A Holographic Path to the Turbulent Side of Gravity

arXiv:1309.7940 [hep-th]

Stephen Green with Federico Carrasco and Luis Lehner

23rd Midwest Relativity MeetingUniversity of Wisconsin, MilwaukeeOctober 25, 2013

Introduction

• AdS / CFT correspondence relates quantum gravity in (d+1) dimensions and quantum field theory on the d-dimensional boundary.

• There exists a purely classical limit, where

(d+1) quantum gravity (d+1) perturbed black brane solutions to Einstein’s equation in anti-de Sitter

d-dimensional QFT d-dimensional fluid

• We are interested in whether studying fluids can teach us anything about general relativity?

AdS/CFT“gravity/fluid correspondence”

Review of “gravity/fluid correspondence”

• Initial-boundary value problem:

• Solve the bulk Einstein equation for some perturbed black brane initial data. Read off the “boundary stress-energy tensor”.

• At late times, the boundary stress-energy takes the form of a relativistic, viscous conformal fluid. This is derived in a derivative expansion; valid for long-wavelength perturbations. (Bhattacharyya et al, 2008)

• From the boundary stress-energy, can re-construct a bulk metric which solves the Einstein equation in the derivative expansion.

⌘µ⌫Boundary:Impose “boundary metric” = (mirror)

Bulk:Asymptotically AdS

black brane

Tµ⌫ ⌘ limr!1

rd�2

8⇡Gd+1(Kµ⌫ �K⌘µ⌫ � (d� 1)⌘µ⌫)

Turbulence?

• Issue: Fluids at high Reynolds number exhibit turbulence, but black brane perturbations described by quasinormal modes for small-amplitude perturbations.

• A few ways around this...

1. Dual fluid is relativistic, viscous, and conformal. Not ordinary Navier-Stokes. Does it exhibit turbulence?

2. Maybe turbulence is outside the regime of validity of the fluid/gravity correspondence?

Turbulence?

• Issue: Fluids at high Reynolds number exhibit turbulence, but black brane perturbations described by quasinormal modes for small-amplitude perturbations.

• A few ways around this...

1. Dual fluid is relativistic, viscous, and conformal. Not ordinary Navier-Stokes. Does it exhibit turbulence?

2. Maybe turbulence is outside the regime of validity of the fluid/gravity correspondence?

Our answer: YES

Turbulence?

• Issue: Fluids at high Reynolds number exhibit turbulence, but black brane perturbations described by quasinormal modes for small-amplitude perturbations.

• A few ways around this...

1. Dual fluid is relativistic, viscous, and conformal. Not ordinary Navier-Stokes. Does it exhibit turbulence?

2. Maybe turbulence is outside the regime of validity of the fluid/gravity correspondence?

Our answer: YES

Our answer: NO.Turbulence remains within the regime of the correspondence, at least for d=3. Inverse energy cascade pushes perturbations to large scales, where the correspondence is most valid.

Boundary stress-energy

• Resulting boundary stress-energy tensor (to 2nd order in derivatives):

where the viscous part is given by

Transport coefficients all functions of the density. In particular, shear viscosity

• Equations of motion all follow from conservation of stress-energy. Also promoteto independent field and reduce order. (cf. Israel-Stewart)

• There is also a dual bulk metric, which is expressed in terms of

T [0+1+2]µ⌫ =

⇢

d� 1(duµu⌫ + ⌘µ⌫) +⇧µ⌫

⌘ =s

4⇡/ T d�1 / ⇢(d�1)/d

⇧µ⌫ = � 2⌘�µ⌫

+ 2⌘⌧⇧

✓hu↵@↵�µ⌫i+

1

d� 1�µ⌫@↵u

↵

◆+ h�1�µ↵�

↵⌫ + �2�µ↵!

↵⌫ + �3!µ↵!

↵⌫ i

shear vorticity

1st order in derivatives2nd order

uµ, ⇢

⇧µ⌫

Black brane quasinormal modes

• Modes in red are the hydrodynamic modes, which are captured by the fluid.

• The shear mode is purely imaginary, and for high T, becomes long lived.

• Higher modes not captured by dual fluid, but they decay more rapidly and are not relevant at long wavelengths.

Class. Quantum Grav. 26 (2009) 163001 Topical Review

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

1

2

3

4

5

6

Figure 14. Quasinormal spectrum of black three-brane gravitational fluctuations in the ‘shear’(left) and ‘sound’ (right) channels, shown in the plane of complex frequency w = !/2"T , forfixed spatial momentum q = q/2"T = 1. Hydrodynamic frequencies are marked with hollow reddots (adapted from [109]).

of the stress–energy tensor in four-dimensional N = 4 SU(Nc) SYM in the limit Nc !", g2

YMNc ! " are given by the QNM frequencies of the gravitational perturbation hµ# ofmetric (125). By symmetry, the perturbations are divided into three groups [109, 415]. Indeed,since the dual gauge theory is spatially isotropic, we are free to choose the momentum of theperturbation along, say, the z-direction, leaving the O(2) rotational symmetry of the (x, y)

plane intact. The perturbations hµ#(r, t, z) are thus classified according to their transformationproperties under O(2). Following [109, 415], we call them the scalar (hxy), shear (htx, hzx orhty, hzy) and sound (htt , htz, hzz, hxx + hyy) channels. (Here we partially fixed the gauge byrequiring hµr=0.) The O(2) symmetry ensures that the equations of motion for perturbations ofdifferent symmetry channels decouple. Linear combinations of perturbations invariant underthe (residual) gauge transformations hµ# ! hµ# # $µ$# # $#$µ form the gauge-invariantvariables

Z1 = qHtx + !Hzx, (135)

Z2 = q2f Htt + 2!qHtz + !2Hzz + [q2(2 # f ) # !2]H, (136)

Z3 = Hxy (137)

in the shear, sound and scalar channels, respectively (here Htt = L2htt/r2f,Hij = L2hij /r2

(i, j %= t), H = L2(hxx +hyy)/2r2). From the equations of motion satisfied by the fluctuations,one obtains three independent second-order ODEs for the gauge-invariant variables Z1, Z2, Z3.

The QNM spectra in all three channels share a characteristic feature: an infinite sequenceof (asymptotically) equidistant frequencies approximated (for q = 0) by a simple formula[99, 100, 107]

!n = 2"T n (±1 # i) + !0, n ! ". (138)

Each frequency has a non-trivial dependence on q [99, 100]. Spectra in the shear and soundchannels are shown in figure 14. In addition to the sequence mentioned above, they contain theso-called hydrodynamic frequencies shown in figure 14 by hollow red dots. The hydrodynamicfrequencies are remarkable, in that their existence and dependence on q are predicted bythe hydrodynamics of the dual field theory. For example, low-frequency, small momenta

53

shear channel sound channel

Plots from Berti, Cardoso, Starinets (2009)

Simulations in d=2+1: Initial data

• We wish to study the shear mode. So we choose initial data with fluid flow orthogonal to the velocity gradient.

• Consider a uniform fluid flow,

linearly perturbed by shear flow,

• Find, for modes , a dispersion relation

• We choose initial data corresponding to shear mode, and study its evolution under the full nonlinear equations. We also include tiny random perturbations, in order to assess stability.

ux

(1) = ux

(1)(t, y)

⇧(1)xy

= ⇧(1)xy

(t, y)

⇢(0) = constant

uµ(0) = (1, 0, 0)

⇧

(0)µ⌫ = 0,

⇠ e�i!t+iky ! ⇡ �i2k2⌘(0)3⇢(0)

= �ik2

4⇡T(0)

same as lowest shear QNM for black brane

Typical turbulent run

• Plotting viscosity field at various times...10

(a) t = 0 (b) t = 350

(c) t = 500 (d) t = 900

(e) t = 2500 (f) t = 7000

FIG. 1. Vorticity field at various times for a turbulent run. (⇢0 = 1010, v0 = .05, D = 10, n = 10)

Typical turbulent run

• Plotting viscosity field at various times...

10

(a) t = 0 (b) t = 350

(c) t = 500 (d) t = 900

(e) t = 2500 (f) t = 7000

FIG. 1. Vorticity field at various times for a turbulent run. (⇢0 = 1010, v0 = .05, D = 10, n = 10)

Typical turbulent run

• Plotting viscosity field at various times...

10

(a) t = 0 (b) t = 350

(c) t = 500 (d) t = 900

(e) t = 2500 (f) t = 7000

FIG. 1. Vorticity field at various times for a turbulent run. (⇢0 = 1010, v0 = .05, D = 10, n = 10)

What’s going on?

11

10 100 1000time

0.01

0.1

||d[xu y]|| 2

(a) Vorticity

0 500 1000 1500time

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

||ui|| 2

uxuy

(b) Velocity

FIG. 2. L2 norms of (a) vorticity, and (b) ui

, as a function of time for turbulent flow (same run as Fig. 1). In (a) there is aninitial exponential decay, followed by a power law during the inverse cascade, followed by a slower exponential decay. In (b) u

y

grows exponentially until it is of similar amplitude to ux

.

FIG. 3. uy

field at t = 300 for the same run as Fig. 1.

Shortly after equipartition is reached between ux and uy, the vorticity field displays a number of turbulent eddies,and the exponential decay of vorticity ceases. It is replaced by a power law decay / t

�↵ with ↵ ' 1.2 ± 0.2 DOWE MENTION OTHER QUANTITIES?. During this decay, these eddies merge into vortices, which continueto merge into larger and larger vortices, as expected for an inverse cascade. There have been several previous studiesof unforced turbulent decay in d = 3, for the case of Navier-Stokes fluids which have also found power law decays[33] CITE OTHERS. Finally, one is left with two counter-rotating vortices, which decay exponentially, albeit moreslowly than the initial exponential decay. This late time behavior is expected from a quasi-normal mode point ofview, as the largest vortices have no further modes they can decay and their slowest decay rate corresponds to theQNM behavior for such longer wavelengths. FIRM THIS UP — MEASURE DECAY RATES; COMPAREWITH QUASINORMAL MODES

[COMPARISON TO 2D TURBULENCE FOR ORDINARY FLUIDS, AND POWER LAWS OBTAINED]

[DEPENDENCE ON DOMAIN]

Initial exponential decay (matches

linear expectations) Power law during turbulent cascade

Late time exponential

Random seed grows exponentially

⇡ t�1

Unstable mode

• is used for monitoring the unstable mode. Here we plot the field during the initial growth period.

• It is also a shear mode, but withlonger wavelength than the initial flow.

11

10 100 1000time

0.01

0.1

||d[xu y]|| 2

(a) Vorticity

0 500 1000 1500time

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

||ui|| 2

uxuy

(b) Velocity

FIG. 2. L2 norms of (a) vorticity, and (b) ui

, as a function of time for turbulent flow (same run as Fig. 1). In (a) there is aninitial exponential decay, followed by a power law during the inverse cascade, followed by a slower exponential decay. In (b) u

y

grows exponentially until it is of similar amplitude to ux

.

FIG. 3. uy

field at t = 300 for the same run as Fig. 1.

Shortly after equipartition is reached between ux and uy, the vorticity field displays a number of turbulent eddies,and the exponential decay of vorticity ceases. It is replaced by a power law decay / t

�↵ with ↵ ' 1.2 ± 0.2 DOWE MENTION OTHER QUANTITIES?. During this decay, these eddies merge into vortices, which continueto merge into larger and larger vortices, as expected for an inverse cascade. There have been several previous studiesof unforced turbulent decay in d = 3, for the case of Navier-Stokes fluids which have also found power law decays[33] CITE OTHERS. Finally, one is left with two counter-rotating vortices, which decay exponentially, albeit moreslowly than the initial exponential decay. This late time behavior is expected from a quasi-normal mode point ofview, as the largest vortices have no further modes they can decay and their slowest decay rate corresponds to theQNM behavior for such longer wavelengths. FIRM THIS UP — MEASURE DECAY RATES; COMPAREWITH QUASINORMAL MODES

[COMPARISON TO 2D TURBULENCE FOR ORDINARY FLUIDS, AND POWER LAWS OBTAINED]

[DEPENDENCE ON DOMAIN]

uy uy

Reynolds number

• The initial decaying shear flow has an associated Reynolds number, which we define as

• This is a function of time, since the flow is decaying due to viscosity.

• Large R: unstable flow; turbulenceSmall R: stable flow; laminar

R ⌘ ⇢�

⌘maxu

x

/ ⇢1/3�maxux

Rc

Critical Reynolds number

Laminar flow

• The random seed perturbation does not grow. remains small. 12

0 500 1000 1500time

1e-10

1e-08

1e-06

1e-04

1e-02

||d[x

u y]|| 2

simulationlinear prediction

(a) Vorticity

0 500 1000 1500time

1e-12

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

||ui|| 2

uxuy

(b) Velocity

FIG. 4. L2 norms of (a) vorticity, and (b) ui

, as a function of time for laminar flow. The norm of uy

remains small throughoutthe decay. (⇢0 = 107, v0 = .02, D = 10, n = 10)

0 1000 2000 3000 4000time

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02max(ux)||uy||2

(a) Velocity

1e-05 1e-04 1e-03

max(ux)

0.00

0.01

0.02

d(ln(||u y|| 2))/dt

(b) Growth rate of kuy

k2

FIG. 5. Intermediate flow: (a) ui

, as a function of time for laminar flow, and (b) growth rate of kuy

k2 versus maxux

. The zeroof (b) corresponds to the critical Reynolds number. (⇢0 = 107, v0 = .015, D = 4, n = 10)

D. Late time decay

At late times... NEED LONGER RUNS

We’d like to answer questions like

• What’s the overall behavior depending on the temperature adopted. How does it depend on perturbation,number of modes, etc.?

• What are the slopes observed? are they consistent with the quasinormal mode slopes?

• Is the power-law something that can be argued from turbulence? and the value consistent with what would beexpected?

uy

Intermediate flow

• R is time-dependent: Can begin at , then drop to before turbulence develops.

• Allows us to extract the critical Reynolds number, when the growth rate is zero. Find:

R > Rc R < Rc

12

0 500 1000 1500time

1e-10

1e-08

1e-06

1e-04

1e-02

||d[x

u y]|| 2

simulationlinear prediction

(a) Vorticity

0 500 1000 1500time

1e-12

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

||ui|| 2

uxuy

(b) Velocity

FIG. 4. L2 norms of (a) vorticity, and (b) ui

, as a function of time for laminar flow. The norm of uy

remains small throughoutthe decay. (⇢0 = 107, v0 = .02, D = 10, n = 10)

0 1000 2000 3000 4000time

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-02max(ux)||uy||2

(a) Velocity

1e-05 1e-04 1e-03

max(ux)

0.00

0.01

0.02

d(ln(||u y|| 2))/dt

(b) Growth rate of kuy

k2

FIG. 5. Intermediate flow: (a) ui

, as a function of time for laminar flow, and (b) growth rate of kuy

k2 versus maxux

. The zeroof (b) corresponds to the critical Reynolds number. (⇢0 = 107, v0 = .015, D = 4, n = 10)

D. Late time decay

At late times... NEED LONGER RUNS

We’d like to answer questions like

• What’s the overall behavior depending on the temperature adopted. How does it depend on perturbation,number of modes, etc.?

• What are the slopes observed? are they consistent with the quasinormal mode slopes?

• Is the power-law something that can be argued from turbulence? and the value consistent with what would beexpected?

Rc ⇡✓

3

2A

◆1/3

6⇡ ⇥ 0.6 (1 + 0.1n)

Bulk black branes

• Translated to the black brane language, our results imply that, for d=3:

For , hydrodynamic shear quasinormal modes are unstable to small perturbations. This leads to turbulence, and a transfer of energy to longer wavelength modes.

• We have the correspondence:

R > Rc

Fluid side

Laminar flow

Turbulent flow

Gravity side

Quasinormal mode regime

New, turbulent phase[observed directly by Adams, Chesler and Liu (2013)]

Mapping from boundary to bulk

• Fluid/gravity correspondence includes a direct mapping from boundary quantities to a bulk metric...

10

(a) t = 0 (b) t = 350

(c) t = 500 (d) t = 900

(e) t = 2500 (f) t = 7000

FIG. 1. Vorticity field at various times for a turbulent run. (⇢0 = 1010, v0 = .05, D = 10, n = 10)

15

(a) r6 CABCD

CABCD (b) r7 ⇤CABCD

CABCD

FIG. 8. Contour plots of principal invariants of the Weyl tensor in the bulk, computed from the zeroth order metric (2.1), fromthe simulation snapshot in Fig. 1e. Notice that (a) is representative of the energy density ⇢, while (b) is representative of thevorticity, as expected from Eq. (B6).

to the enstrophy can be defined away from the boundary. Such a quantity is a key element needed to argue that aninverse energy cascade occurs.

Naturally, as already pointed out in [19], it would be very useful to develop a spacetime definition of the Reynoldsnumber. This would provide an intrinsic way to predict the onset of turbulence in gravity and could thus be appliedin broader contexts. Using the gravity/fluid correspondence, this would also lead to a Reynolds number suitable forrelativistic hydrodynamics. Based upon the form of the bulk metric (2.1), and the fluid Reynolds number (4.1), theform of the Reynolds number for black hole perturbations is, roughly,

RGR / THawking

�����hAB

✓@

@r

◆B�����L, (5.1)

where we have substituted (certain components of) the metric perturbation hAB ⌘ gAB � g

(0)AB for the velocity

fluctuation; and L is the characteristic length scale of the perturbation. Of course, whether or not this is applicablein more general contexts would require further investigation. In particular, a suitable definition of R should be gaugeinvariant.

As a final comment, we point out an important application of the fact that the inverse cascade guarantees the systemstays within the domain of validity of the gravity/fluid correspondence. The relativistic hydrodynamic equations in2 + 1 dimensions are dual to (long-wavelength) perturbed black branes in the bulk. Thus, a Newtonian limit in thebulk – where time derivatives are taken to be one-order subleading to space derivatives and velocities are taken to alsobe small11 – corresponds to a Navier-Stokes limit on the boundary. Since we know that the Navier-Stokes equationadmits global, well-behaved solutions [42], one can surmise that general relativity is similarly well-behaved in thebulk.

B. Connection to ordinary perturbation theory

Linear perturbation theory predicts that small-amplitude metric perturbations can be decomposed into independentmodes which undergo simple exponential decay. On the other hand, the picture described in the previous section

11 For some alternative discussions, see, e.g., [40, 41].

vorticity

Funnels run along tubesconnecting boundary and horizon

Mapping from boundary to bulk

15

(a) r6 CABCD

CABCD (b) r7 ⇤CABCD

CABCD

FIG. 8. Contour plots of principal invariants of the Weyl tensor in the bulk, computed from the zeroth order metric (2.1), fromthe simulation snapshot in Fig. 1e. Notice that (a) is representative of the energy density ⇢, while (b) is representative of thevorticity, as expected from Eq. (B6).

to the enstrophy can be defined away from the boundary. Such a quantity is a key element needed to argue that aninverse energy cascade occurs.

Naturally, as already pointed out in [19], it would be very useful to develop a spacetime definition of the Reynoldsnumber. This would provide an intrinsic way to predict the onset of turbulence in gravity and could thus be appliedin broader contexts. Using the gravity/fluid correspondence, this would also lead to a Reynolds number suitable forrelativistic hydrodynamics. Based upon the form of the bulk metric (2.1), and the fluid Reynolds number (4.1), theform of the Reynolds number for black hole perturbations is, roughly,

RGR / THawking

�����hAB

✓@

@r

◆B�����L, (5.1)

where we have substituted (certain components of) the metric perturbation hAB ⌘ gAB � g

(0)AB for the velocity

fluctuation; and L is the characteristic length scale of the perturbation. Of course, whether or not this is applicablein more general contexts would require further investigation. In particular, a suitable definition of R should be gaugeinvariant.

As a final comment, we point out an important application of the fact that the inverse cascade guarantees the systemstays within the domain of validity of the gravity/fluid correspondence. The relativistic hydrodynamic equations in2 + 1 dimensions are dual to (long-wavelength) perturbed black branes in the bulk. Thus, a Newtonian limit in thebulk – where time derivatives are taken to be one-order subleading to space derivatives and velocities are taken to alsobe small11 – corresponds to a Navier-Stokes limit on the boundary. Since we know that the Navier-Stokes equationadmits global, well-behaved solutions [42], one can surmise that general relativity is similarly well-behaved in thebulk.

B. Connection to ordinary perturbation theory

Linear perturbation theory predicts that small-amplitude metric perturbations can be decomposed into independentmodes which undergo simple exponential decay. On the other hand, the picture described in the previous section

11 For some alternative discussions, see, e.g., [40, 41].

density

Mapping from boundary to bulk

• We believe that the bulk metric is close to an actual solution of Einstein’s equation because:

• With the transfer of energy to large scales via the inverse cascade, if we begin within the range of validity of the fluid/gravity correspondence, we remain within that range.

• Turbulence has been observed directly in simulations on the GR side. [Adams, Chesler, Liu (2013)]

How to reconcile bulk QNM expectations with fluid turbulence?

• The instability of the fluid flow suggests an instability of the black brane quasinormal mode.

• Presence of this “instability” controlled by Reynolds number, which represents, in the equation of motion for the unstable mode, the ratio between nonlinear driving terms and viscous damping terms. If the viscous damping is too weak, nonlinear terms dominate, and mode grows in amplitude.

• In contrast, linearized perturbation theory assumes that the nonlinear terms are smaller than the damping term. Low R regime.

Long lived QNMs High Reynolds number

Extensions / Open Questions

• Higher dimensions: Direct cascade to small scales.

• Black holes rather than black branes: Expect similar behavior.

• Beyond AdS: Other cases of long-lived quasinormal modes might lead to turbulence.

• Gravitational definition of Reynolds number: In situations without a fluid description, could predict onset of turbulence.

Summary

• We have studied conditions for the onset of turbulence in relativistic viscous conformal fluids in d=2+1. Determined numerically the critical Reynolds number for shear flows.

• The fluid/gravity correspondence then implies that the quasinormal mode description for black branes breaks down at high Reynolds number. Turbulent phase entered instead.

• The onset of turbulence at high R is closely related to the presence of long-lived quasinormal modes.

The End