Early thermalization, hydrodynamics and energy loss in AdS/CFT

4 MB·English

by  Paul M. Chesler

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Chapter 1
5
1 Early thermalization, hydrodynamics and energy loss in
0 AdS/CFT
2

n
a Paul M. Chesler
J
Department of Physics, Harvard University, Cambridge, MA 02138, USA
0
2
Wilke van der Schee

] MIT Center for Theoretical Physics, Cambridge, MA 02139, USA
h
-t Gauge/gravity duality has provided unprecedented opportunities to study dy-
l namics in certain strongly coupled gauge theories. This review aims to highlight
c
u severalapplicationstoheavyioncollisionsincludingfar-from-equilibriumdynam-
n ics, hydrodynamics and jet energy loss at strong coupling.
[

1 1. Introduction
v
2 Gauge/gravitydualityequatescertaingaugetheorieswiththeoriesofquantumgrav-
5
ity in one higher dimension.1 This extra dimension has a natural interpretation as
9
4 the renormalization group scale, and has led to the name “holographic duality." In
0 the limit where the gauge theory is strongly coupled and has a large number of
.
1 colors N the dual description reduces to classical supergravity. Hence, challenging
c
0 stronglycoupledquantumdynamicsinthegaugetheorycanbeaccessedbysolving
5
classical partial differential equations. All physics — from microscopic interactions
1
: tomacroscopichydrodynamics—isencodedinthedualclassicaldynamics. Holog-
v
raphy can thereby provide systematic and controlled access to strongly coupled
i
X dynamics via the solution to the classical partial differential equations. Currently
r there is no other theoretical tool capable of accessing strongly coupled real-time
a
dynamics in in a controlled setting.
HeavyioncollisionsatRHICandtheLHChavedemonstratedthattheproduced
quark-gluon plasma is strongly coupled.2 Hence it is of interest to model heavy ion
collisions using holography. However, while there exists many theories with dual
gravitational descriptions — some with QCD-like features such as confinement and
chiral symmetry breaking3,4 — the dual description of QCD is not known (if it
exists). Nevertheless, given the immense challenge of studying strongly coupled
dynamicsinQCDitisinvaluabletohaveamodelwherestronglycoupleddynamics
can be studied in a controlled systematic setting. This is especially true if the
results are valid for a class of strongly coupled gauge theories and hence have some
degree of universality.
1
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2 Paul M. Chesler and Wilke van der Schee
Inthisreviewweshallfocusonthesimplesttheorywithaholographicdual: 3+1
dimensional super-Yang-Mills theory with four supersymmetries ( =4 SYM). In
N
thelimitwheretheSYMisstronglycoupledandhasalargenumberofcolorsN the
c
dualdescriptionreducestoclassicalsupergravityinasymptoticallyfivedimensional
Anti-deSitterspacetimeAdS 1 (seealso5,6). WhilethegroundstateofSYMisvery
5
differentfromQCD—SYMisconformalandcontainsnoparticleswhereasQCDis
confining and has a rich spectrum of hadrons — at temperatures not too far above
the deconfinement transition both theories consist of strongly coupled non-Abelian
plasmas. ItisinthissettingwheredynamicsinSYMcanpotentiallyprovidelessons
for dynamics in QCD.
In what follows we highlight insights obtained via holography in three settings.
First, we discuss relativistic hydrodynamics including computations of the shear
viscosity at strong coupling and the construction of causal viscous hydrodynamics.
We then discuss far-from-equilibrium dynamics and the collisions of shock waves in
SYM and their application to early time dynamics in heavy-ion collisions. Finally,
we give a brief review of progress made in jet quenching and energy loss at strong
coupling.
2. First successes: viscosity and relativistic hydrodynamics
2.1. Viscosity from black hole horizons
Gauge/gravitydualityprovidesaverynaturalwaytodescribethermalstates: they
aredescribedbyblackholes,whicharethesimpleststationarystatesinAdS gravity
5
withanotionoftemperature. Thistemperatureissimplytheblackhole’sHawking
temperature.7 The topology of the horizon is inherited from the gauge theory,
where a thermal state usually is extended infinitely; this is one major difference
from ordinary black holes in our universe, and in fact these black holes in AdS are
5
more appropriately called black branes. Another major difference is the boundary
oftheAdS spacetime,whichcanbethoughtofastheplacewherethegaugetheory
5
lives. This boundary reflects back the thermal radiation, and the black branes are
hence in stable thermal equilibrium, as opposed to black holes in flat spacetime,
which evaporate awaya.
It is also possible to construct gradient expansion solutions to Einstein’s equa-
tions, where the local geometry is approximately that of boosted black brane with
slowly varying temperature and boost velocity.8,9 In this case the corresponding
dual gauge theory state is that of a system in local thermal equilibrium with dy-
namics governed by hydrodynamics. Simply put, in the limit of slowly varying
fields Einstein’s equations reduce to hydrodynamics in one less dimension. The
aAlso,blackholesinouruniversetendtohaveverysmalltemperaturesoforderofmK,asblack
holesofhighertemperaturearehardtoformandwouldquicklyevaporate. Thethermalstabilityof
blackholesinAdSallowsanytemperature,andinfactthetemperatureofblackholesrepresenting
aquark-gluonplasmawouldhavethesameveryhightemperatureoforder1012K.
21stJanuary2015 1:22 WorldScientificReviewVolume-9.75inx6.5in chapter1.4
Early thermalization, hydrodynamics and energy loss in AdS/CFT 3
connection between slowly varying solutions to Einstein’s equations and hydrody-
namics is known as the fluid/gravity correspondence. By matching the gradient
expansion obtained by solving Einstein’s equations to the constitutive relations of
hydrodynamics, transport coefficients such as the viscosity can be computed at
strong coupling.
At first order in gradients the constitutive relation for the stress tensor Tµν in
relativistic neutral hydrodynamics read10
T =eu u +p[e]∆ +π , where, (1)
µν µ ν µν µν
∆ =g +u u and (2)
µν µν µ ν
π = η[e]σ ζ[e]∆ ( u)+ (∂2), with (3)
µν µν µν
− − ∇· O
2
σ =∆ ∆ ( µuν + νuµ) ∆ ∆ αuβ, (4)
µν µα νβ µν αβ
∇ ∇ − 3 ∇
where e is the proper energy density, u the fluid velocity, p[e] is the pressure
µ
(which is a function of the proper energy via an equation of state), π is the shear
µν
tensor, η the shear viscosity and ζ is the bulk viscosity, which vanishes in scale
invariant theories such as SYM. Note that the fluid velocity and energy density are
defined as the time-like eigenvector and associated eigenvalue of the stress tensor
(T uν = eu ) and that σµν is transverse and traceless: uµσ = σµ = 0.
µν − µ µν µ
Alternatively one can say that the fluid velocity is defined such that when boosting
T with velocity u there is no momentum flow, i.e. T = 0, which is called the
µν µ 0(cid:48)i
Landau frameb. The conservation equation µT =0 together with the equation
µν
∇
of state and the viscosity (computed theoretically or measured) now form a closed
system of equations, fully determined by a given initial energy density and fluid
velocity.
The equation of state, viscosity and higher order transport coefficients are not
determined by hydrodynamics itself, and are a property of the microscopic theory
under consideration. In weakly coupled QCD it is possible to use the Boltzmann
equation to perturbatively compute the shear viscosity, which led to12
T3 1
η =κ + ( ), (5)
weak g4log(1/g) O g4
with T the temperature, g the gauge coupling and κ a coefficient depending on
the number of colors and flavors, which can be computed numerically. Non-
perturbatively it is unknown how to compute the viscosity of QCD. However, there
have been attempts to compute the viscosity using lattice techniques (see13 for a
review). It was therefore ground-breaking when a few years after the discovery of
gauge/gravity duality Policastro, Son and Starinets computed the shear viscosity
of a thermal plasma described by strongly coupled SYM theory:14
s π
η = = N2T3, (6)
4π 8 c
bWhen including a charge density with corresponding current and chemical potential it is also
possible to define a frame where there is no charge flow, which is called the Eckart frame.11 In
thisreviewwedonotincludeachargedensity.
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4 Paul M. Chesler and Wilke van der Schee
with s is the entropy density. The factor of 1/4π is surprisingly universal and valid
foranystrongcouplingandlargeN quantumfieldtheorywithagravitationaldual.
c
Forinstanceaddingchargedensitychangesboththeshearviscosityandtheentropy
density, but its ratio is still 1/4π.
Nevertheless, it is possible to have holographic models with different viscosity
to entropy ratios, most notably by including higher derivative corrections in the
Einstein equations. In this case corrections for finite N can actually make the η/s
c
ratiodropbelow1/4π,therebyviolatingtheconjectureduniversallowerbound15 on
the viscosity.16 Nevertheless, general considerations using the quantum mechanical
uncertainty principle still hint at the existence of a lower bound, even though its
value is likely smaller than 1/4π.
HolographycanalsoyieldfinitecouplingandfiniteN correctionstotheviscos-
c
ity. For SYM the results read:17–19
1 15 5 λ1/2
η/s= 1+ ζ(3)+ , (7)
4π λ3/2 16 N2
(cid:18) c (cid:19)
with λ = g2N the ’t Hooft coupling. As expected from (5) the viscosity increases
c
asthecouplingconstantdecreases,andtheincreasecanbesignificantatreasonable
values of the coupling constant (for λ 20 and N 3 the corrections both are
c
∼ ∼
roughly 20%). Current hydrodynamic models of the quark-gluon plasma indeed
favor an η/s in the range 1/4π to 2.5/4π.20–23
2.2. Second order relativistic hydrodynamics
Figure 1. (left) The second order hydrodynamics derived using AdS/CFT (AdS) is compared
withtheolderweaklycoupledMuller-Israel-Stewarthydrodynamcis(IS),whichusesτπ =6η/sT
anddoesnotincludetheλ1orλ2ineqn. 8. Fortheseinitialconditionsthespatialandmomentum
anisotropy (ex and ep) do not depend much on the precise value of the second order transport
coefficients(figurefrom24). (right)Similarhydrodynamicshasbeenusedtoobtainrecentviscosity
over entropy ratios of roughly 0.12 and 0.2 at top RHIC and LHC collisions respectively (LHC
estimateshown,figurefrom23).
Holographyandinparticularthefluid/gravitydualityalsohasledtoamoresystem-
aticunderstandingof(relativistic)hydrodynamicsitself.8,25,26 Asalludedtoabove,
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Early thermalization, hydrodynamics and energy loss in AdS/CFT 5
via the fluid/gravity correspondence it is possible to systematically solve Einstein’s
equations. The gradient expansion solution encodes the constitutive relations of
hydrodynamics to all orders in gradients and associated transport coefficients.8,25
Let us begin by simply stating the results of8,25 for the shear tensor πµν up to the
second order in gradients.
Since SYM is conformal, to write πµν at second order in gradients it is useful
to employ derivative operators which transform covariantly under conformal
µ
D
transformations. For tensors Qµ... which, under a conformal transformation g
ν... µν →
e2φg˜ , transform like Qµ... =e wφQµ..., is defined by26
µν ν... − ν... D
DσQµν...... ≡∂σQµν......+wAσQ(cid:101)µν......+γσµλQλν......+...−γσλνQµλ......−...,
γλ g Aλ δλA δλA ,
µν ≡ µν − µ ν − ν µ
∂ uν
A uν∂ u ν u .
µ ν µ µ
≡ − 3
With this definition derivatives transform covariantly: Qµ... = e wφ Qµ...,
Dλ ν... − Dλ ν...
and for instance Eq. (4) simplifies to σ = u + u . The full second order
µν µ ν ν µ
D D
conformal shear tensor is now given byc: (cid:101)
∆
π = ησ +ητ uλ σ +λ [σ σλ µνσαβσ ]
µν − µν π Dλ µν 1 µλ ν − 3 αβ
∆
+λ [ω σλ +ω σλ]+λ [ω ωλ + µνωαβω ], (8)
2 µλ ν νλ µ 3 µλ ν 3 αβ
where ω = u u . In strongly coupled SYM the second order transport
µν µ ν ν µ
D −D
coefficients are given by τ = (2 ln(2))/2πT, λ = η/2πT = 2λ /ln(2) and
π 1 2
−
λ = 0.8 Note that all these coefficients get considerably more complicated when
3
including a chemical potential; in particular λ becomes non-zero.27
3
Second order hydrodynamics is particularly useful for numerical simulations of
viscous hydrodynamics. This is because high momentum modes in first order rela-
tivistichydrodynamicscanpropagatefasterthanthespeedoflight. Fundamentally
this is not a problem, as hydrodynamics is an expansion in long wavelengths, i.e.
small momenta, and these acausal modes are outside the regime of its applicabil-
ity.28 Nevertheless, these modes make the equations numerically unstable29 and
therefore in numerical simulations a second order term was added to stabilize the
equations, giving Muller-Israel-Stewart hydrodynamics.30,31
Muller-Israel-Stewart hydrodynamics can be seen as a special case of second
orderhydrodynamicswhereλ =0. Oneinsightwhichholographyhelpedsolidifyis
i
that,incontrasttotheMuller-Israel-Stewarttheory,atagivenorderinthegradient
expansiononemustaddeverypossibletensorstructuretotheconstitutiverelations
whichisconsistentwiththesymmetriesoftheunderlyingquantumfieldtheory. This
hadleadtothedevelopmentofcausalviscousrelativistichydrodynamics.24,32,33 In
cMostofthesecomputationshavebeendoneforfluidslivinginacurvedspacetime,25 whichgives
anextratermnotconsideredinthisreview.
21stJanuary2015 1:22 WorldScientificReviewVolume-9.75inx6.5in chapter1.4
6 Paul M. Chesler and Wilke van der Schee
conformal theories such as SYM the equations of second order hydrodynamics give
causal propagation for high momenta modes.
While the second order transport coefficients make simulations much easier to
perform, the precise value of these coefficients fortunately does not seem to have
a big influence on the particle spectra in realistic simulations (see figure 1, left).
Thelatterfeaturedoeshowevercruciallydependonusingconformalhydrodynamic
equations; when using (second order) non-conformal equations results do seem to
depend on the second order coefficients.34 One of the biggest successes of these
equations are estimates for the shear viscosity of the quark-gluon plasma,20–23 as
shown in Fig. 1 (right).
Anothermorerecentexampleofabetterunderstandingofhydrodynamicscomes
from,35 where in a specific boost-invariant setting all transport coefficients were
computednumericallyuptoorder240inthegradientexpansion. Withthosecoeffi-
cientsitcouldbeshownthatthehydrodynamicgradientexpansionisnotnecessarily
convergent, it was found to be an asymptotic series. Moreover, it was understood
why hydrodynamics did not converge, since the divergent hydrodynamic expansion
contains information about the non-hydrodynamic modes. Within AdS/CFT these
modes are included in the so-called quasi-normal modes: vibrations of the black
hole horizon. Strikingly, by a Borel resummation reference35 was able to extract
the precise value of the dominant quasi-normal mode from the hydrodynamic ex-
pansion, thereby showing how hydrodynamics contains information about its own
break-down.
Much work has been devoted to study charged hydrodynamics, extending the
previousformulastoaplasmawithaconservedcurrent,chemicalpotentialandvar-
ious new transport coefficients, see for instance.27,36 More recently there has been
interest in anomalous hydrodynamics, where quantum anomalies lead to modified
hydrodynamic equations. A plasma with a non-trivial axial charge density will de-
velop an electric current J(cid:126) in the direction of an external magnetic field B(cid:126) (chiral
magnetic effect or CME):37,38
e2
J(cid:126)= µ B(cid:126), (9)
2π2 5
where µ is the chemical potential associated with the axial charge. For averaged
5
heavy ion events µ would be zero, but a full event-by-event anomalous hydrody-
5
namic simulation suggests that this effect is measurable in heavy ion collisions.39
Similarly to the analysis above, also for the CME holography has proved extremely
useful to show how this effect is in fact necessarily included already in first order
charged hydrodynamics, and also to estimate the relevant transport coefficients at
strong coupling.40,41
Lastly, inspiration from the symmetries present in AdS has led to non-trivial
analytic solutions of the hydrodynamic equations. An interesting solution was
provided by Gubser,42 where a solution to viscous conformal hydrodynamics with
boost-invariance and a non-trivial expansion in the transverse plane was found.
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Early thermalization, hydrodynamics and energy loss in AdS/CFT 7
Thissolutionfollowsfromasimpleset-upinAdS,butthenleadstothisnon-trivial
analytic and hence very useful hydrodynamic solution. Recently, this approach has
been extended to an exact solution of the relativistic Boltzmann equation.43
Interestingly, aside from the computation of transport coefficients, much of the
development in hydrodynamics above need not rely on gauge/gravity duality. Nev-
ertheless, inspirationfromthedualityhasledtothismorepreciseunderstandingof
hydrodynamics, by using gauge/gravity as a theoretical arena to get new insights.
3. Fast thermalization and the resulting stress tensor
3.1. Simple models
Perhaps an even more remarkable feature of holography is the straightforward pos-
sibility to study far-from-equilibrium dynamics in strongly coupled gauge theories.
How do non-equilibrium systems thermalize and relax to local equilibrium? Us-
ing holography answering this question amounts to studying real time dynamics
in the gravitational dual, which is not much different from conventional problems
in general relativity, such as binary neutron stars, or black hole inspirals.44,45 As
is well-known these latter problems can still be very challenging from a numerical
point of view, but compared to non-perturbative quantum field theory it is defi-
nitely an enormous simplification. For a review of the use of numerical relativity in
holography see for example.46
The first studies of holographic thermalization computed how perturbed black
holes relax to homogeneous black holes, computing the so-called quasi-normal
modes.47 These are determined by expanding the perturbation of the metric in
a sum of modes proportional to e−iωit, with complex frequency ωi. The Einstein
equations and boundary conditions at the horizon and boundary then dictate that
thereisadiscretespectrum,wherebythemodealwaysdecays(i.e. (ω )<0). The
i
(cid:61)
mode with the slowest decay will then determine the late-time decay back to the
thermal black hole, and hence can be said to determine the thermalization time in
thissetting. Dimensionalanalysisinthisscaleinvarianttheorydictatesthatω T
i
∼
, but the coefficient has to be computed numerically, giving (ω ) 8.6T.47
0
(cid:61) ≈
The estimate from quasi-normal modes could imply a thermalization time
t < 1/T, which for a temperature of 1000 MeV would be about 0.2 fm/c.
therm
Comparedtoperturbativemethodsthisisveryfast,whichshouldbeexpectedfora
systematstrongcoupling. Ontheotherhand,eventhoughthecouplingisassumed
to be infinite, there is still a finite answer for the thermalization time, leading to
the intuition that strong interactions still have some microscopic physics present as
the scale 1/T.
Nevertheless, thethequasi-normalmodeanalysisisonlyvalidnearequilibrium:
the state is assumed to be a small perturbation of the final thermal state. In heavy
ion collisions the initial state is very far-from-equilibrium, so solving the full non-
linearEinsteinequationsisofgreatimportance. Oneofthefirststudiessolvingthis
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8 Paul M. Chesler and Wilke van der Schee
numerically is presented in,48,49 where quark-gluon plasma formation and thermal-
ization was studied in both homogeneous and boost invariant set-ups (see fig. 2).
Thegaugetheorydynamicswas,bysymmetry,effectively1+0dimensionalandthe
dualgravitationaldynamicswas1+1dimensional. Inthesepapersnon-equilibrium
stateswerecreatedbystartinginthegroundstateandturningonbackgroundfields.
It was found that even in the case where the initial state is the vacuum, a thermal
state can be achieved in times of order 1/T with T the final state temperature.
Anintriguingfindingin49 wasthatwhenthesystemfirststartstobehavehydro-
dynamically, viscouseffectswereorderone: thefirstorderviscouscorrectiontothe
constitutiverelations(1)isasbigastheleadingterm. Nevertheless,evenwithsuch
large viscous corrections the hydrodynamic gradient expansion was seen to be well
behaved. Moreover,theapplicabilityofhydrodynamicswasseennottobegoverned
by convergence of the hydrodynamic gradient expansion. Instead, the applicabil-
ity of hydrodynamics was seen to be governed by the decay of non-hydrodynamic
degrees of freedom.
The approach in49 was extended in.50 Here the main feature was the possibility
tochoseseveralinitialstatesandtoevolvethemwithoutturningonanybackground
fields. For all initial states they found a thermalization time less than 1/T, with T
the temperature at the moment of thermalization.
Of particular utility, this paper also introduced a dimensionless quantitiy,
F(w) τ∂ w, where w T τ can be thought of as dimensionless time and
≡ w τ ≡ eff
T is the temperature associated with the energy density at that time. The di-
eff
mensionless function F(w) is completely determined within hydrodynamics and for
instance equals 2/3 for ideal hydrodynamics. As F(w) is unique within hydrody-
namics it is clear that differences between F(w) found for the different evolutions
in AdS/CFT must be due to genuine non-hydrodynamic effects, as is convincingly
shown on the left of Fig. 3.
Therightoffigure3showsasimilarstudyinahomogeneoussetting, withmore
than 1000 initial states, also leading to thermalization within a time of 1/T.51,52
The main new result in this paper was to compare the full non-linear thermaliza-
tionprocesswiththelinearizedquasi-normalmodeapproachoutlinedinthesecond
paragraph of this section. Quite surprisingly it was found that for general initial
states the evolution was always well described, within 20%, by the linearized ap-
proximation. This then also confirmed that the quasi-normal mode estimates of
strongly coupled thermalization times were indeed accurate also at the non-linear
level.
Lastly, these studies can be corrected for finite coupling effects, which has re-
centlybeendonein.53,54 Notsurprisingly,finitecouplingcorrectionsslowdownthe
thermalization, as in these papers becomes apparent through a smaller imaginary
part of the lowest quasi-normal mode.
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Early thermalization, hydrodynamics and energy loss in AdS/CFT 9
Figure 2. A simple and often used model of a heavy-ion collision was proposed by Bjorken,55
whereheassumedthataheavy-ioncollisionisapproximatelyboost-invariant,atleastnearz=0.
Thismeansthatallphysicsonlydependsonpropertimeτ =√t2 z2,andhencethatallphysics
−
experiencedbyobservers,suchasatemperaturefieldillustratedhere,isindependentofitsframe.
Thetworedlineswouldillustratetwosuchframes,whichindeedhasequaltemperaturesatequal
τ (figurefrom56).
Figure3. Theleftfigureshowsboostinvariantthermalization,wheretheevolutionof29random
isneictoianlds-taantedstohfiFrd(-wor)d≡erwτhy∂dτrwodaysnaamfuinccst(iofinguorfewfr≡omT5e0f)f.τItisccaonmbpeasreeednwtihthatptrheedyicatilolnthsefrrommalfiizresti-n,
atimewhereτ (cid:46)0.5/Teff,whereTeff isthetemperatureassociatedwiththeenergydensityat
thattime. Therightfiguredidasimilaranalysisinahomogeneousset-up,usingover1000initial
states. Also there thermalization always happens within t(cid:46)1/T. Here it is also shown that the
difference between the linearized and fully non-linear Einstein equations is small, almost always
lessthan20%(figurefrom51).
3.2. Colliding shock waves
All models presented above contained enough spacetime symmetry to make the
gauge theory dynamics effectively 1+0 dimensional. Here we shall review results
where the gauge theory dynamics are allowed to be 1+1 and 1+2 dimensional. In
holography the energy-momentum in the gauge theory directly corresponds to the
metric in the gravitational theory, so to simulate highly energetic colliding nuclei
it is therefore natural to consider collisions of gravitational shock waves, moving at
the speed of light.57–62 The shock waves can be viewed as analogous to the shock
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10 Paul M. Chesler and Wilke van der Schee
waves constructed in flat space by Dray and ’t Hooft,63 which consist of taking a
Schwarzschild black hole and boosting this to the speed of light, keeping its total
energy fixed. The shock waves are dual to lumps of energy in the gauge theory,
which propagate at the speed of light. When the shock waves collide they can
produce a black hole which will then relax to equilibrium. Likewise, in the dual
gauge theory, the collision will result in the formation of a non-Abelian plasma
which will subsequently thermalize.
One interesting result from the aforementioned studies of shock wave collisions
was the entropy production, estimated by the area of the trapped surface of the
formedblackholeatthetimeofcollision. Thistrappedsurfacegivesalowerbound
on the area of the black hole horizon, which is directly related to the produced
entropy in the gauge theory. Ref.59 estimated that:
√s 2/3
S S 35000 NN , (10)
≥ trapped ≈ 200GeV
(cid:18) (cid:19)
where S is the entropy produced in a central gold nucleus-nucleus collision with a
nucleon-nucleoncollisionofenergy√s andS istheentropyofthetrapped
NN trapped
surface. The entropy increase during the evolution of the quark-gluon plasma is
not too big due too to the relatively small viscosity, but can in fact be sizable as
highly anisotropic plasmas increase the entropy production. Nevertheless Eq. (10)
gives a lower bound on the final entropy produced, which is directly related to the
measurable total charged particle multiplicity N =S/7.5.64 In56 the estimate
tot,ch
(10)wasmadesomewhatstrongerbydoingcalculationsnumericallytilllatertimes,
but two important features remained. Firstly the numerical value is about right
for RHIC collisions, albeit on the high side, which is impressive as Eq. (10) is a
direct calculation, without having any input from other experiments. Secondly,
the 2/3 power law as a function of √s is a robust outcome of a scale invariant
NN
theory. Nevertheless, with the LHC results now known,65 the experimental data
favorsapowerof1/2,andEq.(10)thereforeoverestimatedthemultiplicityatLHC
collisions. This can be taken as an indication that QCD is not scale invariant at
energies probed between RHIC and LHC.
Thepreviouscalculationscouldnotsaymuchaboutthedynamicsafterthecolli-
sion,whichrequiressolvingthefullnon-linearEinsteinequations. Itwasthereforea
majorbreakthroughwhenarealcollisioncouldbesimulatedwithingravity.66 This
study collides planar gravitational shock waves, thereby simplifying the problem
by neglecting transverse dynamics. The initial conditions for the shock wave are
determined by the stress tensor
T (z )=κh(z ), (11)
±± ± ±
where z = t z are null coordinates along the collision axis and κ = N2/2π2 is
± ± c
a measure of the degrees of freedom of the gauge theory. All other components of
the stress tensor are zero, and within pure gravity this fully determines the metric