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 Meeting
University of Wisconsin, Milwaukee
October 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
AdS/CFT
d-dimensional QFT
(d+1) perturbed black brane solutions
to Einstein’s equation in anti-de Sitter
“gravity/fluid correspondence”
d-dimensional fluid
• We are interested in whether studying fluids can teach us anything about
general relativity?
Review of “gravity/fluid correspondence”
• Initial-boundary value problem:
Bulk:
Asymptotically AdS
black brane
Boundary:
Impose “boundary metric” =
⌘µ⌫ (mirror)
• Solve the bulk Einstein equation for some perturbed black brane initial data. Read off the
rd 2
“boundary stress-energy tensor”.
Tµ⌫ ⌘ lim
(Kµ⌫ K⌘µ⌫ (d 1)⌘µ⌫ )
r!1 8⇡Gd+1
• 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.
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 NavierStokes. 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 NavierStokes. Does it exhibit turbulence?
Our answer: YES
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 NavierStokes. Does it exhibit turbulence?
Our answer: YES
2. Maybe turbulence is outside the regime of validity of the fluid/gravity
correspondence?
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):
[0+1+2]
Tµ⌫
=
⇢
d
1
(duµ u⌫ + ⌘µ⌫ ) + ⇧µ⌫
where the viscous part is given by
⇧µ⌫ =
2⌘
1st order in derivatives
µ⌫
✓
+ 2⌘⌧⇧ hu↵ @↵
µ⌫ i +
1
d
1
µ⌫ @↵ u
↵
◆
2nd order
+h
1 µ↵ ⌫
↵
+
2 µ↵ !⌫
shear
Transport coefficients all functions of the density.
s
In particular, shear viscosity
⌘=
/ T d 1 / ⇢(d
4⇡
↵
+
3 !µ↵ !⌫
↵
i
vorticity
1)/d
• Equations of motion all follow from conservation of stress-energy. Also promote ⇧µ⌫
to independent field and reduce order. (cf. Israel-Stewart)
• There is also a dual bulk metric, which is expressed in terms of
uµ , ⇢
Black brane quasinormal modes
• Modes
in red are the hydrodynamic modes, which are captured
by the fluid.
Class. Quantum Grav. 26 (2009) 163001
Topical Review
6
6
5
5
4
4
3
3
2
2
1
1
0
-6 -5 -4 -3 -2 -1 0
1
2
shear channel
• The shear
3
4
5
6
Plots from Berti, Cardoso, Starinets (2009)
0
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
6
sound channel
Figure 14. Quasinormal spectrum of black three-brane gravitational fluctuations in the ‘shear’
(left)
‘sound’ (right)
channels, shown
the plane
of complex
frequency wlong
= ω/2π
T , for
mode
isand
purely
imaginary,
andinfor
high
T, becomes
lived.
fixed spatial momentum q = q/2π T = 1. Hydrodynamic frequencies are marked with hollow red
dots (adapted from [109]).
• Higher modes not captured by dual fluid, but they decay more rapidly and are not
relevant
long wavelengths.
of theat
stress–energy
tensor in four-dimensional N = 4 SU (Nc ) SYM in the limit Nc →
2
∞, gYM
Nc → ∞ are given by the QNM frequencies of the gravitational perturbation hµν of
metric (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 the
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, ⇢(0) = constant
uµ(0) = (1, 0, 0)
⇧(0)
µ⌫ = 0,
linearly perturbed by shear flow, ux(1) = ux(1) (t, y)
⇧(1)
xy
• Find, for modes ⇠ e
i!t+iky
=
⇧(1)
xy (t, y)
, a dispersion relation ! ⇡
same as lowest shear
QNM for black brane
2k 2 ⌘(0)
i
=
3⇢(0)
k2
i
4⇡T(0)
• 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.
Typical turbulent run
• Plotting viscosity field at various times...
10
(a) t = 0
(b) t = 350
Typical turbulent run
• Plotting viscosity field at various times...
(a) t = 0
(b) t = 350
(c) t = 500
(d) t = 900
Typical turbulent run
• Plotting viscosity field at various times...
(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?
Initial exponential
decay (matches
linear expectations)
Power law during
⇡t
turbulent cascade
Random seed
grows exponentially
11
1
1e-02
1e-03
0.1
ux
uy
||ui||2
||d[xuy]||2
1e-04
1e-05
1e-06
1e-07
0.01
1e-08
10
100
time
(a) Vorticity
1000
Late time exponential
0
500
time
1000
1500
(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 an
initial exponential decay, followed by a power law during the inverse cascade, followed by a slower exponential decay. In (b) uy
grows exponentially until it is of similar amplitude to u .
1e-06
1e-07
0.01
1e-08
Unstable mode
10
100
1000
time
0
500
(a) Vorticity
time
1
(b) Velocity
FIG. 2. L2 norms of (a) vorticity, and (b) ui , as a function of time for turbulent flow (same run as Fig
initial exponential decay, followed by a power law during the inverse cascade, followed by a slower expon
y
grows exponentially until it is of similar amplitude to ux .
• uy is used for monitoring the unstable mode. Here we plot the
during the initial growth period.
u
field
• It is also a shear mode, but with
longer wavelength than the initial flow.
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 numbe
and the exponential decay of vorticity ceases. It is replaced by a power law decay / t ↵ wit
WE MENTION OTHER QUANTITIES?. During this decay, these eddies merge into vor
Reynolds number
• The initial decaying shear flow has an associated Reynolds number, which we
define as
⇢
R⌘
max ux / ⇢1/3 max ux
⌘
• This is a function of time, since the flow is decaying due to viscosity.
• Large R: unstable flow; turbulence
Small R: stable flow; laminar
Critical Reynolds number
Rc
Laminar flow
• The random seed perturbation does not grow. uy remains small.
12
1e+00
simulation
linear prediction
1e-02
1e-02
1e-04
1e-04
||ui||2
||d[xuy]||2
ux
uy
1e-06
1e-06
1e-08
1e-08
1e-10
1e-10
0
500
time
(a) Vorticity
1000
1500
1e-12
0
500
time
1000
1500
(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 throughout
the decay. (⇢0 = 107 , v0 = .02, D = 10, n = 10)
1e-10
1e-10
0
500
time
1000
1e-12
1500
0
500
Intermediate flow
(a) Vorticity
time
1000
1500
(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 throughout
the decay. (⇢0 = 107 , v0 = .02, D = 10, n = 10)
• R is time-dependent: Can begin at R > Rc , then drop to R < Rc before
turbulence develops.
0.02
1e-02
max(ux)
||uy||2
1e-03
d(ln(||uy||2))/dt
1e-04
1e-05
1e-06
1e-07
1e-08
0.01
0.00
1e-05
1e-04
1e-03
1e-09
0
1000
2000
3000
time
4000
max(ux)
(b) Growth rate of kuy k2
(a) Velocity
FIG.
5. Intermediate
(a) ui , as
a function
of time
for laminar flow,
and (b) growth
of ku
max uxis
. The zero
y k2 versus rate
• Allows
us toflow:
extract
the
critical
Reynolds
number,
whenratethe
growth
7
◆1/3number. (⇢0 = 10 , v0 = .015, D = 4, n = 10)
of (b) corresponds to the critical✓Reynolds
zero. Find:
Rc ⇡
3
2A
6⇡ ⇥ 0.6 (1 + 0.1n)
At late times... NEED LONGER RUNS
D.
Late time decay
Bulk black branes
• Translated to the black brane language, our results imply that, for d=3:
For R > Rc , 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:
Fluid side
Gravity side
Laminar flow
Quasinormal mode regime
Turbulent flow
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
a500bulk metric...
(c) t =
(d) t = 900
(e) t = 2500
vorticity
FIG. 1. Vorticity field at various times for a turbulent run. (⇢
(a) r6 CABCD C ABCD
15
(f) t = 7000
0
= 1010 , v0 = .05, D = 10, n = 10)
Funnels run along tubes
(b) r7 ⇤ CABCD C ABCD
connecting boundary and horizon
FIG. 8. Contour plots of principal invariants of the Weyl tensor in the bulk, computed from the zeroth order metric (2.1), from
the simulation snapshot in Fig. 1e. Notice that (a) is representative of the energy density ⇢, while (b) is representative of the
vorticity, as expected from Eq. (B6).
Mapping from boundary to bulk
density
(a) r6 CABCD C ABCD
FIG. 8. Contour plots of principal invariants of the Weyl tensor in the b
the simulation snapshot in Fig. 1e. Notice that (a) is representative of
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.
Long lived QNMs
High Reynolds number
• In contrast, linearized perturbation theory assumes that the nonlinear terms
are smaller than the damping term. Low R regime.
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 longlived quasinormal modes.
The End