Michał P. Heller Universiteit van Amsterdam, the Netherlands

& National Centre for Nuclear Research, Poland

Numerical solutions of AdS gravity: new lessons about dual equilibration processes at strong coupling

Introduction

Numerical holography

1/12

numerical relativity + holography = new window on far-from-equilibrium physics.

Why interesting? ab initio calculations in a class of interacting quantum field theories.

My main motivation will be the creation of quark-gluon plasma in Heavy Ion Collisions.

Hence I will consider the Poincare patch of AdS4+1

In this talk I will discuss 3 solutions of with planar horizons.

In common: applications to HIC & using the ingoing Eddington-Finkelstein coordinates.

Rab 1

2Rgab

6

L2gab = 0

states in a large-Nc CFT at strong coupling

Setups

2/12

initial data (solves constraints)

event horizon (present from the start) boundary; we demand here that if

then

note that in the setups I am going to study .

Why the ingoing Eddington-Finkelstein coordinates? Manifestly regular on the horizon + attractive integrations scheme (if no caustics).

constant “time” slices

!

=22

N2c

(for N=4 SYM)

fluid-gravity duality++ 1309.1439 [hep-th] Chesler & Yaffe

Zd

3x hTtti = 1

hµ(u, x) = µ +4GN

L

3hTµ(x)i · u4 + . . .

ds

24+1 =

L

2

u

2

du

2 + hµ(u, x) dxµdx

Isotropization at strong coupling

MPH, D. Mateos, W. van der Schee & D. Trancanelli1202.0981 [hep-th] PRL 108 (2012) 191601:MPH, D. Mateos, W. van der Schee & M. Triana1304.5172 [hep-th] JHEP 1309, 026 (2013):

Holographic isotropizationMPH, D. Mateos, W. van der Schee & D. Trancanelli1202.0981 [hep-th] PRL 108 (2012) 191601:

3/12

One of the simplest equilibration processes to study holographically is described by

It is identically traceless and conserved. EOMs are .Rab 1

2Rgab

6

L2gab = 0

Symmetries of the stress tensor lead to a general metric ansatz

ds

2 = fttdt2 + 2ftrdtdr + frrdr

2 + 2e

2Bdx

21 + 2

e

B(dx22 + dx

23)

We fix almost all the gauge freedom by adopting (the ingoing EF coordinates)

ds

2 = 2dtdr Adt

2 + 2e

2Bdx

21 + 2

e

B(dx22 + dx

23)

We can solve Einstein’s equations near the boundary and obtain*B =

1

r4

b4(t) +

1

rb04(t) +

2

12r6b004(t) +

1

4r3b(3)4 (t) + . . .

P (t) =

3

82N2

c b4(t)with

hTµi = diag

E , 1

3E 2

3P(t),

1

3E +

1

3P(t),

1

3E +

1

3P(t)

µ

Equilibration dynamics

absorption by the horizon

MPH, D. Mateos, W. van der Schee & D. Trancanelli1202.0981 [hep-th] PRL 108 (2012) 191601:

4/12

rhr

ds

2 = 2dtdr Adt

2 + 2e

2Bdx

21 + 2

e

B(dx22 + dx

23)

initial data: and

t T

3N2c

82E r4B(t, r)

B(t = 0, r)hT00i = E

hTµi = diag

E , 1

3E 2

3P(t),

1

3E +

1

3P(t),

1

3E +

1

3P(t)

µ

Fast relaxation

5/12

0.2 0.4 0.6 0.8 1.0 1.2 T t

-25-20-15-10-5

DPêE

0.2 0.4 0.6 0.8 1.0 1.2 T t

-0.4

-0.2

0.0

0.2

0.4DPêE

Figure 5. (Top) t

iso

is the di↵erence between the isotropization time predicted by the full andthe linear equations. The height of each bar in the histogram indicates the number of initial statesfor which the evolution yielded values in the corresponding bin. The total number of initial statesis more than 800. We see both that holographic isotropization proceeds quickly, at most over atime scale set by the inverse temperature, and that the linearized Einstein’s equations correctlyreproduce the isotropization time with a 20% accuracy in most cases. Note that the histogram isbased on a di↵erent sample of initial states than those originally considered in [1]. In particular,we incorporated the binary search algorithm absent in [1] and were stricter about the maximumviolation of the constraint that we allowed.(Botom) Close inspection of one of the few profiles for which the linearized approximation seeminglyfails by more than 20% (tiso/tiso = 0.5) shows that it is the imperfect isotropization criteriumwhich leads to the mismatch rather than the failure of the linear approximation. Indeed, the leftplot shows that, on the scale of the initial anisotropy, the linear result yields a good approximation.However, the isotropization criterium makes no reference to this scale, and results in a 50% di↵erencein the isotropization times, indicated by the arrows on the right plot. See [9] for a related discussionof subtleties involved in defining the thermalization (or more accurately hydrodynamization) timein a similar setup.

– 16 –

!Checked for circa 103 different n-eq initial conditions

tiso

:P(t t

iso

)

E

0.1

(RHIC c=0-5%: ) 0.25 fm 500MeV = 0.630801.4361 [nucl-th] W. Broniowski et al.

Shock wave collisions and hydrodynamization

1305.4919 [hep-th] PRL 111 (2013) 181601: J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee1312.2956 [hep-th]: J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee

Towards a holographic „heavy ion collision”general issue: which holographic initial conditions are closest to the experiment?

practical viewpoint: collide two lumps of matter moving at relativistic speeds.

6/12

0

u

z

t

1011.3562 [hep-th] P. Chesler & L. Yaffe[hep-th/0512162] R. Janik & R. Peschanski

Gravitational shock wave solutions

0

u

ds

2 =1

u

2(du2 + µdx

µdx

) + u

2h(x)dx

2

Chesler & Yaffe 1011.3562 [hep-th]Janik & Peschanski [hep-th/0512162]

Poincare patch vacuum AdS

shock wave disturbance moving with the speed of light

Solution of Einstein’s equations with the negative CC for any longitudinal profile h(x)

z

t

dual stress tensor:

7/12

Let’s consider now . But, in a CFT, what matters is:

(in real HIC and corresponds to Pb at RHIC)e 1/2 eCY 0.64e =

h(t± z) = 4 exp(t± z)2/22

hT tti = hT zzi = ±hT tzi = N2c

22h(t z)

Dynamical crossovereleft = 2 eCY

shocks coalesce and explode hydro-dynamically (similar to the Landau picture)

hydro applicable only at mid-rapidities and late enough!!!

3

t

z

S4

t

z

S4

FIG. 2. Energy flux for collisions of thick (left) and thin (right) shocks. The dotted curves show the location of the maxima ofthe flux.

zz

t

FIG. 3. 3P loc

L Eloc for thick (left) and thin (right) shocks. The white areas indicate the vacuum regions outside the light cone.The grey areas indicate regions where hydrodynamics deviates by more than 100%. The dotted curves indicate the location ofthe maxima of the energy flux, as in Fig. 2.

the energy flux in this region is less than 10% of the max-imum incoming flux, as illustrated by Fig. 2(left). At latetimes, the velocity of the receding shocks can be read o↵from the same figure as the inverse slope of the dottedline. This is not constant in time, but at late times itreaches a maximum of about v 0.88. The validity ofthe hydrodynamic description can be seen in Fig. 3(left)and Fig. 4(left column). Hydrodynamics becomes appli-cable even earlier than t

max

, and the region where it isapplicable extends from z = 0 to the location of the re-ceding maxima. This is intuitive since gradients becomesmaller as the width of the shocks increases. We concludethat the thick-shocks collisions results in hydrodynamicexpansion with initial conditions in which all the veloci-ties are close to zero, in close similarity with the Landau

model [5].

The thin shocks illustrate the transparency scenario.In this case the shocks pass through each other and,although their shape gets altered, they keep moving atv 1, as seen in Fig. 2(right). The most dramatic modifi-cation in their shape is a region of negative E and P

L

thattrails right behind the receding shocks. While the nega-tive E only develops away from the center of the collision,the negative P

L

is already present at z = 0, as shown moreclearly in the top-right plot of Fig. 4. These features arecompatible with the general principles of Quantum FieldTheory [6], since the ‘negative region’ is far from equi-librium and highly localized near a bigger region withpositive energy and pressure. In the case of thin shocks,we see from Fig. 3(right) and Fig. 4(right column) that

3

t

z

S4

t

z

S4

FIG. 2. Energy flux for collisions of thick (left) and thin (right) shocks. The dotted curves show the location of the maxima ofthe flux.

zz

t

FIG. 3. 3P loc

L Eloc for thick (left) and thin (right) shocks. The white areas indicate the vacuum regions outside the light cone.The grey areas indicate regions where hydrodynamics deviates by more than 100%. The dotted curves indicate the location ofthe maxima of the energy flux, as in Fig. 2.

the energy flux in this region is less than 10% of the max-imum incoming flux, as illustrated by Fig. 2(left). At latetimes, the velocity of the receding shocks can be read o↵from the same figure as the inverse slope of the dottedline. This is not constant in time, but at late times itreaches a maximum of about v 0.88. The validity ofthe hydrodynamic description can be seen in Fig. 3(left)and Fig. 4(left column). Hydrodynamics becomes appli-cable even earlier than t

max

, and the region where it isapplicable extends from z = 0 to the location of the re-ceding maxima. This is intuitive since gradients becomesmaller as the width of the shocks increases. We concludethat the thick-shocks collisions results in hydrodynamicexpansion with initial conditions in which all the veloci-ties are close to zero, in close similarity with the Landau

model [5].

The thin shocks illustrate the transparency scenario.In this case the shocks pass through each other and,although their shape gets altered, they keep moving atv 1, as seen in Fig. 2(right). The most dramatic modifi-cation in their shape is a region of negative E and P

L

thattrails right behind the receding shocks. While the nega-tive E only develops away from the center of the collision,the negative P

L

is already present at z = 0, as shown moreclearly in the top-right plot of Fig. 4. These features arecompatible with the general principles of Quantum FieldTheory [6], since the ‘negative region’ is far from equi-librium and highly localized near a bigger region withpositive energy and pressure. In the case of thin shocks,we see from Fig. 3(right) and Fig. 4(right column) that

devia

tion

from

visco

us h

ydro

maximum ofthe energy flux

„low energy” „high energy” 2

E4

t

z

E4

t

z

PL4

t

z

PL4

t

z

PT 4

t

z

PT 4

t

z

FIG. 1. Energy and pressures for collisions of thick (left column) and thin (right column) shocks. The grey planes lie at theorigin of the vertical axes.

2. A dynamical cross-over. Fig. 1 shows the energydensity and the pressures for thick and thin shock colli-sions. In the case of E and P

L

one can see the incomingshocks at the back of the plots, the collision region in thecenter, and the receding maxima at the front. The in-coming shocks are absent in the case of P

T

, as expected.A simultaneous rescaling of and w that keeps w fixedwould change the overall scales on the axes of these fig-ures but would leave the physics unchanged.

The thick shocks illustrate the full-stopping scenario.

As the shocks start to interact the energy density getscompressed and ‘piles up’, comes to an almost completestop, and subsequently explodes hydrodynamically. In-deed, at the time t

max

0.58 at which the energy den-sity reaches its maximum in the top-left plot, the energydensity profile is very approximately a rescaled version ofone of the incoming Gaussians, with about three times itsheight (see table I) and 2/3 its width. At this time, 90%of the energy is contained in a region of size z 2.4w inwhich the flow velocity is everywhere v 0.1. Similarly,

Dispels the myth that strong coupling necessarily leads to immediate stopping*

eright = 0.125 eCY

8/12

1305.4919 [hep-th] PRL 111 (2013) 181601: J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee

hT tti = N2c

22E

Hydrodynamization in a shock wave collision

9/12

Hydrodynamics: hTµi = E + P(E)uµu + P(E) µ +µ

perfect fluid dissipative

CFT

= µ + . . .

We use and compare and with hydro prediction. hT zzi hT??ihTµ iu = E uµ

Surprise: large anisotropy at the onset of hydrodynamics due to the shear tensor!

at the collision axis (z = 0)

hT tti3 E0

hT zziE0

hT??iE0

dotted: hydro prediction

t

thyd Thyd = 0.26

see also 0906.4426 & 1011.3562 Chesler & Yaffe and 1103.3452 MPH, Janik & Witaszczyk

1305.4919 [hep-th] PRL 111 (2013) 181601: J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee

The nature of hydrodynamics

MPH, R. A. Janik & P. Witaszczyk1302.0697 [hep-th] PRL 110 (2013) 211602:

Hydrodynamic series at high ordersMPH, R. A. Janik & P. Witaszczyk1302.0697 [hep-th] PRL 110 (2013) 211602:

10/12

Question: what is the nature of hydrodynamic gradient expansion?

Idea: use the fluid-gravity duality to compute subsequent gradient terms on-shell.

To make it operational, we used the boost-invariant flow. Why?

t = cosh y and z = sinh y

uµ@µ = @ and hT i = E() = 3

8N2

c 2T ()4

Gradient expansion solving ODEs in the bulk go to 240th order in grads.

1

T ()rµu 1

T ()

1

!

With the restriction of transversality and tracelessness, there are eight possible contribu-

tions to the stress-energy tensor:

∇⟨µ ln T ∇ν⟩ ln T, ∇⟨µ∇ν⟩ ln T, σµν(∇·u), σ⟨µλσν⟩λ

σ⟨µλΩν⟩λ, Ω⟨µ

λΩν⟩λ, uαRα⟨µν⟩βuβ, R⟨µν⟩ .(3.6)

By direct computations we find that there are only five combinations that transform

homogeneously under Weyl tranformations. They are

Oµν1 = R⟨µν⟩ − (d − 2)

!

∇⟨µ∇ν⟩ ln T −∇⟨µ ln T ∇ν⟩ ln T"

, (3.7)

Oµν2 = R⟨µν⟩ − (d − 2)uαRα⟨µν⟩βuβ , (3.8)

Oµν3 = σ⟨µ

λσν⟩λ , Oµν4 = σ⟨µ

λΩν⟩λ , Oµν5 = Ω⟨µ

λΩν⟩λ . (3.9)

In the linearized hydrodynamics in flat space only the term Oµν1 contributes. For conve-

nience and to facilitate the comparision with the Israel-Stewart theory we shall use instead

of (3.7) the term⟨Dσµν ⟩ +

1

d − 1σµν(∇·u) (3.10)

which, with (3.5), reduces to the linear combination: Oµν1 − Oµν

2 − (1/2)Oµν3 − 2Oµν

5 . It is

straightforward to check directly that (3.10) transforms homogeneously under Weyl transfor-

mations.

Thus, our final expression for the dissipative part of the stress-energy tensor, up to second

order in derivatives, is

Πµν = −ησµν

+ ητΠ

#

⟨Dσµν ⟩ +1

d − 1σµν(∇·u)

$

+ κ%

R⟨µν⟩ − (d − 2)uαRα⟨µν⟩βuβ

&

+ λ1σ⟨µ

λσν⟩λ + λ2σ⟨µ

λΩν⟩λ + λ3Ω⟨µ

λΩν⟩λ .

(3.11)

The five new constants are τΠ, κ, λ1,2,3. Note that using lowest order relations Πµν = −ησµν ,

Eqs.(3.5) and Dη = −η∇·u, Eq. (3.11) may be rewritten in the form

Πµν = −ησµν − τΠ

#

⟨DΠµν ⟩ +d

d − 1Πµν(∇·u)

$

+ κ%

R⟨µν⟩ − (d − 2)uαRα⟨µν⟩βuβ

&

+λ1

η2Π⟨µ

λΠν⟩λ − λ2

ηΠ⟨µ

λΩν⟩λ + λ3Ω⟨µ

λΩν⟩λ .

(3.12)

This equation is, in form, similar to an equation of the Israel-Stewart theory (see Section 6).

In the linear regime it actually coincides with the Israel-Stewart theory (6.1). We emphasize,

however, that one cannot claim that Eq. (3.12) captures all orders in the momentum expansion

(see Section 6).

– 8 –

+ . . .

Hydrodynamic series at high ordersMPH, R. A. Janik & P. Witaszczyk1302.0697 [hep-th] PRL 110 (2013) 211602:

at large orders factorial growth of gradient contributions with order

T 00 = () 1X

n=2

n(2/3)n (T1rµu

2/3)

First evidence that hydrodynamic expansion has a zero radius of convergence!

at low orders behavior is different

2

longitudinal direction. This symmetry can be made man-ifest upon passing to curvilinear proper time - rapidityy coordinates related to the lab frame time x

0 and posi-tion along the expansion axis x1 via

x

0 = cosh y and x

1 = sinh y. (1)

In the case of (3+1)-dimensional conformal field theoryplasma, the most general stress tensor obeying the sym-metries of the problem in coordinates (, y, x1

, x

2) reads

T

µ = diag(, pL, pT , pT )

µ , (2)

where the energy density is a function of proper timeonly and the longitudinal pL and transverse pT pressuresare fully expressed in terms of the energy density [9]

pL =

0 and pT = +1

2

0. (3)

Note that, in the proper time - rapidity coordinates (1),there is no momentum flow in the stress tensor (2) andso the flow velocity is trivial and takes the form u =@ . Hydrodynamic constituent relations lead, then, togradient expanded energy density of the form

=3

8N

2

c 2

1

4/3

2

+

3

1

2/3+

4

1

4/3+ . . .

, (4)

where the choice of 2

sets an overall energy scale, in par-ticular for the quasinormal frequencies (7) and 9). Theprefactor was chosen to match the N = 4 super Yang-Mills theory at large-Nc and strong coupling. In the fol-lowing, we choose the units by setting

2

=

4.Large- expansion of the energy density in powers of

2/3, as in (4), is equivalent to the hydrodynamic gra-dient expansion and arises from expressing gradients ofvelocity (rµu

1) in units of the e↵ective tempera-ture (T

1/4

1/3). The value of the coecient

3

is related to the shear viscosity , whereas

4

is a sumof two transport coecients: relaxation time

and theso-called

1

[10]. Higher order contributions to the en-ergy density are expected to be linear combinations of sofar unidentified transport coecients. Note also that theexpansion (4) is sensitive to both linear and nonlineargradient terms.

As explained in [11, 12] (see also Supplemental ma-terial), higher order contributions to the energy density(4) can be obtained by solving Einstein’s equations witha negative cosmological constant for the metric ansatz ofthe form

ds

2 = 2ddrAd

2+2

e

2Bdy

2+2

e

B(dx2

1

+dx

2

2

), (5)

where the warp factors A, and B are functions of rand constructed in the gradient expansion as requiredby the fluid-gravity duality. At leading order, the warpfactors are that of a locally boosted black brane and this

solution gets systematically corrected in

2/3 expansion,as is the case with the energy density in the dual fieldtheory (4).The background expanded in

2/3 around a locallyboosted black brane is slowly evolving and captures onlyhydrodynamic degrees of freedom. One can, in ad-dition, consider the incorporation of nonhydrodynamic(fast evolving) degrees of freedom by linearizing Ein-stein’s equations on top of the hydrodynamic solution,i.e. B = B

hydro

+ B, and similarly for A and , andlooking for B corresponding to (at very large time) theexponentially decaying contribution to the stress tensordepending only on . For the static background analo-gous calculation would lead to the spectrum of nonhydro-dynamic quasinormal modes carrying zero momentum,which is known to be the same as the spectrum of zeromomentum quasinormal modes for the massless scalarfield [13].In the leading order of the gradient expansion, the re-

sulting modes, on the gravity side, indeed essentially re-duce to the scalar quasinormal modes but obtain an ad-ditional factor of 3

2

and are damped exponentially in

2

3

[14]. Upon including viscous correction, the modes ob-tain a further nontrivial powerlike preexponential factor

↵qnm exp (i

3

2!qnm

2/3). (6)

Explicit gravity calculation for the lowest mode yield

!qnm = 3.11952.7467, ↵qnm = 1.5422+0.5199 i. (7)

The frequency !qnm agrees with the frequency of thelowest nonhydrodynamic scalar quasinormal mode andwas calculated before in [14], whereas the prediction of↵qnm is a new result specific to the dissipative modifi-cations of the expanding black hole geometry (see theSupplemental Material for further details). In the fol-lowing, we will be able to reproduce numerically (7) justfrom the large order behavior of the hydrodynamic series.Large order behavior of hydrodynamic energy

density. Numerical implementation of the methods out-lined in [11, 12] allow for ecient calculation of hydrody-namic series given by (4), up to a very large order, sinceone is e↵ectively solving a set of linear ODE’s (comingfrom Einstein’s equations) at each order. Using spectralmethods we iteratively solved these equations in the largetime expansion reconstructing the energy density up tothe order 240, i.e. up to the term

242

in (4). To the bestof our knowledge this is the first approach allowing us toaccess information about the large order behavior of thehydrodynamic series in any physical system or model.As a way of monitoring the accuracy of our procedures

we compared normalized values of evaluated Einstein’sequations at each order of the

2/3 expansion to theratio of coecients of gradient-expanded energy densityto gradient expanded warp factors. This ensures thatour results for the energy density are reliable. We also

(n!)1/n (2n)1/2n

e· n

11/12

n2

1/n

Summary

12/12

0.2 0.4 0.6 0.8 1.0 1.2 T t

-25-20-15-10-5

DPêE

0.2 0.4 0.6 0.8 1.0 1.2 T t

-0.4

-0.2

0.0

0.2

0.4DPêE

Figure 5. (Top) t

iso

is the di↵erence between the isotropization time predicted by the full andthe linear equations. The height of each bar in the histogram indicates the number of initial statesfor which the evolution yielded values in the corresponding bin. The total number of initial statesis more than 800. We see both that holographic isotropization proceeds quickly, at most over atime scale set by the inverse temperature, and that the linearized Einstein’s equations correctlyreproduce the isotropization time with a 20% accuracy in most cases. Note that the histogram isbased on a di↵erent sample of initial states than those originally considered in [1]. In particular,we incorporated the binary search algorithm absent in [1] and were stricter about the maximumviolation of the constraint that we allowed.(Botom) Close inspection of one of the few profiles for which the linearized approximation seeminglyfails by more than 20% (tiso/tiso = 0.5) shows that it is the imperfect isotropization criteriumwhich leads to the mismatch rather than the failure of the linear approximation. Indeed, the leftplot shows that, on the scale of the initial anisotropy, the linear result yields a good approximation.However, the isotropization criterium makes no reference to this scale, and results in a 50% di↵erencein the isotropization times, indicated by the arrows on the right plot. See [9] for a related discussionof subtleties involved in defining the thermalization (or more accurately hydrodynamization) timein a similar setup.

– 16 –

!

MPH, D. Mateos, W. van der Schee & D. Trancanelli1202.0981 [hep-th] PRL 108 (2012) 191601:

n2

1/n

2

E4

t

z

E4

t

z

PL4

t

z

PL4

t

z

PT 4

t

z

PT 4

t

z

FIG. 1. Energy and pressures for collisions of thick (left column) and thin (right column) shocks. The grey planes lie at theorigin of the vertical axes.

2. A dynamical cross-over. Fig. 1 shows the energydensity and the pressures for thick and thin shock colli-sions. In the case of E and P

L

one can see the incomingshocks at the back of the plots, the collision region in thecenter, and the receding maxima at the front. The in-coming shocks are absent in the case of P

T

, as expected.A simultaneous rescaling of and w that keeps w fixedwould change the overall scales on the axes of these fig-ures but would leave the physics unchanged.

The thick shocks illustrate the full-stopping scenario.

As the shocks start to interact the energy density getscompressed and ‘piles up’, comes to an almost completestop, and subsequently explodes hydrodynamically. In-deed, at the time t

max

0.58 at which the energy den-sity reaches its maximum in the top-left plot, the energydensity profile is very approximately a rescaled version ofone of the incoming Gaussians, with about three times itsheight (see table I) and 2/3 its width. At this time, 90%of the energy is contained in a region of size z 2.4w inwhich the flow velocity is everywhere v 0.1. Similarly,

Summary

1305.4919 [hep-th] PRL 111 (2013) 181601: J. Casalderrey-Solana, MPH, D. Mateos & W. van der Schee

1302.0697 [hep-th] PRL 110 (2013) 211602: MPH, R. A. Janik & P. Witaszczyk

+ hydrodynamization

extrahttp://goo.gl/BBYh7J