parts of this review (for example, the section about hydrodynamics) should be
useful for readers with a string theory or general relativity background who are interested in the
connection between string theory, gravity, and hydrodynamics. The perspectives here are shaped
by our personal taste and therefore may appear narrow, but the authors believe that this review
may serve as the starting point to explore the much richer original literature.
In this review we use the “mostly plus” metric signature − + ++.
2 HYDRODYNAMICS
From the modern perspective, hydrodynamics [13] is best thought of as an effective theory, describ-
ing the dynamics at large distances and time-scales. Unlike the familiar effective field theories (for
example, the chiral perturbation theory), it is normally formulated in the language of equations of
motion instead of an action principle. The reason for this is the presence of dissipation in thermal
media.
In the simplest case, the hydrodynamic equations are just the laws of conservation of energy and
momentum,
µν = 0 . (1)
To close the system of equations, we must reduce the number of independent elements of T µν .
This is done through the assumption of local thermal equilibrium: If perturbations have long wave-
lengths, the state of the system, at a given time, is determined by the temperature as a function
of coordinates T (x) and the local fluid velocity uµ, which is also a function of coordinates uµ(x).
Because uµu
µ = −1, only three components of uµ are independent. The number of hydrodynamic
variables is four, equal to the number of equations.
In hydrodynamics we express T µν through T (x) and uµ(x) through the so-called constitutive
equations. Following the standard procedure of effective field theories, we expand in powers of
spatial derivatives. To zeroth order, T µν is given by the familiar formula for ideal fluids,
T µν = (ǫ + P )uµuν + Pgµν , (2)
where ǫ is the energy density, and P is the pressure. Normally one would stop at this leading
order, but qualitatively new effects necessitate going to the next order. Indeed, from Eq. 2 and the
thermodynamic relations dǫ = TdS, dP = sdT , and ǫ+P = Ts (s is the entropy per unit volume),
one finds that entropy is conserved [14]
∂µ(su
µ) = 0 . (3)
Thus, to have entropy production, one needs to go to the next order in the derivative expansion.
At the next order, we write
T µν = (ǫ + P )uµuν + Pgµν − σµν , (4)
where σµν is proportional to derivatives of T (x) and uµ(x) and is termed the dissipative part of
T µν . To write these terms, let us first fix a point x and go to the local rest frame where ui(x) = 0.
4 Son, Starinets
In this frame, in principle one can have dissipative corrections to the energy-momentum density
T 0µ. However, one recalls that the choice of T and uµ is arbitrary, and thus one can always redefine
them so that these corrections vanish, σ00 = σ0i = 0, and so at a point x,
T 00 = ǫ, T 0i = 0 . (5)
The only nonzero elements of the dissipative energy-momentum tensor are σij. To the next-to-
leading order there are extra contributions whose forms are dictated by rotational symmetry:
σij = η
∂iuj + ∂jui −
δij∂ku
+ ζδij∂ku
k . (6)
Going back to the general frame, we can now write the dissipative part of the energy-momentum
tensor as
σµν = PµαP νβ
∂αuβ + ∂βuα −
gαβ∂λu
+ ζgαβ∂λu
, (7)
where Pµν = gµν + uµuν is the projection operator onto the directions perpendicular to uµ.
If the system contains a conserved current, there is an additional hydrodynamic equation related
to the current conservation,
µ = 0 . (8)
The constitutive equation contains two terms:
jµ = ρuµ − DPµν∂να , (9)
where ρ is the charge density in the fluid rest frame and D is some constant. The first term
corresponds to convection, the second one to diffusion. In the fluid rest frame, j = −D∇ρ, which
is Fick’s law of diffusion, with D being the diffusion constant.
2.1 Kubo’s Formula For Viscosity
As mentioned above, the hydrodynamic equations can be thought of as an effective theory describing
the dynamics of the system at large lengths and time scales. Therefore one should be able to use
these equations to extract information about the low-momentum behavior of Green’s functions in
the original theory.
For example, let us recall how the two-point correlation functions can be extracted. If we couple
sources Ja(x) to a set of (bosonic) operators Oa(x), so that the new action is
S = S0 +
Ja(x)Oa(x) , (10)
then the source will introduce a perturbation of the system. In particular, the average values
of Oa will differ from the equilibrium values, which we assume to be zero. If Ja are small, the
perturbations are given by the linear response theory as
〈Oa(x)〉 = −
GRab(x − y)Jb(y) , (11)
where GRab is the retarded Green’s function
iGRab(x − y) = θ(x0 − y0)〈[Oa(x), Ob(y)]〉 . (12)
Viscosity, Black Holes, and QFT 5
The fact that the linear response is determined by the retarded (and not by any other) Green’s
function is obvious from causality: The source can influence the system only after it has been
turned on.
Thus, to determine the correlation functions of T µν , we need to couple a weak source to T µν and
determine the average value of T µν after this source is turned on. To find these correlators at low
momenta, we can use the hydrodynamic theory. So far in our treatment of hydrodynamics we have
included no source coupled to T µν . This deficiency can be easily corrected, as the source of the
energy-momentum tensor is the metric gµν . One must generalize the hydrodynamic equations to
curved space-time and from it determine the response of the thermal medium to a weak perturbation
of the metric. This procedure is rather straightforward and in the interest of space is left as an
exercise to the reader.
Here we concentrate on a particular case when the metric perturbation is homogeneous in space
but time dependent:
gij(t,x) = δij + hij(t), hij ≪ 1 (13)
g00(t,x) = −1, g0i(t,x) = 0 . (14)
Moreover, we assume the perturbation to be traceless, hii = 0. Because the perturbation is spatially
homogeneous, if the fluid moves, it can only move uniformly: ui = ui(t). However, this possibility
can be ruled out by parity, so the fluid must remain at rest all the time: uµ = (1, 0, 0, 0). We now
compute the dissipative part of the stress-energy tensor. The generalization of Eq. 7 to curved
space-time is
σµν = PµαP νβ
η(∇αuβ + ∇βuα) +
ζ − 2
gαβ∇ · u
. (15)
Substituting uµ = (1, 0, 0, 0) and gµν from Eq. 13, we find only contributions to the traceless spatial
components, and these contributions come entirely from the Christoffel symbols in the covariant
derivatives. For example,
σxy = 2ηΓ
xy = η∂0hxy . (16)
By comparison with the expectation from the linear response theory, this equation means that we
have found the zero spatial momentum, low-frequency limit of the retarded Green’s function of
T xy:
GRxy,xy(ω,0) =
dt dx eiωtθ(t)〈[Txy(t,x), Txy(0,0)]〉 = −iηω + O(ω2) (17)
(modulo contact terms). We have, in essence, derived the Kubo’s formula relating the shear viscosity
and a Green’s function:
η = − lim
ImGRxy,xy(ω,0) . (18)
There is a similar Kubo’s relation for the charge diffusion constant D.
2.2 Hydrodynamic Modes
If one is interested only in the locations of the poles of the correlators, one can simply look for the
normal modes of the linearized hydrodynamic equations, that is, solutions that behave as e−iωt+ik·x.
Owing to dissipation, the frequency ω(k) is complex. For example, the equation of charge diffusion,
∂tρ − D∇2ρ = 0, (19)
6 Son, Starinets
corresponds to a pole in the current-current correlator at ω = −iDk2.
To find the poles in the correlators between elements of the stress-energy tensor one can, without
loss of generality, choose the coordinate system so that k is aligned along the x3-axis: k = (0, 0, k).
Then one can distinguish two types of normal modes:
1. Shear modes correspond to the fluctuations of pairs of components T 0a and T 3a, where a =
1, 2. The constitutive equation is
T 3a = −η∂3ua = −
ǫ + P
0a , (20)
and the equation for T 0a is
0a − η
ǫ + P
0a = 0 . (21)
That is, it has the form of a diffusion equation for T 0a. Substituting e−iωt+ikx
into the
equation, one finds the dispersion law
ω = −i η
ǫ + P
k2 . (22)
2. Sound modes are fluctuations of T 00, T 03, and T 33. There are now two conservation equations,
and by diagonalizing them one finds the dispersion law
ω = csk −
η + ζ
ǫ + P
, (23)
where cs =
dP/dǫ. This is simply the sound wave, which involves the fluctuation of the en-
ergy density. It propagates with velocity cs, and its damping is related to a linear combination
of shear and bulk viscosities.
In CFTs it is possible to use conformal Ward identities to show that the bulk viscosity vanishes:
ζ = 0. Hence, we shall concentrate our attention on the shear viscosity η.
2.3 Viscosity In Weakly Coupled Field Theories
We now briefly consider the behavior of the shear viscosity in weakly coupled field theories, with
the λφ4 theory as a concrete example. At weak coupling, there is a separation between two length
scales: The mean free path of particles is much larger than the distance scales over which scatterings
occur. Each scattering event takes a time of order T−1 (which can be thought of as the time required
for final particles to become on-shell). The mean free path ℓmfp can be estimated from the formula
ℓmfp ∼
, (24)
where n is the density of particles, σ is the typical scattering cross section, and v is the typical
particle velocity. Inserting the values for thermal λφ4 theory, n ∼ T 3, σ ∼ λ2T−2, and v ∼ 1, one
finds
ℓmfp ∼
. (25)
The viscosity can be estimated from kinetic theory to be
η ∼ ǫℓmfp , (26)
Viscosity, Black Holes, and QFT 7
where ǫ is the energy density. From ǫ ∼ T 4 and the estimate of ℓmft, one finds
η ∼ T
. (27)
In particular, the weaker the coupling λ, the larger the viscosity η. This behavior is explained by
the fact that the viscosity measures the rate of momentum diffusion. The smaller λ is, the longer
a particle travels before colliding with another one, and the easier the momentum transfer.
It may appear counterintuitive that viscosity tends to infinity in the limit of zero coupling λ → 0:
At zero coupling there is no dissipation, so should the viscosity be zero? The confusion arises owing
to the fact that the hydrodynamic theory, and hence the notion of viscosity, makes sense only on
distances much larger than the mean free path of particles. If one takes λ → 0, then to measure
the viscosity one has to do the experiment at larger and larger length scales. If one fixes the size
of the experiment and takes λ → 0, dissipation disappears, but it does not tell us anything about
the viscosity.
As will become apparent below, a particularly interesting ratio to consider is the ratio of shear
viscosity and entropy density s. The latter is proportional to T 3; thus
. (28)
One has η/s ≫ 1 for λ ≪ 1. This is a common feature of weakly coupled field theories. Extrapolat-
ing to λ ∼ 1, one finds η/s ∼ 1. We shall see that theories with gravity duals are strongly coupled,
and η/s is of order one. More surprisingly, this ratio is the same for all theories with gravity duals.
To compute rather than estimate the viscosity, one can use Kubo’s formula. It turns out that
one has to sum an infinite number of Feynman graphs to even find the viscosity to leading order.
Another way that leads to the same result is to first formulate a kinetic Boltzmann equation for the
quasi-particles as an intermediate effective description, and then derive hydrodynamics by taking
the limit of very long lengths and time scales in the kinetic equation. Interested readers should
consult Refs. [15,16] for more details.
3 AdS/CFT CORRESPONDENCE
3.1 Review Of AdS/CFT Correspondence At Zero Temperature
This section briefly reviews the AdS/CFT correspondence at zero temperature. It contains only the
minimal amount of materials required to understand the rest of the review. Further information
can be found in existing reviews and lecture notes [17,18].
The original example of AdS/CFT correspondence is between N = 4 supersymmetric Yang-Mills
(SYM) theory and type IIB string theory on AdS5×S5 space. Let us describe the two sides of the
correspondence in some more detail.
The N = 4 SYM theory is a gauge theory with a gauge field, four Weyl fermions, and six real
scalars, all in the adjoint representation of the color group. Its Lagrangian can be written down
explicitly, but is not very important for our purposes. It has a vanishing beta function and is a
conformal field theory (CFT) (thus the CFT in AdS/CFT). In our further discussion, we frequently
use the generic terms “field theory” or CFT for the N = 4 SYM theory.
8 Son, Starinets
On the string theory side, we have type IIB string theory, which contains a finite number of
massless fields, including the graviton, the dilaton Φ, some other fields (forms) and their fermionic
superpartners, and an infinite number of massive string excitations. It has two parameters: the
string length ls (related to the slope parameter α
′ by α′ = l2s) and the string coupling gs. In the
long-wavelength limit, when all fields vary over length scales much larger than ls, the massive modes
decouple and one is left with type IIB supergravity in 10 dimensions, which can be described by
an action [19]
SSUGRA =
2κ210
−g e−2Φ (R + 4 ∂µΦ∂µΦ + · · ·) , (29)
where κ10 is the 10-dimensional gravitational constant,
κ10 =
8πG = 8π7/2gsl
s , (30)
and · · · stay for the contributions from fields other than the metric and the dilaton. One of these
fields is the five-form F5, which is constrained to be self-dual. The type IIB string theory lives is a
10-dimensional space-time with the following metric:
ds2 =
(−dt2 + dx2) + R
dr2 + R2dΩ25 . (31)
The metric is a direct product of a five-dimensional sphere (dΩ25) and another five-dimensional
space-time spanned by t, x, and r. An alternative form of the metric is obtained from Eq. (31) by
a change of variable z = R2/r,
ds2 =
(−dt2 + dx2 + dz2) + R2dΩ25 . (32)
Both coordinates r and z are known as the radial coordinate. The limiting value r = ∞ (or z = 0)
is the boundary of the AdS space.
It is a simple exercise to check that the (t,x, r) part of the metric is a space with constant
negative curvature, or an anti de-Sitter (AdS) space. To support the metric (31) (i.e., to satisfy the
Einstein equation) there must be some background matter field that gives a stress-energy tensor in
the form of a negative cosmological constant in AdS5 and a positive one in S
5. Such a field is the
self-dual five-form field F5 mentioned above.
Field theory has two parameters: the number of colors N and the gauge coupling g. When the
number of colors is large, it is the ’t Hooft coupling λ = g2N that controls the perturbation theory.
On the string theory side, the parameters are gs, ls, and radius R of the AdS space. String theory
and field theory each have two dimensionless parameters which map to each other through the
following relations:
g2 = 4πgs, (33)
g2Nc =
. (34)
Equation (33) tells us that, if one wants to keep string theory weakly interacting, then the gauge
coupling in field theory must be small. Equation (34) is particularly interesting. It says that the
large ’t Hooft coupling limit in field theory corresponds to the limit when the curvature radius of
Viscosity, Black Holes, and QFT 9
space-time is much larger than the string length ls. In this limit, one can reliably decouple the
massive string modes and reduce string theory to supergravity. In the limit gs ≪ 1 and R ≫ ls,
one has classical supergravity instead of string theory. The practical utility of the AdS/CFT
correspondence comes, in large part, from its ability to deal with the strong coupling limit in gauge
theory.
One can perform a Kaluza-Klein reduction [20] by expanding all fields in S5 harmonics. Keeping
only the lowest harmonics, one finds a five-dimensional theory with the massless dilaton, SO(6)
gauge bosons, and gravitons [21]:
S5D =
8π2R3
R5D − 2Λ −
∂µΦ∂µΦ −
F aµνF
aµν + · · ·
. (35)
In AdS/CFT, an operator O of field theory is put in a correspondence with a field φ (“bulk” field)
in supergravity. We elaborate on this correspondence below; here we keep the operator and the field
unspecified. In the supergravity approximation, the mathematical statement of the correspondence
Z4D[J ] = e
iS[φcl] . (36)
On the left is the partition function of a field theory, where the source J coupled to the operator
O is included:
Z4D[J ] =
Dφ exp
iS + i
d4xJO
. (37)
On the right, S[φcl] is the classical action of the classical solution φcl to the field equation with the
boundary condition:
φcl(z, x)
= J(x) . (38)
Here ∆ is a constant that depends on the nature of the operator O (namely, on its spin and
dimension). In the simplest case, ∆ = 0, and the boundary condition becomes φcl(z=0) = J .
Differentiating Eq. (36) with respect to J , one can find the correlation functions of O. For example,
the two-point Green’s function of O is obtained by differentiating Scl[φ] twice with respect to the
boundary value of φ,
G(x − y) = −i〈TO(x)O(y)〉 = − δ
2S[φcl]
δJ(x)δJ(y)
φ(z=0)=J
. (39)
The AdS/CFT correspondence thus maps the problem of finding quantum correlation functions in
field theory to a classical problem in gravity. Moreover, to find two-point correlation functions in
field theory, one can be limited to the quadratic part of the classical action on the gravity side.
The complete operator to field mapping can be found in Refs. [5, 17]. For our purpose, the
following is sufficient:
• The dilaton Φ corresponds to O = −L = 1
F 2µν + · · ·, where L is the Lagrangian density.
• The gauge field Aaµ corresponds to the conserved R-charge current Jaµ of field theory.
• The metric tensor corresponds to the stress-energy tensor T µν . More precisely, the partition
function of the four-dimensional field theory in an external metric g0µν is equal to
Z4D[g
µν ] = exp(iScl[gµν ]) , (40)
10 Son, Starinets
where the five-dimensional metric gµν satisfies the Einstein’s equations and has the following
asymptotics at z = 0:
ds2 = gµνdx
µdxν =
(dz2 + g0µνdx
µdxν) . (41)
From the point of view of hydrodynamics, the operator 1
F 2 is not very interesting because its
correlator does not have a hydrodynamic pole. In contrast, we find the correlators of the R-charge
current and the stress-energy tensor to contain hydrodynamic information.
We simplify the graviton part of the action further. Our two-point functions are functions of
the momentum p = (ω,k). We can choose spatial coordinates so that k points along the x3-axis.
This corresponds to perturbations that propagate along the x3 direction: hµν = hµν(t, r, x
3). These
perturbations can be classified according to the representations of the O(2) symmetry of the (x1, x2)
plane. Owing to that symmetry, only certain components can mix; for example, h12 does not mix
with any other components, whereas components h01 and h31 mix only with each other. We assume
that only these three metric components are nonzero and introduce shorthand notations
φ = h12, a0 = h
0, a3 = h
3 . (42)
The quadratic part of the graviton action acquires a very simple form in terms of these fields:
Squad =
8π2R3
d4x dr
gµν∂µφ∂νφ −
4g2eff
gµαgνβfµνfαβ
, (43)
where fµν = ∂µaν − ∂νaµ, and g2eff = gxx. In deriving Eq. (43), our only assumption about the
metric is that it has a diagonal form,
ds2 = gttdt
2 + grrdr
2 + gxxdx
2 , (44)
so it can also be used below for the finite-temperature metric.
As a simple example, let us compute the two-point correlation function of T xy, which corresponds
to φ in gravity. The field equation for φ is
−g gµν∂νφ) = 0 . (45)
The solution to this equation, with the boundary condition φ(p, z = 0) = φ0(p), can be written as
φ(p, z) = fp(z)φ0(p) , (46)
where the mode function fp(z) satisfies the equation
fp = 0 (47)
with the boundary condition fp(0) = 1. The mode equation (47) can be solved exactly. Assuming
p is spacelike, p2 > 0, the exact solution and its expansion around z = 0 is
fp(z) =
(pz)2K2(pz) = 1 −
(pz)2 − 1
(pz)4 ln(pz) + O((pz)4) . (48)
The second solution to Eq. (47), (pz)2I2(pz), is ruled out because it blows up at z → ∞.
Viscosity, Black Holes, and QFT 11
We now substitute the solution into the quadratic action. Using the field equation, one can
perform integration by parts and write the action as a boundary integral at z = 0. One finds
φ(x, z)φ′(x, z)|z→0 =
(2π)4
φ0(−p)F(p, z)φ0(p)|z→0 , (49)
where
F(p, z) = N
f−p(z)∂zfp(z) . (50)
Differentiating the action twice with respect to the boundary value φ0 one finds
〈TxyTxy〉p = −2 lim
F(p, z) = N
p4 ln(p2) . (51)
Note that we have dropped the term ∼ p4 ln z, which, although singular in the limit z → 0, is a
contact term [i.e., a term proportional to a derivative of δ(x) after Fourier transform]. Removing
such terms by adding local counter terms to the supergravity action is known as the holographic
renormalization [22]. It is, in a sense, a holographic counterpart to the standard renormalization
procedure in quantum field theory, here applied to composite operators.
For time-like p, p2 < 0, there are two solutions to Eq. (47) which involve Hankel functions H(1)(z)
and H(2)(z) instead of K2(z). Neither function blows up at z → ∞, and it is not clear which should
be picked. Here we encounter, for the first time, a subtlety of Minkowski-space AdS/CFT, which is
discussed in great length in subsequent sections. At zero temperature this problem can be overcome
by an analytic continuation from space-like p. However, this will not work at nonzero temperatures.
3.2 Black Three-Brane Metric
At nonzero temperatures, the metric dual to N = 4 SYM theory is the black three-brane metric,
ds2 =
(−fdt2 + dx2) + R
dr2 + R2dΩ25 , (52)
with f = 1 − r40/r4. The event horizon is located at r = r0, where f = 0. In contrast to the usual
Schwarzschild black hole, the horizon has three flat directions x. The metric (52) is thus called a
black three-brane metric.
We frequently use an alternative radial coordinate u, defined as u = r20/r
2. In terms of u, the
boundary is at u = 0, the horizon at u = 1, and the metric is
ds2 =
(πTR)2
(−f(u)dt2 + dx2) + R
4u2f(u)
du2 + R2dΩ25 . (53)
The Hawking temperature is determined completely by the behavior of the metric near the
horizon. Let us concentrate on the (t, r) part of the metric,
ds2 = −4r0
(r − r0)dt2 +
4r0(r − r0)
dr2 . (54)
Changing the radial variable from r to ρ,
r = r0 +
, (55)
12 Son, Starinets
and the metric components become nonsingular:
ds2 =
dρ2 − 4r
ρ2dt2
. (56)
Note also that after a Wick rotation to Euclidean time τ , the metric has the form of the flat metric
in cylindrical coordinates, ds2 ∼ dρ2 + ρ2dϕ2, where ϕ = 2r0R−2τ . To avoid a conical singularity
at ρ = 0, ϕ must be a periodic variable with periodicity 2π. This fact matches with the periodicity
of the Euclidean time in thermal field theory τ ∼ τ + 1/T , from which one finds the Hawking
temperature:
. (57)
One of the first finite-temperature predictions of AdS/CFT correspondence is that of the ther-
modynamic potentials of the N = 4 SYM theory in the strong coupling regime. The entropy is
given by the Bekenstein-Hawking formula S = A/(4G), where A is the area of the horizon of the
metric (52); the result can then be converted to parameters of the gauge theory using Eqs. (30),
(33), and (34). One obtains
N2T 3 , (58)
which is 3/4 of the entropy density in N = 4 SYM theory at zero ’t Hooft coupling.
We now try to generalize the AdS/CFT prescription to finite temperature. In the Euclidean
formulation of finite-temperature field theory, field theory lives in a space-time with the Euclidean
time direction τ compactified. The metric is regular at r = r0: If one views the (τ, r) space as
a cigar-shaped surface, then the horizon r = r0 is the tip of the cigar. Thus, r0 is the minimal
radius where the space ends, and there is no point in space with r less than r0. The only boundary
condition at r = r0 is that fields are regular at the tip of the cigar, and the AdS/CFT correspondence
is formulated as
Z4D[J ] = Z5D[φ]|φ(z=0)→J . (59)
4 REAL-TIME AdS/CFT
In many cases we must find real-time correlation functions not given directly by the Euclidean path-
integral formulation of thermal field theory. One example is the set of kinetic coefficients expressed,
through Kubo’s formulas, via a certain limit of real-time thermal Green’s functions. Another related
example appears if we want to directly find the position of the poles in the correlation functions
that would correspond to the hydrodynamic modes.
In principle, some real-time Green’s functions can be obtained by analytic continuation of the
Euclidean ones. For example, an analytic continuation of a two-point Euclidean propagator gives
a retarded or advanced Green’s function, depending on the way one performs the continuation.
However, it is often very difficult to directly compute a quantity of interest in that way. In particular,
it is very difficult to get the information about the hydrodynamic (small ω, small k) limit of real-time
correlators from Euclidean propagators. The problem here is that we need to perform an analytic
continuation from a discrete set of points in Euclidean frequencies (the Matsubara frequencies)
ω = 2πin, where n is an integer, to the real values of ω. In the hydrodynamic limit, we are
interested in real and small ω, whereas the smallest Matsubara frequency is already 2πT .
Viscosity, Black Holes, and QFT 13
Therefore, we need a real-time AdS/CFT prescription that would allow us to directly compute
the real-time correlators. However, if one tries to naively generalize the AdS/CFT prescription, one
immediately faces a problem. Namely, now r = r0 is not the end of space but just the location of
the horizon. Without specifying a boundary condition at r = r0, there is an ambiguity in defining
the solution to the field equations, even as the boundary condition at r = ∞ is set.
As an example, let us consider the equation of motion of a scalar field in the black hole back-
ground, ∂µ(g
µν∂νφ) = 0. The solution to this equation with the boundary condition φ = φ0 at
u = 0 is φ(p, u) = fpφ0(p), where fp(u) satisfies the following equation in the metric (53):
f ′′p −
1 + u2
f ′p +
fp = 0 . (60)
Here the prime denotes differentiation with respect to u, and we have defined the dimensionless
frequency and momentum:
, q =
. (61)
Near u = 0 the equation has two solutions, f1 ∼ 1 and f2 ∼ u2. In the Euclidean version of
thermal AdS/CFT, there is only one regular solution at the horizon u = 1, which corresponds to a
particular linear combination of f1 and f2. However, in Minkowski space there are two solutions,
and both are finite near the horizon. One solution termed fp behaves as (1−u)−iw/2, and the other
is its complex conjugate f∗p ∼ (1 − u)iw/2. These two solutions oscillate rapidly as u → 1, but the
amplitude of the oscillations is constant. Thus, the requirement of finiteness of fp allows for any
linear combination of f1 and f2 near the boundary, which means that there is no unique solution
to Eq. (60).
4.1 Prescription For Retarded Two-Point Functions
Physically, the two solutions fp and f
p have very different behavior. Restoring the e
−iωt phase in
the wave function, one can write
e−iωtfp ∼ e−iω(t+r∗) , (62)
e−iωtf∗p ∼ eiω(t−r∗) , (63)
where the coordinate
ln(1 − u)
was introduced so that Eqs. (62) and (63) looked like plane waves. In fact, Eq. (62) corresponds to
a wave that moves toward the horizon (incoming wave) and Eq. (63) to a wave that moves away
from the horizon (outgoing wave).
The simplest idea, which is motivated by the fact that nothing should come out of a horizon,
is to impose the incoming-wave boundary condition at r = r0 and then proceed as instructed by
the AdS/CFT correspondence. However, now we encounter another problem. If we write down
the classical action for the bulk field, after integrating by parts we get contributions from both the
boundary and the horizon:
(2π)4
φ0(−p)F(p, z)φ0(p)
. (65)
14 Son, Starinets
If one tried to differentiate the action with respect to the boundary value φ0, one would find
G(p) = F(p, z)|zH0 + F(−p, z)|
0 . (66)
From the equation satisfied by fp and from f
p = f−p, it is easy to show that the imaginary part
of F(p, z) does not depend on z; hence the quantity G(p) in Eq. (66) is real. This is clearly not
what we want, as the retarded Green’s functions are, in general, complex. Simply throwing away
the contribution from the horizon does not help because F(−p, z) = F∗(p, z) owing to the reality
of the equation satisfied by fp.
A partial solution to this problem was suggested in Ref. [7]. It was postulated that the re-
tarded Green’s function is related to the function F by the same formula that was found at zero
temperature:
GR(p) = −2 lim
F(p, z) . (67)
In particular, we throw away all contributions from the horizon. This prescription was established
more rigorously in Ref. [8] (following an earlier suggestion in Ref. [23]) as a particular case of a
general real-time AdS/CFT formulation, which establishes the connection between the close-time-
path formulation of real-time quantum field theory with the dynamics of fields in the whole Penrose
diagram of the AdS black brane. Here we accept Eq. (67) as a postulate and proceed to extract
physical results from it.
It is also easy to generalize this prescription to the case when we have more than one field. In
that case, the quantity F becomes a matrix Fab, whose elements are proportional to the retarded
Green’s function Gab.
4.2 Calculating Hydrodynamic Quantities
As an illustration of the real-time AdS/CFT correspondence, we compute the correlator of Txy.
First we write down the equation of motion for φ = hxy :
φ′′p −
1 + u2
φ′p +
w2 − q2f
φp = 0 . (68)
In contrast to the zero-temperature equation, now ω and k enter the equation separately rather
than through the combination ω2 − k2. Thus the Green’s function will have no Lorentz invariance.
The equation cannot be solved exactly for all ω and k. However, when ω and k are both much
smaller than T , or w, q ≪ 1, one can develop series expansion in powers of w and q. There are
two solutions that are complex conjugates of each other. The solution that is an incoming wave at
u = 1 and normalized to 1 at u = 0 is
fp(z) = (1 − u2)−iw/2 + O(w2, q2) . (69)
The kinetic term in the action for φ is
S = −π
2N2T 4
φ′2 . (70)
Applying the general formula (67), one finds the retarded Green’s function of Txy,
GRxy,xy(ω, k) = −
π2N2T 4
iw , (71)
Viscosity, Black Holes, and QFT 15
and, using Kubo’s formula for η, the viscosity,
N2T 3 . (72)
It is instructive to compute other correlators that have poles corresponding to hydrodynamic
modes. As a warm-up, let us compute the two-point correlators of the R-charge currents, which
should have a pole at ω = −iDk2, where D is the diffusion constant. We first write down Maxwell’s
equations for the bulk gauge field. Let the spatial momentum be aligned along the x3-axis: p =
(ω, 0, 0, k). Then the equations for A0 and A3 are coupled:
wA′0 + qfA
3 = 0 , (73)
A′′0 −
(q2A0 + wqA3) = 0 , (74)
A′′3 +
A′3 +
(w2A3 + wqA0) = 0 . (75)
One can eliminate A3 and write down a third-order equation for A0,
A′′′0 +
(uf)′
A′′0 +
w2 − q2f
A′0 = 0 . (76)
Near u=1 we find two independent solutions, A′0 ∼ (1− u)±iw/2, and the incoming-wave boundary
condition singles out (1 − u)−iw/2. One can substitute A′0 = (1 − u)−iw/2F (u) into Eq. (76). The
resulting equation can be solved perturbatively in w and q2. We find
A′0 = C(1 − u)−iw/2
1 + u
+ q2 ln
1 + u
. (77)
Using Eq. (74) one can express C through the boundary values of A0 and A3 at u = 0:
q2A0 + wqA3
iw − q2
. (78)
Differentiating the action with respect to the boundary values, we find, in particular,
〈J0J0〉p =
iω − Dk2 , (79)
where
. (80)
The correlator given by Eq. (79) has the expected hydrodynamic diffusive pole, and D is the
R-charge diffusion constant.
Similarly, one can observe the appearance of the shear mode in the correlators of the metric
tensor. We note that the shear flow along the x1 direction with velocity gradient along the x3
direction involves T01 and T31, hence the interesting metric components are a0 = h
0 and a3 = h
Two of the field equations are
a′0 −
a′3 = 0 , (81)
a′′3 −
1 + u2
a′3 +
(w2a3 + wqa0) = 0 . (82)
16 Son, Starinets
They can be combined into a single equation:
a′′′0 −
a′′0 +
2uf − q2f + w2
a′0 = 0 . (83)
Again, the solution can be found perturbatively in w and q:
a′0 = C(1 − u)−iw/2
u − iw
1 − u − u
1 + u
(1 − u)
. (84)
Applying the prescription, one finds the retarded Green’s functions. For example,
Gtx,tx(ω, k) =
iω −Dk2 , (85)
where
N2T 3, D = 1
. (86)
Thus, we found that the correlator contains a diffusive pole ω = −iDk2, just as anticipated from
hydrodynamics. Furthermore, the magnitude of the momentum diffusion constant D also matched
our expectation. Indeed, if one recalls the value of η from Eq. (72) and the entropy density from
Eq. (58), one can check that
D = η
ǫ + P
. (87)
5 THE MEMBRANE PARADIGM
Let us now look at the problem from a different perspective. The existence of hydrodynamic modes
in thermal field theory is reflected by the existence of the poles of the retarded correlators computed
from gravity. Are there direct gravity counterparts of the hydrodynamic normal modes?
If the answer to this question is yes, then there must exist linear gravitational perturbations of
the metric that have the dispersion relation identical to that of the shear hydrodynamic mode,
ω ∼ −iq2, and of the sound mode, ω = csq− iγq2. It turns out that one can explicitly construct the
gravitational counterpart of the shear mode. (It should be possible to find a similar construction
for the sound mode, but it has not been done in the literature; for a recent work on the subject,
see [24].) Our discussion is physical but somewhat sketchy; for more details see Ref. [25].
First, let us construct a gravity perturbation that corresponds to a diffusion of a conserved charge
(e.g., the R-charge in N = 4 SYM theory). To keep the discussion general, we use the form of the
metric (44), with the metric components unspecified. Our only assumptions are that the metric is
diagonal and has a horizon at r = r0, near which
g00 = −γ0(r − r0), grr =
r − r0
. (88)
The Hawking temperature can be computed by the method used to arrive at Eq. (57), and one
finds T = (4π)−1(γ0/γr)
We also assume that the action of the gauge field dual to the conserved current is
Sgauge =
4g2eff
FµνFµν
, (89)
Viscosity, Black Holes, and QFT 17
where geff is an effective gauge coupling that can be a function of the radial coordinate r. For
simplicity we set geff to a constant in our derivation of the formula for D; it can be restored by
replacing
√−g → √−g/g2eff in the final answer.
The field equations are
g2eff
−g Fµν
= 0 . (90)
We search for a solution to this equation that vanishes at the boundary and satisfies the incoming-
wave boundary condition at the horizon.
The first indication that one can have a hydrodynamic behavior on the gravity side is that
Eq. (90) implies a conservation law on a four-dimensional surface. We define the stretched horizon
as a surface with constant r just outside the horizon,
r = rh = r0 + ε, ε ≪ r0 , (91)
and the normal vector nµ directed along the r direction (i.e., perpendicularly to the stretched
horizon). Then with any solution to Eq. (90), one can associate a current on the stretched horizon:
jµ = nνF
. (92)
The antisymmetry of Fµν implies that jµ has no radial component, jr = 0. The field equation (90)
and the constancy of nν on the stretched horizon imply that this current is conserved: ∂µj
µ = 0. To
establish the diffusive nature of the solution, we must show the validity of the constitutive equation
ji = −D∂ij0.
Such constitutive equation breaks time reversal and obviously must come from the absorptive
boundary condition on the horizon. The situation is analogous to the propagation of plane waves to
a non-reflecting surface in classical electrodynamics. In this case, we have the relation B = −n×E
between electric and magnetic fields. In our case, the corresponding relation is
Fir = −
r − r0
, (93)
valid when r is close to r0. This relates ji ∼ Fir to the parallel to the horizon component of the
electric field F0i, which is one of the main points of the “membrane paradigm” approach to black
hole physics [26, 27]. We have yet to relate ji to j0 ∼ F0r, which is the component of the electric
field normal to the horizon. To make the connection to F0r, we use the radial gauge Ar = 0, in
which
F0i ≈ −∂iA0 . (94)
Moreover, when k is small the fields change very slowly along the horizon. Therefore, at each
point on the horizon the radial dependence of the scalar potential A0 is determined by the Poisson
equation,
−g grrg00∂rA0) = 0 , (95)
whose solution, which satisfies A0(r = ∞) = 0, is
A0(r) = C0
g00(r
′)grr(r
−g(r′)
. (96)
18 Son, Starinets
This means that the ratio of the scalar potential A0 and electric field F0r approaches a constant
near the horizon:
g00grr
g00grr√−g (r) . (97)
Combining the formulas ji ∼ F0i ∼ ∂iA0, and A0 ∼ F0r ∼ j0, we find Fick’s law ji = −D∂ij0, with
the diffusion constant
√−g00grr
−g00grrg2eff√
−g (r) . (98)
Thus, we found that for a slowly varying solution to Maxwell’s equations, the corresponding
charge on the stretched horizon evolves according to the diffusion equation. Therefore, the gravity
solution must be an overdamped one, with ω = −iDk2. This is an example of a quasi-normal mode.
We also found the diffusion constant D directly in terms of the metric and the gauge coupling geff .
The reader may notice that our quasinormal modes satisfy a vanishing Dirichlet condition at
the boundary r=∞. This is different from the boundary condition one uses to find the retarded
propagators in AdS/CFT, so the relation of the quasinormal modes to AdS/CFT correspondence
may be not clear. It can be shown, however, that the quasi-normal frequencies coincide with the
poles of the retarded correlators [28,29].
We can now apply our general formulas to the case of N = 4 SYM theory. The metric components
are given by Eq. (52). For the R-charge current geff = const, Eq. (98) gives D = 1/(2πT ), in
agreement with our AdS/CFT computation. For the shear mode of the stress-energy tensor we
have effectively g2eff = gxx, so D = 1/(4πT ), which also coincides with our previous result. In both
cases, the computation is much simpler than the AdS/CFT calculation.
6 THE VISCOSITY/ENTROPY RATIO
6.1 Universality
In all thermal field theories in the regime described by gravity duals the ratio of shear viscosity η
to (volume) density of entropy s is a universal constant equal to 1/(4π) [h̄/(4πkB), if one restores
h̄, c and the Boltzmann constant kB ].
One proof of the universality is based on the relationship between graviton’s absorption cross
section and the imaginary part of the retarded Green’s function for Txy [31]. Another way to prove
the universality [32] is via the direct AdS/CFT calculation of the correlation function in Kubo’s
formula (18).
We, however, follow a different method. It is based on the formula for the viscosity derived from
the membrane paradigm. A similar proof was given by Buchel & Liu [30].
The observation is that the shear gravitational perturbation with k = 0 can be found exactly by
performing a Lorentz boost of the black-brane metric (52). Consider the coordinate transformations
r, t, xi → r′, t′, x′i of the form
r = r′ ,
t′ + vy′√
1 − v2
≈ t′ + vy′ ,
Viscosity, Black Holes, and QFT 19
y′ + vt′√
1 − v2
≈ y′ + vt′ ,
xi = x
i , (99)
where v < 1 is a constant parameter and the expansion on the right corresponds to v ≪ 1. In the
new coordinates, the metric becomes
ds2 = g00dt
′ 2 + grrdr
′ 2 + gxx(r)
(dx′ i)2 + 2v(g00 + gxx)dt
′dy′ . (100)
This is simply a shear fluctuation at k = 0. In our language, the corresponding gauge potential is
a0 = vg
xx(g00 + gxx) . (101)
This field satisfies the vanishing boundary condition a0(r = ∞) = 0 owing to the restoration
of Poincaré invariance at the boundary: g00/gxx → −1 when r → ∞. This clearly has a much
simpler form than Eq. (96) for the solution to the generic Poisson equation. The simple form of
solution (101) is valid only for the specific case of the shear gravitational mode with g2eff = gxx.
We have also implicitly used the fact that the metric satisfies the Einstein equations, with the
stress-energy tensor on the right being invariant under a Lorentz boost.
Equation (97) now becomes
= −1 + g
xxg00
∂r(gxxg00)
gxx(r0)
. (102)
The shear mode diffusion constant is
D = a0
gxx(r0)
. (103)
Because D = η/(ǫ + P ), and ǫ + P = Ts in the absence of chemical potentials, we find that
. (104)
In fact, the constancy of this ratio has been checked directly for theories dual to Dp-brane [25],
M -brane [11], Klebanov-Tseytlin and Maldacena-Nunez backgrounds [30], N = 2∗ SYM theory
[33] and others. Curiously, the viscosity to entropy ratio is also equal to 1/4π in the pre-AdS/CFT
“membrane paradigm” hydrodynamics [34]: there, for a four-dimensional Schwarzschild black hole
one has η
= 1/16πGN , while the Bekenstein-Hawking entropy is s = 1/4GN .
As remarked in Sec. 2, the ratio η/s is much larger than the one for weakly coupled theories.
The fact that we found the ratio to be parametrically of order one implies that all theories with
gravity duals are strongly coupled.
In N = 4 SYM theory, the ratio η/s has been computed to the next order in the inverse ’t Hooft
coupling expansion [35]
135ζ(3)
8(g2N)3/2
. (105)
The sign of the correction can be guessed from the fact that in the limit of zero ’t Hooft coupling
g2N → 0, the ratio diverges, η/s → ∞.
20 Son, Starinets
6.2 The Viscosity Bound Conjecture
From our discussion above, one can argue that
(106)
in all systems that can be obtained from a sensible relativistic quantum field theory by turning on
temperatures and chemical potentials.
The bound, if correct, implies that a liquid with a given volume density of entropy cannot be
arbitrarily close to being a perfect fluid (which has zero viscosity). As such, it implies a lower bound
on the viscosity of the QGP one may be creating at RHIC. Interestingly, some model calculations
suggest that the viscosity at RHIC may be not too far away from the lower bound [36,37].
One place where one may think that the bound should break down is superfluids. The ability
of a superfluid to flow without dissipation in a channel is sometimes described as “zero viscosity”.
However, within the Landau’s two-fluid model, any superfluid has a measurable shear viscosity
(together with three bulk viscosities). For superfluid helium, the shear viscosity has been measured
in a torsion-pendulum experiment by Andronikashvili [38]. If one substitutes the experimental
values, the ratio η/s for helium remains larger than h̄/4πkB ≈ 6.08 × 10−13 K s for all ranges of
temperatures and pressures, by a factor of at least 8.8.
As discussed in Sec. 2.3, the ratio η/s is proportional to the ratio of the mean free path and the
de Broglie wavelength of particles,
∼ ℓmfp
. (107)
For the quasi-particle picture to be valid, the mean free path must be much larger than the de
Broglie wavelength. Therefore, if the coupling is weak and the system can be described as a
collection of quasi-particles, the ratio η/s is larger than 1.
We have found is that, within the N = 4 SYM theory and, more generally, theories with gravity
duals, even in the limit of infinite coupling the ratio η/s cannot be made smaller than 1/(4π).
7 CONCLUSION
In this review, we covered only a small part of the applications of AdS/CFT correspondence to
finite-temperature quantum field theory. Here we briefly mention further developments and refer
the reader to the original literature for more details.
In addition to N = 4 SYM theory, there exists a large number of other theories whose hydrody-
namic behavior has been studied using the AdS/CFT correspondence, including the worldvolume
theories on M2- and M5-branes [11], theories on Dp branes [25], and little string theory [39]. In
all examples the ratio η/s is equal to 1/(4π), which is not surprising because the general proofs of
Sec. 6 apply in these cases.
We have concentrated on the shear hydrodynamic mode, which has a diffusive pole (ω ∼ −ik2).
One can also compute correlators which have a sound-wave pole from the AdS/CFT prescription
[10]. One such correlator is between the energy density T 00 at two different points in space-time.
The result confirms the existence of such a pole, with both the real part and imaginary part having
exactly the values predicted by hydrodynamics (recall that in conformal field theories the bulk
viscosity is zero and the sound attenuation rate is determined completely by the shear viscosity).
Viscosity, Black Holes, and QFT 21
Some of the theories listed above are conformal field theories, but many are not (e.g., the Dp-
brane worldvolume theories with p 6= 3). The fact that η/s = 1/(4π) also in those theories implies
that the constancy of this ratio is not a consequence of conformal symmetry. Theories with less
than maximal number of supersymmetries have been found to have the universal value of η/s, for
example, the N = 2∗ theory [40], theories described by Klebanov-Tseytlin, and Maldacena-Nunez
backgrounds [30]. A common feature of these theories is that they all have a gravitational dual
description. The bulk viscosity has been computed for some of these theories [41,39].
Besides viscosity, one can also compute diffusion constants of conserved charges by using the
AdS/CFT correspondence. Above we presented the computation of the R-charge diffusion constant
in N = 4 SYM theory; for similar calculations in some other theories see Ref. [11,25].
Recently, the AdS/CFT correspondence was used to compute the energy loss rate of a quark in
the fundamental representation moving in a finite-temperature plasma [42,43,44,45]. This quantity
is of importance to the phenomenon of “jet quenching” in heavy-ion collisions.
So far, the only quantity that shows a universal behavior at the quantitative level, across all
theories with gravitational duals, is the ratio of the shear viscosity and entropy density. Recently,
it was found that this ratio remains constant even at nonzero chemical potentials [46,47,48,49,50].
What have we learned from the application of AdS/CFT correspondence to thermal field theory?
Although, at least at this moment, we cannot use the AdS/CFT approach to study QCD directly,
we have found quite interesting facts about strongly coupled field theories. We have also learned
new facts about quasi-normal modes of black branes. However, we have also found a set of puzzles:
Why is the ratio of the viscosity and entropy density constant in a wide class of theories? Is there
a lower bound on this ratio for all quantum field theories? Can this be understood without any
reference to gravity duals? With these open questions, we conclude this review.
Acknowledgments
The work of DTS is supported in part by U.S. Department of Energy under Grant No. DE-FG02-
00ER41132. Research at Perimeter Institute is supported in part by the Government of Canada
through NSERC and by the Province of Ontario through MEDT.
Literature Cited
1. Kirzhnits DA, Linde AD. Macroscopic consequences of the Weinberg model. Phys. Lett. B
42:471 (1972)
2. Gyulassy M, and McLerran L, New forms of QCD matter discovered at RHIC. Nucl. Phys. A
750:30 (2005)
3. Maldacena JM. The large N limit of superconformal field theory and supergravity. Adv. Theor.
Math. Phys. 2:231 (1998)
4. Gubser SS, Klebanov IR, Polyakov AM. Gauge theory correlators from noncritical string theory.
Phys. Lett. B 428:105 (1998)
5. Witten E. Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2:253 (1998)
6. Policastro G, Son DT, Starinets AO. Shear viscosity of strongly coupled N = 4 supersymmetric
Yang-Mills plasma. Phys. Rev. Lett. 87:081601 (2001)
22 Son, Starinets
7. Son DT, Starinets AO. Minkowski-space correlators in AdS/CFT correspondence: Recipe and
applications. J. High Energy Phys. 0209:042 (2002)
8. Herzog CP, Son DT. Schwinger-Keldysh propagators from AdS/CFT correspondence. J. High
Energy Phys. 0303:046 (2003)
9. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. J.
High Energy Phys. 0209:042 (2002)
10. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. II:
Sound waves. J. High Energy Phys. 0212:054 (2002)
11. Herzog CP. The hydrodynamics of M-theory. J. High Energy Phys. 0212:026 (2002)
12. Herzog CP. The sound of M-theory. Phys. Rev. D 68:024013 (2003)
13. Forster D. Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Reading:
Benjamin (1975)
14. Landau LD, Lifshitz EM. Fluid Mechanics, Oxford: Pergamon Press (1987)
15. Jeon S. Hydrodynamic transport coefficients in relativistic scalar field theory. Phys. Rev. D
52:3591 (1995)
16. Jeon S, Yaffe LG. From quantum field theory to hydrodynamics: transport coefficients and
effective kinetic theory. Phys. Rev. D 53:5799 (1996)
17. Aharony O et al. Large N field theories, string theory and gravity. Phys. Rep. 323:183 (2000)
18. Klebanov IR. TASI lectures: Introduction to the AdS/CFT correspondence.
arXiv:hep-th/0009139
19. Polchinski J. String theory, Cambridge: Cambridge University Press (1998).
20. Appelquist T, Chodos A, Freund PG. Modern Kaluza-Klein theories, Menlo Park: Addison-
Wesley (1987)
21. Kim HJ, Romans LJ, van Nieuwenhuizen P. The mass spectrum of chiral N = 2 D = 10
supergravity on S5. Phys. Rev. D 32:389 (1985)
22. Bianchi M, Freedman DZ, Skenderis K. Holographic renormalization. Nucl. Phys. B 631:159
(2002)
23. Maldacena JM. Eternal black holes in anti-de-Sitter. J. High Energy Phys. 0304:021 (2003)
24. Saremi O. Shear waves, sound waves on a shimmering horizon. arXiv:hep-th/0703170
25. Kovtun P, Son DT, Starinets AO. Holography and hydrodynamics: Diffusion on stretched
horizons. J. High Energy Phys. 0310:064 (2003)
26. Damour T. Black Hole Eddy Currents. Phys. Rev. D 18: 3598 (1978)
27. Thorne, KS, Price RH, Macdonald DA. Black Hole: The Membrane Paradigm, New Haven:
Yale University Press (1986)
28. Kovtun PK, Starinets AO. Quasinormal modes and holography. Phys. Rev. D 72:086009 (2005)
29. Starinets AO. Quasinormal spectrum and black hole membrane paradigm. Unpublished
30. Buchel A, Liu JT. Universality of the shear viscosity in supergravity. Phys. Rev. Lett., 93:090602
(2004)
31. Kovtun P, Son DT, Starinets AO. Viscosity in strongly interacting quantum field theories from
black hole physics. Phys. Rev. Lett. 94:111601 (2004)
32. Buchel A. On universality of stress-energy tensor correlation functions in supergravity, Phys.
Lett. B, 609:392 (2005).
33. Buchel A. N = 2* hydrodynamics. Nucl. Phys. B 708:451 (2005)
http://arXiv.org/abs/hep-th/0009139
http://arXiv.org/abs/hep-th/0703170
Viscosity, Black Holes, and QFT 23
34. Damour T. Surface effects in black hole physics. Proceedings of the Second Marcel Grossmann
Meeting on General Relativity. Ed. R. Ruffini, Amsterdam: North Holland (1982)
35. Buchel A, Liu JT, Starinets AO. Coupling constant dependence of the shear viscosity in N = 4
supersymmetric Yang-Mills theory. Nucl. Phys. B 707:56 (2005)
36. Teaney D. Effect of shear viscosity on spectra, elliptic flow, and Hanbury Brown-Twiss radii.
Phys. Rev. C 68:034913 (2003)
37. Shuryak E. Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid? Prog.
Part. Nucl. Phys. 53:273 (2004)
38. Andronikashvili E. Zh. Eksp. Teor. Fiz. 18:429 (1948)
39. Parnachev A, Starinets A. The silence of the little strings. J. High Energy Phys. 0510:027 (2005)
40. Buchel A, Liu JT, Thermodynamics of the N = 2∗ flow. J. High Energy Phys. 0311:031 (2003)
41. Benincasa P, Buchel A, Starinets AO. Sound waves in strongly coupled non-conformal gauge
theory plasma. Nucl. Phys. B 733:160 (2006)
42. Herzog CP et al. Energy loss of a heavy quark moving through N = 4 supersymmetric Yang-
Mills plasma. J. High Energy Phys. 0607:013 (2006)
43. Liu H, Rajagopal K, Wiedemann UA. Calculating the jet quenching parameter from AdS/CFT,
Phys. Rev. Lett. 97:182301 (2006)
44. Casalderrey-Solana J, Teaney D. Heavy quark diffusion in strongly coupled N = 4 Yang Mills.
Phys. Rev. D 74:085012 (2006)
45. Gubser, SS. Drag force in AdS/CFT. Phys. Rev. D 74:126005 (2006)
46. Son DT, Starinets AO. Hydrodynamics of R-charged black holes. J. High Energy Phys. 0603:052
(2006)
47. Mas J, Shear viscosity from R-charged AdS black holes. J. High Energy Phys. 0603:016 (2006)
48. Maeda K, Natsuume M, Okamura T. Viscosity of gauge theory plasma with a chemical potential
from AdS/CFT. Phys. Rev. D 73:066013 (2006)
49. Saremi O. The viscosity bound conjecture and hydrodynamics of M2-brane theory at finite
chemical potential. J. High Energy Phys. 0610:083 (2006)
50. Benincasa P, Buchel A, Naryshkin R. The shear viscosity of gauge theory plasma with chemical
potentials. Phys. Lett. B 645:309 (2007)
	INTRODUCTION
	HYDRODYNAMICS
	Kubo's Formula For Viscosity
	Hydrodynamic Modes
	Viscosity In Weakly Coupled Field Theories
	AdS/CFT CORRESPONDENCE
	Review Of AdS/CFT Correspondence At Zero Temperature
	Black Three-Brane Metric
	REAL-TIME AdS/CFT
	Prescription For Retarded Two-Point Functions
	Calculating Hydrodynamic Quantities
	THE MEMBRANE PARADIGM
	THE VISCOSITY/ENTROPY RATIO
	Universality
	The Viscosity Bound Conjecture
	CONCLUSION
