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March 1, 2019

A Gravity Dual of RHIC Collisions

Edward Shuryaka1, Sang-Jin Sinb2 and Ismail Zaheda3

a Department of Physics and Astronomy, SUNY Stony-Brook, NY 11794

b Department of Physics, Hanyang University, Seoul 133-791, Korea

Abstract

In the context of the AdS/CFT correspondence we discuss the gravity dual of a

heavy-ion-like collision in a variant of N = 4 SYM. We have provided a simple physical

picture of efficient thermalization mechanism by considering the dual process, namely,

the formation of AdS black hole. We estimated the initial entropy and temperature

as a function of Qs. We also consider the cooling procedure by considering the brane

motion in ads black hole background. While the cooling of the edges of the fireball is

luminal or 1/τ , that of the core is slower due to the strongly interacting character of

sQGP. Our analysis suggests that the cooling of the fireball is 1/√τ which is slower

than Bjorken 3d cooling 1/τ . The fireball freezes when the dual black hole background

is replaced by confining background through the Hawking-Page transition.

1E-mail:shuryak@tonic.physics.sunysb.edu

2E-mail: sjsin@hanyang.ac.kr

3E-mail:zahed@zahed.physics.sunysb.edu

1

1 Introduction and Summary

Following Fermi [1], who first suggested that collision of strongly interacting matter will pro-

duce a thermal state, Landau [2] observed that the system would follow an adiabatic cooling

path through transitory thermal states, with the amount of entropy being conserved. He

further pointed out that the evolution should then be described by (ideal) hydrodynamics.

Indeed, one of the key feature of the ’strongly interacting’ Quark Gluon Plasma (sQGP)

is precisely the observation of a hydrodynamical expansion in the form of radial and el-

liptic flow at RHIC. On the other hand, the AdS/CFT correspondence [3] has provided a

framework for discussing a strongly coupled regime of this gauge theory in terms of their

gravity dual description. The equilibrium finite temperature problem using a black-hole

background was discussed in [4]. This approach has provided results on bulk thermody-

namics [5] and transport coefficients [6] that are surprisingly close to what is measured in

current heavy-ion collisions at RHIC. In addition, the efficiency of the AdS/CFT picture to

explain [7] the Jet quenching phenomena is an encouraging signal to use AdS/CFT in this

strongly coupled Quark-Gluon Plasma (sQGP: [8, 9, 10]) created by RHIC.

The purpose of this paper is to address the complex issues of entropy formation and

cooling in a heavy-ion collision using the gravity dual description. QCD posses asymptotic

freedom, due to which at short distances the interaction cannot be strong: thus Landau

scenario can only be applicable after some ‘parton thermalization’ time. Strongly coupled

N=4 SUSY YM theory is simpler in this respect, and we thus ask if one can prove that

Landau hydrodynamics works in this theory, perhaps with some modifications and correc-

tions. If this goal can be achieved, one can then return to more QCD-like theories with the

asymptotic freedom and chiral-deconfinement phase transition. So we would like to start

with the simplest nontrivial geometries corresponding to the infinite spatial extension, only

eventually returning to much more complicated problem of high energy collision of finite

size objects.

What can the gravity dual of a RHIC heavy-ion collision be? We use the AdS/CFT

framework along the lines suggested in [11, 12, 7, 13]. It should be a process of black

hole formation followed by Hawking-Page transition, which from the boundary point of

view correspond to the thermalization and confinement-deconfinement phase transition.

Although the secondary scattering of partons at the boundary is a quantum mechanical

process, in the AdS/CFT framework, its dual must be completely classical.

There are three important steps we want to emphasize. The first step is the creation

of closed string states out of open string process in the boundary. As we will argue in the

2

main text, each scattering vertices pinch off a closed strings so that thousands of partons

scatter to make a crowd of closed strings in AdS bulk which we regards as interacting

particles falling into the center of the AdS starting from certain heights corresponding to

their energy.

The second step is the understanding of the thermalization and entropy generation

procedure of strongly interacting quantum system in terms of classical many body dynamics.

Some of them will try to escape the crowds, but due to the strong interaction between them

the rests will contract. Since the interaction between the closed string is dual to the strongly

interacting particles at the boundary, the string-string interaction must be also very strong

and self-gravitating, while the cascade of evaporation-contraction processes must happen

continuously. This provides a very efficient mechanism of contraction of the core, which

eventually leads to the usual gravitational collapse and formation of the small black hole

by the core. Notice, however, there is a strong confining potential in the AdS bulk so that

none of the evaporated particle can go far to infinity. They will fall again into the center

and will eventually be absorbed by the already formed (small) black hole. This is the step

of the black hole growth. Due to the strong nature of the interaction, we expect that the

whole process should happen on the order of one dynamical time.

There is a clear distinction between coherent parton-parton scattering and incoherent

macroscopic (heavy-ion) collision of large number of partons. In the former, information

is conserved, which is a hallmark of quantum mechanics, while in the latter the informa-

tion is lost and entropy is generated. While entropy generation maybe traced back to the

incoherence due to the many binary scattering in a RHIC heavy-ion collision, it is readily

understood in the gravity dual description: As the particles evaporate from the crowds, the

contracting core is losing information and finally become a (small) black hole. Although the

lost information will be back to the black hole (due to the AdS gravity) and the black hole

grows, the entire information is hidden inside the horizon. This is an interesting procedure

showing how the system can loose information from the observer’s sight.

The third step is the understanding of the late stages of the cooling. Since the

entropy of the fireball seems to be preserved while the temperature cools down, we model

the situation by a moving brane in AdS black hole background. We found it useful to

first neglect the effect of local temperature variation whose holographic image would be

brane bending due to the non-supersymmetric background. The black hole is at uniform

(but time dependent) temperature T (τ). This approach is strongly related to the idea

of brane cosmology [14, 15, 16]. Indeed, in the cosmic evolution the universe expands or

3

contracts with a time-dependent metric and temperature. After that one can try to get

some correction due to the brane bending, assuming that the cooling is adiabatic.

Summarizing, the initial entropy deposition is followed by a cosmic evolution which is

the gravity dual of the cooling of the fire-ball in the heavy-ion collision by hydrodynamical

expansion till freeze out. Due to the induced metric on the probe brane, the 4d hydrody-

namics pertinent to the fire-ball expansion on the brane will be different from that of flat

spacetime, T νµ,ν = 0, by the covariantization T ν

µ;ν = 0.

2 Elements of RHIC physics

This discussion is intended to be elementary to shorten up the vocabulary gaps between the

string community and the heavy ion community interested in the gauge-gravity problems

through the AdS/CFT correspondence.

Collision: Experimentally we use the heaviest (and fully ionized) nuclei (mostly

Au197 at RHIC) with as large energy per nucleon as possible (the relativistic gamma factors

γ ∼ 100 in center of mass, to be increase further at LHC soon.)

One may ignore the complexities of nuclear physics and QCD evolution, and focus

solely on the partonic wave function of hadrons or nuclei before the collision. More precisely,

as coherence is lost anyway, one needs to know the mean squared amplitudes of the pertinent

harmonics of the comoving gluon field with the so called saturation scale Qs or equivalently

the transverse density of partons Q2s. At RHIC Qs is about 1.5 GeV for a typical Feynman

x = 10−2. It will be higher at LHC say Qs = 6− 8 GeV at lower x. A model currently used

to describe the low-x part of the nuclear wavefunction prior to the RHIC collision is the

color glass condensate (CGC). It is rooted on a weak coupling argument in QCD contrary

to what is stated in [13].

Equilibration: This is a transition from the CGC to thermal quarks and gluons.

Solutions of classical Yang-Mills, both for random fields [17] and sphalerons [18] have ac-

tually produced thermal-looking spectra but more is to be understood, perhaps along the

discussion of plasma instabilities [19].

Hydrodynamics: This is a key aspect of RHIC physics. Maintaining collective

flow for systems containig just ∼ 100 − 1000 particles is a nontrivial issue [20], and would

not happen for usual liquids like water. Thus the matter produced at RHIC is now refered

to as a strongly coupled quark-gluon plasma (sQGP) or liquid. Indeed it exhibits both bulk

thermodynamical parameters and transport coefficients (viscosity) that are surprisingly

4

close to what the AdS/CFT correpondence predicted for strongly coupled N=4 SUSY YM

theory. The short time behavior of the hydrodynamical expansion is close to the 1-d Bjorken

regime whereby the temperature depends only on the proper time τ =√t2 − z2. For central

collisions the expansion becomes axially symmetric before turning to a full 3d spherical

expansion. For non-central collisions there is azimuthal anisotropy which is successfully

described by hydrodynamics.

Freezeouts: This corresponds to chemical and thermal freezeouts whereby the

change in the composition is turned off (chemical) and the particles decouple (thermal)

with free streaming. Both freezeouts follow from the same condition νexpansion = νreaction,

where we have used the covariant definition of the expansion rate νexpansion = ∂µuµ.

In cosmology, the expansion is so slow that not only strong (pp) scattering survives,

but even weak equilibrium through p+e ↔ ν+n does, untill T ≈ 1 MeV. Photons freezeout

at much lower temperatures T ≈ 0.1 eV. At RHIC chemical freezeout corresponds to the end

of particle changing reactions such as 2π → 4π, while kinetic freezeout corresponds to the

last elastic collision such as 2π → 2π. Experimentally both freezeouts are reasonably well

measured, the former from matter composition while the latter from particle spectra [21].

While the critical temperature in QCD Tch ≈ 176 MeV is independent on the collision

centrality, the freezeout temperatures depend on the system size. For instance, the kinetic

freezeout temperature Tkin does depend on the system size, and goes down for the largest

fireballs (central collisions) to about 90 MeV. Thus the whole range of temperatures at

RHIC is about 4-fold, from the initial Ti ≈ 350 MeV to the kinetic freezeout Tkin ≈ 90

MeV. The energy density changes by about 2 orders of magnitude.

The main reason for the rapid freezeout of a hadronic gas is the Goldstone nature of

the pions. The self-interaction through derivatives makes it difficult to generate soft pions.

At low temperature, the pion gas collision rates can be calculated from the leading chiral

interaction (Weinberg-Tomozawa). Specifically, the elastic rate is [22]

νππ =T 5

12f4π

(1)

The strong T dependence follows from dimensional arguments. The inelastic rates can be

found in [23].

At RHIC detailed numerical calculations show that the proper time spent in the

sQGP phase (T > Tc) the “mixed phase” (T ≈ Tc) and the hadronic phase (T < Tc) are all

comparable. However at LHC the sQGP should dominate. For simplicity, we may ignore

5

the complications inherent to the running coupling in QCD, the confinement-deconfinement

transition and the pion dynamics by restricting the discussion to the early phase of the

collision dominated by the sQGP. If the latter phase is close to strongly coupled N=4 SUSY

matter at finite temperature, as two of us discussed recently [7], it is then useful to use the

duality insights to bear on the bulk and kinetic properties of the sQGP.

3 RHIC collision and dual black hole formation

Recently two of us have suggested [7] that real-time dynamics such as jet quenching in

RHIC has a gravity dual in the form of a gravitational wave falling on the black hole. The

opacity length was found to be independent of the jet energy at strong coupling [7]. In a

related but different picture[13], it was further suggested that 5d black holes are formed

through the gravitational colliding shock waves, following the original work of t’Hooft in

flat space case [24]. This suggestion, however, has a few shortcomings:

1. The argument for the formation of a black-hole works even for parton-parton scatter-

ing, while no high energy p-p collision shows the evidence of transverse hydrodynam-

ical expansion.

2. The argument based on shock-wave duality has led to a mass dependent freezeout

temperature T ∼ mπ. This conclusion is not supported by facts (e.g.the freezeout

temperature for pions and kaons is the same) and also there is no reason T should have

anything to do with the quark masses, to which pion and kaon masses are proportional.

Below, we provide a different duality arguments for chemical and hadron freezeout

temperatures.

3. The formulation does not say where in AdS space the black holes are formed. Thus

it does not lead to the√s-independent total stopping of gluons as suggested by [7].

4. The assumption of a fixed temperature for the resulting black-hole is unrealistic. There

is no particular temperature of the fireball, but rather an adiabatic path on the phase

diagram.

After local equilibration is achieved in a heavy ion collision, the matter expands and

the temperature depends on both the location and time. At RHIC this dependence was

estimated using ideal relativistic hydrodynamics. What uniquely characterizes the fireball is

its initial entropy. This is conserved in the adiabatic expansion of the fireball. Adiabaticity

6

is a consequence of the fact that the micro-motion and the macro-expansion operate at very

different time scales. This observation of entropy conservation and the cooling of the fireball

is apparently paradoxical from a gravity dual point of view. The fixed entropy corresponds

to fixed black hole area and hence fixed black hole temperature, which in turn implies the

fixed gauge theory temperature according to Witten [4].

The resolution of this paradox is to assume that we are on a test brane inside the AdS

bulk and the black hole is moving away from it, toward the gravity brane creating the AdS

background. This motion of the black hole is natural, according to the law of AdS gravity.

It is crucial that this black hole is not a “black brane” used in constant T applications:

as it is created from closed string objects, it cannot have the Abelian charges (coupled to

vector fields in classical supergravity in the bulk) which normally compensate gravitational

attraction between branes.

One can view the situation as the AdS with cut off where the boundary is at the

finite distance instead of infinity. Then the moving brane sees bigger and bigger scale

factor of the bulk metric as it moves away, resulting in the cosmic expansion on the brane

world. This is nothing but the brane world cosmology addressed in [14, 15, 16, 25, 26, 27].

In this way, we identified the gravity dual of the expansion and cooling of the fireball as

the cosmic gravitational expansion in the AdS-black-hole background. In order words, we

approximated the Little Bang as the Big Bang. In the Little Bang, the temperature has

space and time dependence, while in the Big Bang we are neglecting the spatial dependence

looking only to the time dependence. We will return to this point later.

What is the initial location of the brane in the radial direction of the AdS space?

Following the scale-radius duality [28], we suggest that it should be identified as the energy

scale (√s of the scattering experiment on the brane) since that is the only relevant energy

scale in this problem. Notice that this scale is far greater than the Hawking’s temperature,

which is to be identified as the initial fireball temperature on the brane.

Is this set up natural from the view point of formation of the the AdS black hole?

As we will argue in more detail below, a lot of closed string states will be formed out of

open string (gauge theory) processes, some of which may be in a small black hole state.

Since closed strings can be detached from the brane, they fall into the center of the AdS.

The re-scattering of the partons and gluons can be viewed as the dual of the interaction of

the closed stings, which may include eating other closed string states (particles) by small

black holes as well as the merging of the small black holes. Due to the property of AdS

metric, a droplet is contracted rather than enlongated during the infall. This significantly

7

gauge d=4 theory string/gravity in d=10

initial gluons, CGC Aichelburg-Sexl-type shock waves

Thermalization → T black hole formation with Tbh = T

the entropy area of the horizon

rescattering (q − q) production of closed string (gravitons)

rescattering (g − g) interactions between closed strings

fireball expansion falling b.h. in AdS

further equilibration merger of gravitons to black hole.

fireball with finite extention b.h off the center

global equilibration with uniform temp. b.h at the center

cooling and expansion probe brane motion in the 5-th direction

ideal hydrodynamics stationary black hole

hydro with viscosity growing black hole

deconfinement hitting the MKK scale

kinetic freezeout cutoff of gravitons

Table 1: A vocabulary of dual phenomena in gauge and gravity formulations.

enhances the chance of large black hole formation. The small black holes is dual to local

thermalization. When a large black hole is formed at the center, the whole boundary (brane)

system is equilibrated.

A remark is in order. In pure N=4 SUSY the interaction is the same at all scales.

Thus a very large expanding fireball of “CFT plasma” will never freezeout, and will expand

hydrodynamically forever till zero temperature is reached. Freezeout can be reached in the

gravity dual by switching to confining D-brane metrics, which is known as a Hawking-Page

transition.

With these considerations, the gravity dual of the RHIC collision can be set up by

considering the physical process together with the general dictionary of ADS/CFT listed

below. See table 1.

A gravity dual of this geometry is shown schematically in Fig 1. Even though matter

is fully stopped on the boundary (or probe brane), the initial entropy build up will cause

the formation of a fireball and its expansion. Its gravity dual is a detached set of small

black-holes and closed string states that fall into the AdS space. The falling black hole can

be described by the moving brane, which is the original set up mentioned above.

8

(a) (b) (c)

Figure 1: Schematic view of 3 stages of wall-wall collision. The horizontal coordinate is the 5th

dimension r, the vertical line is the longitudinal coordinate x3. (a) indicates the time before the

collision, when two walls of matter (open circles) approach each other; (b) is the time shortly after

the collision, when violent collisions cause color rearrangements and massive production of closed

strings, which flake out into the radial direction of AdS space (the shaded area); (c) A large black

hole(s) is formed (the black circle) at the center and falls into AdS flying away from the brane. Some

closed strings (the dashed lines) still fall into it.

3.1 Collision geometries

The problem we are addressing is quite formidable, so one obviously wants to simplify

at least the geometry of the problem as much as possible. A standard way to do so is

to imagine that colliding bodies may have infinite extensions in some directions, with the

solution naturally independent on the corresponding variables. Let us call the number of

“non-contracted” variables d.

The simplest geometry (i) would be a spherical collapse: One may imagine a spherical

shell of matter collapsing into itself with an initial radial velocity v and Lorentz factor

γ = 1/√1− v2. A fireball which is produced in this case is expanding in a spherically

symmetric way, producing a “Little Bang” like at RHIC, only in a much simpler spherical

geometry#1 The next geometry (ii) to consider is a collapse of a cylindrical shell, leaving

one “non-contracted” variable, d = 1. The gravity dual to it should have a black hole

with one less dimension. The geometry (iii) with d = 2, is a collision of two infinite 2d

walls. This is close to what happens at RHIC, where the colliding Au nuclei are Lorentz

contracted by a factor hundred into two thin pancakes. A variant of this are light-like

wall-wall collisions that pass through each other causing surface/string rearrangements in

the minimal impact parameter region much like the parton-parton scattering approach

originally suggested in [11, 12].

#1This has been considered by one of us many years ago [29] for e+e- collisions and prior to QCD. However,

due to asymptotic freedom this condition cannot be created experimentally: e+e- collisions in fact result in

2 jets, propagating from the collision points in random directions rather than a spherical expansion.

9

The realistic #2 case (iv) corresponding to RHIC is a collision of finite-size objects

(although as large as practically possible). Due to relativistic boosts the nuclei get flatten

in the collision direction x3. Furthermore, for non-central collisions the overlap region is

not axially symmetric but has an almond-like shape. Its gravity dual presumably would

create a black hole with a horizon of some ellipsoidal shape, with different dimensions in all

directions.

3.2 Fermi-Landau model and the entropy formation

In QCD and other asymptotically free theories we know that at small distances (close to

the origin) the interaction is weak. In the collision the constituent partons would literally

fly through each other. Thus the issue of entropy formation at RHIC is complex and, as

one may have suspected, not unanimous.

In contrast, in strongly coupled N=4 SUSY YM theory, there is no relation between

the coupling and the scale. At strong coupling, one may think that the colliding matter is

stopped, and that most of the entropy is produced promptly at this stage. Thus, we use for

this case the Fermi-Landau (FL) model [1, 2] as a benchmark for further comparison.

The main assumption of FL is that matter can be stopped in a Lorentz-contracted

size R = R0/γ, where R0, γ are the original size of the colliding objects and their Lorentz

factor. The volume in which it is supposed to happen is

V ∼ R30/γ

3−d (2)

where d is the number of “non-contracted” coordinates introduced in the preceeding sub-

section. The first step of the argument is to evaluate the temperature at this stopped stage.

The energy density is

ǫ = E/V ∼ γ4−d ∼ T 4 (3)

where the last equality is from the EoS of matter. Therefore the temperature grows with

the collision energy as

T ∼ γ1−d/4 (4)

The next step gives the amount of entropy produced:

S ≈ T 3V ≈ γd/4 (5)

#2The gauge theory under consideration is still not QCD but a strongly coupled N=4 SUSY YM

10

D

C

D

A

B

C

D

A

B

A

B

C

Figure 2: At each interaction vertex of two scattering mesons a closed string must pop up. This is

a unique feature of AdS space that does not take place easily in flat space.

One can see, that in the spherical collapse (i), there is no entropy growth because d = 0.

The lesson from it is that only the cases with less trivial geometry provide some interesting

predictions.

Despite the differences between the FL model and QCD, the entropy prediction for

the wall-on-wall case (iii), S ∼ γ1/2 ∼ s1/4, agrees with the observed multiplicity growth

quite well. We will return to the discussion of this point later.

3.3 Creation of Closed Strings

Short of solving the pertinent gravity equations in bulk with light-like wall-wall initial

conditions as described geometrically above, we will provide some heuristic arguments for

how the black-hole may form in bulk. The key idea is that as these walls come closer to

each other on the boundary they involve parton-parton and hadron-hadron scattering.

The many parton collisions at the boundary trigger i. elastic collisions which are dual

to massive closed strings; ii. inelastic collisions which are dual surface flips. An example

of the former process is shown in Fig. 3. Since the minimal string is not a straightline

connecting two sources on the boundary (infinite warping factor), the string must stretch

inside the AdS space [30, 31]. As two mesonic composites come together, the recombination

from AB + CD to AD + BC should happen just before B and C touch each other, since

that is energetically favored. For example, when the separation (in boundary) of AB and

BC are both L and that of BC is ǫ, then for small enough ǫ the difference of total lengths

of the string is

lAB + lCD − lAD − lBC = −2c

L+

c

2L+ ǫ+

c

ǫ> 0, (6)

where c is just a constant. Thus in a hadron-hadron scattering process, recombination of

the string must arise at the vertex (where B and C coincide) generating a closed string.

This is a remarkable feature of AdS space with no analogue in flat geometry. Although the

11

Figure 3: Multiple interaction vertices create a shower of massive closed strings in AdS space. The

strings flake and fall towards the AdS center. They may yield an AdS blackhole in bulk under certain

conditions.

above example is for pure AdS, we expect the mechanism to be universal regardless of the

geometry in the IR region if the UV region remains AdS.

Each of these liberate closed strings that fall in the AdS space under AdS gravity.

Some of the the closed string states could be in a black hole state if sufficiently much energy

is contained in a small enough region. However, the creation of the such black holes does

not correspond to the thermalized state on the boundary, which is dual to the large black

hole at the center of the AdS space. To form a global black hole located at the center of the

AdS space, we need many other pieces of closed strings in order to stop the falling small

black hole(s) at the center of the AdS space. This distinction leads to the difference in

heavy ion system and the small pp system contrary to [13].

The efficient creation of the particles provide a mechanism to convert the deposited

collision kinetic energy to mass resulting in lowering the temperature scale, which in turn

increase the strength of the interaction by the running coupling. From the boundary lan-

guage, the increase of number density of particle and the increase of interaction strength is

the key point to get the efficiency in the thermalization. The key point for the formation

of the large black-hole is that enough number of closed strings should be flaked out of the

boundary collision and they should be assembled by AdS gravity into a black hole with

horizon larger than the string size. In the latter, quantum gravity/ string theory properties

appears as well as the size R of the AdS space.

Following the initial flaking of the closed strings in the curved AdS space and the

black-hole formation, we still expect further elastic and inelastic collisions to take place at

the boundary implying further mass fall-off in bulk. The result is a slight increase in the area

of the black-hole which is the gravity dual of the dissipative processes following the onset of

the fire ball at the boundary (e.g. viscosity) which increase the entropy further. In bulk these

phenomena are quantum and correspond to graviton scattering on the black-hole which is

incoherent since the black-hole boundary is non-hermitean. Indeed, the viscosity coefficient

12

η was shown to be proportional to the horizon area with a universal ratio η/s = h/4π [6].

3.4 Estimates

For simplicity, we assume that the black-hole has been formed but that the flaking of closed

strings is still taking place, and ask: under what conditions the flaking strings can be

captured by the black-hole? what is the typical accumulated entropy? what is the typical

time for this entropy formation? Most of the arguments in this subsection are heuristic.

From the AdS black hole metric we have

ds2 = −fdt2 + f−1dr2 + r2dΩ25, with f = 1−G5M/r2 + (r/R)2. (7)

where we have ignored the distorsions caused by the boundary brane on the black-hole. The

horizon size of the black hole is

rbh = R

(

(

G5M

R2+

1

4

)1/2

− 1

2

)1/2

. (8)

Hence rbh = (R2G5M)1/4 := b for a large black hole, while rbh =√G5M for a small black

hole. #3 The temperature of the large black hole is given by Tbh = b/πR2, while that of the

small black hole is Tbh = R/2πb2 ∼ 1/√G5M . The large AdS black hole does not evaporate

while small black holes can. However, the hawking temperature goes up as it evaporate

while the fireball cools as it evolve. Therefore small black hole seems to be improper to

describe the RHIC fireball. Therefore throughout this paper we identify the RHIC fireball

with large AdS black hole.

We can express the mass and entropy in terms of Hawking’s temperature

M = R6T 4/G5,S

V3=

π3

2

R2 b3

G10≈ T 3. (9)

On the other hand, using G−15 = G−1

10 R5 = M8

pR5, Mp = 1/lsg

1/4s , R4 = gsNcl

4s , we can

express Hawking’s temperature in term of mass

T =b

π R2≈ 1

π√Nc

(

M

R3

)1/4

. (10)

The time the entropy is reached corresponds to typically the falling time in AdS space

τ =π

2R. (11)

#3Large black hole means G5M/R2≫ 1 with f ∼ (r/R)2(1 − b4/r4), and small black hole means

G5M/R2≪ 1 with f ∼ 1−G5M/r2.

13

In a typical RHIC experiment, hundreds of nucleons or thousands of quarks are

involved as shown above. As the interaction is almost simultaneous, thousands of interaction

vertices are involved and a shower of massive closed strings are created and fall into the

center of the AdS space. Let N = π R2N Q2

s be the number of such collisions with Q2s their

transverse density and π R2N the transverse nucleon size. In field theory charged quantas

moving with rapidity Y are surrounded by extra quanta distributed at smaller rapidities

dy = dx/x. In QED the Weiszacker-Williams approximation yields a flat distribution of

these quantas versus y, i.e. dN/dy constant. In QCD dN/dy is not constant and behaves

approximately as eα(t)(Y −y). HERA data suggests α ≈ 1/4 at t ≈ −1GeV2. In weak-

coupling the BFKL approximation gives αBFKL(0) = (4αN/π) ln 2, while at strong coupling

arguments based on AdS/CFT duality yield [12] αAdS(t) ≈ 7/96 + 0.23 t. In our case, we

will use a transverse parton density dN/dydx⊥ = Q2s(y) with Qs in general y-dependent.

Since the 5th AdS coordinate r is orthogonal to the boundary collision axis, only

momenta of closed strings with typically Qs are relevant. Then the typical total energy of the

closed strings is M ≈ ∫

dydN/dy Qs, with all strings assumed to be created instantaneously

at the impact. The energy of the closed string must be identified as the radial coordinate in

the AdS space. The strings flake towards the center of the curved AdS space under gravity

and arrive at the central region simultaneously. The average energy per string is ǫ ≈ Qs.

The total energy is therefore of order NQs. A black-hole forms when the horizon radius is

bigger than the size of the closed string. For the large black hole, the horizon distance is

rbh = (R2G5M)1/4, where G5 is the 5 dimensional Newton’s constant. Hence the black hole

formation condition is

Q−1s ≤ (R2G5M)1/4 . (12)

In terms of N(s) the number of pair collisions at the boundary, (12) reads

N ≥ R−2G5Q−5s . (13)

We recall that M ≈ Q3s and N ≈ Q2

s ≈ sα. The entropy generated at the surface by the

RHIC collision depends on the collision energy as follows

S ≈ T 3 V3 ≈ s9α/8−1/2 (14)

or an initial temperature T ≈ s9α/8.

14

4 The cooling of the RHIC fireball

In this section we discuss the cooling and expansion of the fireball on the brane. The

expansion of the fire ball, if we do not consider the mutual interaction, can be described

by the holographic description as in [7]. Indeed, in [7] we discussed the holographic image

of the gluon propagation following Mikhailov’s work [32]. the propagation of the gluon

corresponds to the falling of massless particle in AdS5 along the trajectory passing through

the center of the AdS space (trajectory connecting antipodal pair of the boundary).

The front of the fireball has some similarity to the propagation of gluon in free space

since it is composed of high energy particles which do not experience rescattering. Therefore

the front of the fireball should expand with the velocity of ligh. Hence the apparent size a

of the fireball should expand with a(τ) ≈ τ , which is consistent with Bjorken’s solution [33].

However, this is not the full story. Inside the bulk of the fireball, rescattering processes

substantially slow down the expansion. These processes are at the origin of equilibration

which is dual to the AdS black hole.

So the the fireball should have have at least two regions with different characteristics:

an equilibrated core and a less equilibrated surface zooming light-like. The latter expansion

is linear in proper time. The expansion of the core is in general more involved. However,

the late stages of the process can be addressed quite reliably. Indeed, at late times the

formation of the black-hole is complete and can be described by a static rather than dy-

namic background. We will assume that the probe brane remains undeformed. Since the

black hole is moving away from the probe brane we may describe this situation as a probe

brane receeding from the static AdS black hole (brane). The AdS black hole background

provides a net force on the receeding brane according to the Dirac-Born-Infeld action which

in turn determines its time evolution. The induced metric gives a direct reading on the

cooling/expansion of the fireball.

First, consider the simplest case where there is a black 3-brane sitting in 10d space

and in its near horizon geometry a parallel probe 3-brane is moving away from it. The black

3-brane is actually the geometry dual to the finite temperature gauge field theory with a

global temperature. In our first model we assume that the black hole is ’large’ #4 This

approximation is sensible for the core of the fire ball, or equivalently for a fireball that is

covering the whole universe. This will be analyzed in section 4.1. In reality, the black hole is

not so large and it is also moving away from the probe (boundary) brane. Correspondingly,

#4This means that it is large enough so that its effect can be approximated by the near horizon geometry

of the black brane.

15

in the dual gauge theory the fireball is not equilibrated uniformly and temperature changes

in time. We will analyze the situation in section 4.2.

When the gauge theory is confining at low temperatures, the pertinent background

is that of a doubly rotated black brane metric [4], where the euclidean time is compactified

to give a bubble of nothing. Therefore as the cooling proceed, the background must change

from black hole to confining background. This shift is known as Hawking-Page phase

transition. It is a dual to the confinement-deconfinement transition at the boundary. In

our set up, the probe brane feels the BH background at first. After some time when it

moved far enough distance from the BH, there is a background shift from BH to confining

background.

As shown in [14, 16, 15] moving branes in the background of black brane induces a

metric that describe a cosmological expansion on the probe brane. The corresponding 4-d

equations of motion on the brane should be equivalent to the Einstein equations. Note that

the hydro-like equations for matter T µν;ν = 0 are part of Einstein equations. This suggests

important generalization for the hydrodynamics due to the induced metric on the probe

brane.

4.1 Big Bang on a moving brane

We will discuss quantitatively the late stage cooling at RHIC using the brane cosmology.

In [14], it was shown that the brane motion in the AdS space can be interpreted as an

expansion of the universe. The rate of the expansion in a black hole background can be

precisely that of the radiation dominated universe although no source is assumed on the

brane. In [16] it was suggested that such radiation may be attributed to Hawking radiation.

In [15], similar equation of motion can be obtained by considering the effective action for

the probe brane. For review, see for example, [27]

To discuss the fate of the of the RHIC fireball, we notice that the temperature de-

creases while the entropy is fixed, thereby fixing the size of the black hole horizon. The

cooling is interpreted as the increased distance of our world from the black hole hori-

zon. This means that the brane motion, which has been used for the brane cosmology

[25, 14, 15, 16, 26, 27] is the appropriate set up to discuss the expansion and cooling of the

fireball.

The warping factor gx on the brane provides the scale factor of the fireball. The two

are tied by the equations of motion of the brane. Thus the brane move in the warped metric,

which is interpreted as a cosmological expansion (big-bang), the dual of the little bang in

16

RHIC collision. The cooling of the strongly interacting fluid formed in RHIC collision is

dual to a brane moving in the black hole background.

Although the real expansion is mostly 1 dimensional, we believe that the thermally

equilibrated expansion is 3 dimensional in nature. This is because the 1 dimensional expan-

sion is driven by the ultra relativistic motion of the initial particles whose speed can not be

caught up by the interactions.

We consider a class of metric given by the near horizon limit of non-extremal Dp

branes:

ds2 = g00dt2 + g(r)d ~xp

2 + grr(r)dr2 + gSdΩ8−p , (15)

where g = (r/R)(7−p)/2, |g00| = (r/R)(7−p)/2(1 − (b/r)7−p) = g−1rr and gS = r2(R/r)(7−p)/2.

The dilaton is given by

e2φ =

(

R

r

)(7−p)(3−p)/2

. (16)

If we neglect the brane bending effect and consider the configuration of zero angular

momentum of the brane around the sphere, the DBI action for the Dp brane

Sp = −Tp

∫

dp+1ξe−φ√

− det γαβ − Tp

∫

Cp+1, (17)

can be written as

Sp = −Tp

∫

dp+1ξe−φgp/2√

|g00| − grr r2. (18)

Since there is no explicit time dependence

E = p · q − L =gp/2e−φ

√

|g00| − grr r2− C, (19)

with C = (r/R)7−p, is a constant of motion. Using the equation of motion

grr r2 + g00 + gp|g00|e−2φ/(C + E)2 = 0, (20)

the induced metric can be written as

ds2 = −g200gpe−2φ

(C + E)2dt2 + gdx2. (21)

Defining the proper (cosmic) time τ by

dτ = |g00|gp/2e−φ/(C + E)dt, (22)

17

the induced metric can be written as a zero curvature Friedman-Robertson form

ds2 = −dτ2 + a2(τ)dx2, (23)

where a2 = g(r(τ)). The equation of motion can be rewritten in terms of a and τ

(

a

a

)2

=

(

(C + E)2e2φ

|g00|grrgp− 1

grr

)

(

g′

2g

)2

, (24)

with g′ = dg/dr. Then, the equation of motion in terms of a and τ is given by

(

a

a

)2

=

(

7− p

4

)2

a2(3−p)/(7−p)

[

(E

a4+ 1)2 −

(

1− b7−p

R7−p

1

a4

)]

, (25)

where we have used the fact C = (r/R)7−p = a4. Notice that the effect of the RR-flux field

C is to provide a strong enough repulsion force to cancel the confining AdS gravity.

As a → ∞ (late evolution), we have

a(τ) ≈ τ (7−p)/(11−p). (26)

The scale factor evolution a(τ) captures the cooling of the fire-ball at the boundary through

its holographic dual:

T (a) =Tbh√

|g00|≈ Tbh

a(τ), (27)

with

Tbh =(7− p)

4πb·(

b

R

)(7−p)/2

(28)

as the black hole temperature. The local temperature is the black hole temperature observed

by the observer in the probe brane. This is the actual temperature of the fireball. As the

brane moves away from the black hole, the brane world (the fireball) expands and cools

according to T (a) = Tbh/a.

There are two interesting cases: p = 3 and p = 4. For p = 3,

a(τ) ∼ √τ , T ∼ 1√

τ. (29)

The reason for considering p = 4 is that one of its direction (say x4) in a confining theory

is compactified. After the compactification the p = 4 and p = 3 are identical. Without

compactification a ≈ τ3/7 which is a stronger warping.

This result is to be compared with cooling in D-space. Indeed, the entropy for a

(perfect) gas is just S ≈ TD VD. For a relativistic d-space hydrodynamical expansion we

18

expect VD ≈ VD−d τd. For fixed entropy, the temperature falls like T ≈ 1/τd/D. Bjorken

1-space expansion (29) corresponds to D = 3 and d = 1, therefore T ≈ 1/τ1/3. Fully 3-space

expansion corresponds to T ≈ 1/τ . The AdS case with T ≈ 1/√τ is faster than Bjorken

in 1space but slower than perfect hydrodynamical expansion in 3space. It is like fractal

with d = D/2 = 3/2. One may summarize these result by saying that strong interactions

slow down the expansion of the fireball just as gravity does in the dual picture. We note

that since the viscosity is quantum with η/(S/V3) ≈ h/4π its effects are not present in our

estimates. Their consideration follow from perturbation theory and are easily seen to delay

the cooling.

When T cools enough such that T < ΛQCD, there must be a Hawking-Page transi-

tion [35] and the background metric is replaced by

ds2 = (r/R)3/2(−dt2 + d~x2 + f2(r)dx24) + (R/r)3/2(dr2/f1 + r2dΩ2

4), (30)

where f2 = 1− (rKK/r)3 refers to the compactified direction. Witten [4, 34] suggested that

the transition to this metric maybe interpreted as the confinement/deconfinement phase

transition. The equation of motion in the new background can be calculated. Though

minor, there are a few differences in detail of the calculation, but rather surprisingly, the

final outcome is precisely the same with the substitution b → rKK. For p = 4,

(

a

a

)2

=9

16a−2/3

[

(

E

a4+ 1

)2

−(

1− r3KK

R3

1

a4

)]

. (31)

As we discussed before, The front factor 9/(16a2/3) disappears if x4 is compactified (which

is effectively p = 3). The phase transition point in terms of the brane position occurs when

the warping becomes aF at

T (aF ) ≈ TKK , (32)

where the Kaluza-Klein temperature is given by TKK = 3r1/2KK/(4πR3/2). Thus

aF =Tbh

TF=

√

b

rKK(33)

One may interpret the phase transition as hiding of the black hole horizon behind the KK

singularity r = rKK ≈ 1/ΛQCD. After this phase transition, hadron creation begins and

the fireball ultimatly freezes out when the pions decouple.

19

r(t)

ρ

r(t)

Figure 4: Sketches of the brane motion for the Big Bang (a) and Little bang (b) geometries. (a)

refers to a brane moving away from the large “black brane” with a time-dependent distance r(t) in

the 5th dimension. (b) refers to an asymptotically flat brane with the brane to black hole distance

r⊥(t) and ρ(t) the effective spherical size of the fireball on the brane. Hawking’s radiation has a

homogeneous time-dependent temperature in (a), while it depends on ρ(t) in (b).

4.2 Little Bang on a moving brane

This case is schematically illustrated in Fig. 4 (b). The RHIC collision is taking place in the

(near plane) brane inside the AdS. After the collision, the black hole forms inside the ads

space and the the observer on the brane sees a spherically expanding fire ball with its center

at the point closest to the black hole. The size expansion of the fireball is discussed in the

previous section under the assumption of infinite fireball. The issue to be addressed here is

the shape of the fireball, namely the spatial profile of the finite fireball. This is readily given

by the warping factor of induced metric if the embedding of the probe brane is given. The

temperature at the center of the fire ball is given by the warping factor as determined above

using the distance r from the black hole center to the fire ball center. The temperature at

all other points in the moving brane is warped further since the effective distance now is√

r⊥(t)2 + ρ2, and vanishes asymptotically.

While the precise determination of imbedding of moving brane is very involved, its

shape in the region far from the black hole can be readily discussed approximately by

neglecting the bending effect. More precisely, from the metric

ds2 = −fdt2 + f−1dr2 + r2dΩ25, with f = 1−G5M/r2 + (r/R)2, (34)

20

6

8

10

12

14 0

5

10

15

0.05

0.1

0.15

6

8

10

12

14

Figure 5: Plot of temperature with range 0 ≤ ρ ≤ 15 as brane position r⊥(t) move from 5 to 15.

Bending of the brane is not considered assuming that the brane is far from the black hole. We set

numerical values for G5M = 1, R = 1, T0

√

f(r0) = 1. Notice that the warping factor (∼ r2), which

will overcompensate the apparent shrinking in fireball size, is not taken into account here.

temperature is approximately given by

T (t) = T0

√

f(r0)/√

1−G5M/(r⊥(t)2 + ρ2) + (r⊥(t)2 + ρ2)/R2, (35)

where T0 is the temperature of black hole at a reference distance say r = r0. Compared

with previous subsection, the temperature running has spatial dependence ρ as well as

the distance from the black hole, r⊥(t). It represent the temperature profile which has

a peak at the fireball center, ρ = 0, and decreases as ρ increases. See figure 5. The

upshot of this analysis is that the presence of a distance black hole in bulk produces small

additional forces on moving matter in the moving brane. Even without considering the

bending effect of the brane, the presence of blackhole is detected through the metric. For

instance, hydrodynamical flow of matter on the brane is now described by

T νµ;ν = ∂νT

νµ − Γα

µνTνα + Γν

ανTαµ = 0 (36)

The source of the Christoffels Γ (gravitational force) are two fold: one is the expansion

induced by the motion of the brane inside a warped background, the other is due to the

presence of the black hole. These modifications to ideal hydrodynamics are not small even

at late stages as far as the strong character of the interaction sustains, namely as far as the

ads setup is valid. However, these effects are small far from the fireball center at all times.

21

These analysis will provide yet another route to non-ideal hydrodynamics for N=4 SYM

theory at strong coupling. In particular to the calculation of the transport coefficients. This

will be reported elsewhere.

5 Discussion

Recent heavy ion collisions at RHIC have suggested that the promptly released partonic

phase is strongly interacting in the form of a sQGP. Two of us have argued recently that a

good starting point for adressing key issues of the sQGP is N=4 SYM at strong coupling.

In this paper we have suggested that the entirety of the RHIC collision process from the

prompt entropy release to the freezeout stage can be mapped by duality to black hole

formation and evolution in AdS space. In other words RHIC little bang and cooling is dual

to a cosmological big bang with a flying black hole as a proceed.

We have provided simple physical picture of black hole formation and thermalization

from string theory point of view and gave a rough estimates for the black hole formation

and entropy generation based on string flaking in AdS space. The typical string sizes are of

the order of the inverse saturation length 1/Qs. While the cooling of the edges of the fireball

is luminal or 1/τ , that of the core is slower due to the strongly interacting sQGP. We have

suggested that due to the strong interaction of the fireball liquid, the expansion is slower

than expected from the ideal gas model. The cooling of the fireball is 1/√τ which is slower

than Bjorken 3d cooling 1/τ . The strong nature of interaction slows down the expansion rate

hence the cooling is slower than expected from the Bjorken solution. Cooling freezes when

the background is replaced by confining background through the hawking-page transition.

Non-cosmological like expansions with realistic fire ball geometries on the bound-

ary are more involved to analyze. We have suggested that their asymptotic stages can

be mapped on black hole perturbation theory resulting in non-ideal hydrodynamics from

conventional Einstein gravity. We will report on these issues and others in future.

Acknowledgments

The work of SJS was supported by KOSEF Grant R01-2004-000-10520-0 and by SRC

Program of the KOSEF with grant number R11 - 2005- 021. The work of ES and IZ was

partially supported by the US-DOE grants DE-FG02-88ER40388 and DE-FG03-97ER4014.

22

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