Gravity and the Shape of Turbulence

Home , Event horizon

Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook

Yaron Oz (Tel-Aviv University)

February 2010 (CERN)

With B. Keren-Zur, C. Eling, G. Falkovich, I. Fouxon, X. Liu, Y. Neiman PRL 101 (2008) 261602, JHEP 0903, 120 (2009), PLB 680, 496 (2009), arXiv:0906.4999 (JHEP), arXiv:0909.3574, JFM 644, 465 (2010), arXiv:1002.0804 and to appear. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Outline

Turbulence and Black Holes

Fluid Dynamics and Turbulence

Relativistic CFT Hydrodynamics

Black Hole Horizon Dynamics

Outlook Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Turbulence

Turba is a Latin word for crowd. Turbulence originally refers to the disorderly motion of a crowd. Scientiﬁcally it refers to a complex and unpredictable motions of a ﬂuid. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Turbulence

• The dynamics of ﬂuids is a long standing challenge that remained as an unsolved problem for centuries. • Understanding its main features, chaos and turbulence, is likely to provide an understanding of the principles and non-linear dynamics of a large class of systems far from equilibrium. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Black Hole Dynamics

• We consider a conceptually new viewpoint to study these features using black hole dynamics. • The existence of horizon is crucial: fields can fall into the black hole but cannot emerge, this breaks time reversal symmetry and allows Einstein equations to describe dissipative effects. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Black Hole Dynamics Since the gravitational field is characterized by a curved geometry, the gravity variables provide a geometrical framework for studying the dynamics of fluids: A geometrization of turbulence. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Non-Relativistic Fluid: The Navier-Stokes Equations

• The fundamental formulation of the nonlinear dynamics of non-relativistic ﬂuids is given by the incompressible Navier-Stokes (NS) equations

∂t vi + vj ∂j vi = −∂i P + ν∂jj vi + fi (1)

• vi (x, t), i = 1, ..., d, (d ≥ 2) obeying ∂i vi = 0 is the velocity vector ﬁeld, P(x, t) is the ﬂuid pressure divided by the density, ν is the (kinematic) viscosity and fi (x, t) are the components of an externally applied (random) force. • The equations can be studied mathematically in any space dimensionality d, with two and three space dimensions having an experimental realization. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Reynolds Number and Turbulence

• An important dimensionless parameter is the Reynolds number LV Re = (2) ν where L and V are, respectively, a characteristic scale and velocity of the flow. • Experimental and numerical analysis data show that for Re ≪ 1, the flows are regular (Laminar). For a Reynolds number in the range between 1 and 100 the flow exhibits a complicated (chaotic) structure, while for Re ≫ 100 the flow is highly irregular (turbulent) with a complex spatio-temporal pattern formed by the turbulent velocity field. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Turbulence in Nature

• Most ﬂows in nature are turbulent. This is simple to see by −6 m2 noting that the viscosity of water is ν ≃ 10 sec , while that −5 m2 of air is ν ≃ 1.5 × 10 sec . Thus, a medium size river has 7 a Reynolds number Re ∼ 10 . Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Statistical Approach

• In the turbulent regime the velocity field exhibits a highly complex spatial and temporal structure and appears to be a random process. • A single realization of a solution to the NS equations is considered to be unpredictable. • Thus, although the NS equation is deterministic (without a random force), a statistical approach to the turbulent flows seems to provide an appropriate description. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Statistical Approach • For instance, instead of making attempts to understand the whole picture of a single-time velocity field, one considers the statistics of velocity difference between points separated by a fixed distance. The statistics is defined by averaging over space.

V(x) V(y) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Energy Cascade In the energy cascade picture, energy is transmitted to the ﬂuid by large eddies at a scale L, that transmit the energy to smaller scales by breaking to smaller eddies due to instability, until the viscous scale l is reached, where the energy dissipates due to friction.

L 3 K 41Theory : ∼ R 4 l e Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Anomalous Scaling

• There is experimental and numerical evidence that in the range of distance scales l ≪ r ≪ L (the inertial range) the ﬂows exhibit a universal behavior, e.g. the space-averaged equal-time correlators of velocity differences in the inertial range are characterized by critical exponents. • For instance, the longitudinal n-point functions scale as (r ≡ x − y)

n r ξn Sn(r) ≡ h (v(x) − v(y)) · i∼ r (3) r

The 1941 exact scaling result of Kolmogorov ξ3 = 1 (anomaly in time-reversal symmetry), agrees well with the experimental data, while the other exponents are measurable real numbers. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook K41 Theory

• Longitudinal n-point functions

r n n Sn(r) ≡ h (v(x) − v(y)) · i∼ r 3 (4) r • Energy spectrum − E(k) ∼ k 5/3 (5) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Intermittency

A central assumption of K41 theory is the self-similarity of the velocity ﬁeld at the inertial range. This misses intermittency. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Anomalous Exponents from Experiment A major open problem is to calculate the anomalous exponents. While in two space dimensions the anomalous exponents ξn seem to follow the Kolmogorov linear scaling and are given by rational numbers, this is not the case in three space dimensions.

Anomalous Exponents

n/3

ξn 1

1 2 3 n Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Anomalous Exponents: Analytic Treatment?

• The highly non-linear interactions at all scales and the absence of a small expansion parameter make the analytical treatment a formidable task. • Clearly, new ideas are desperately needed. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook New Predictions

• Recently, we showed that the Kolmogorov relation follows not from energy conservation but from the ability to write the equations in conservation form in terms of currents and sources. • This allows us to derive an analog for a compressible case, and in particular to relativistic hydrodynamics, where the energy cascade picture does not exist. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Incompressible Turbulence: A New Experimental Prediction (with Falkovich and Fouxon)

• In non-relativistic incompressible turbulence:

Dr hv(r)p(r)v 2(0)i = d Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook A Large d Conjecture

• The anomalous exponents ξn of incompressible Navier-Stokes turbulence approach unity in the limit of inﬁnite number of space dimensions d as in Burgers Turbulence ∂t v + v∂x v − ν∂xx v = f • This may provide a starting point for a large d expansion around Burgers statistics rather than Gaussian statistics as has been attempted before. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Third Millennium Problem

• The NS and Euler equations are nonlinear partial differential evolution equations. A major open problem posed by these equations is the understanding and control of their solutions. • Of particular importance is the short distance behavior and existence of singularities in the solutions, i.e. starting from smooth initial data, with a bounded energy condition and a smooth external force, can the solutions develop a ﬁnite-time singularity. • This is known not to be the case in two space dimensions, but is not known in three space dimensions ("The Third Millennium Problem" announced by the Clay Mathematics Institute). Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Shock Wave Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Finite Time Singularities

• Numerical computations appear to exhibit blowup for solutions of the Euler equations, but the extreme numerical instability of the equations makes it very hard to draw reliable conclusions. • From a physical viewpoint, such a ﬁnite-time blowup would mean a breakdown of the large distance effective theory that is supposed to describe ﬂuid dynamics, and a need for an ultraviolet (short distance) information for a complete description of the dynamics. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Knudsen Number

• Hydrodynamics applies under the condition that the correlation length of the ﬂuid lcor is much smaller than the characteristic scale L of variations of the macroscopic ﬁelds. • In order to characterize this, one introduces the dimensionless Knudsen number

Kn ≡ lcor /L (6)

• Since the only dimensionfull parameter is the characteristic temperature of the ﬂuid T , one has by dimensional analysis, lcor = (~c/kB T )G(λ) (7) where λ denotes all the dimensionless parameters of the CFT. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Stress-Energy Tensor

• The stress-energy tensor of the CFT obeys

µν µ ∂νT = 0, Tµ = 0 (8)

• The equations of relativistic hydrodynamics are determined by the constitutive relation expressing T µν in terms of the temperature T (x) and the four-velocity ﬁeld uµ(x) µ satisfying uµu = −1. • The constitutive relation has the form of a series in the small parameter Kn ≪ 1,

∞ µν µν µν l T (x)= Tl (x), Tl ∼ (Kn) , (9) Xl=0

µν µ where Tl (x) is determined by the local values of u and T and their derivatives of a ﬁnite order. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Stress-Energy Tensor

• Keeping only the ﬁrst term in the series gives ideal hydrodynamics and the stress-energy tensor reads

4 Tµν = T [ηµν + 4uµuν] (10)

• The dissipative hydrodynamics is obtained by keeping l = 1 term in the series. The stress-energy tensor reads

4 Tµν = T [ηµν + 4uµuν] − cησµν (11)

where 2 σ = (∂ u +∂ u +u uρ∂ u +u uρ∂ u − ∂ uα[η +u u ] µν µ ν ν µ ν ρ µ µ ρ ν 3 α µν µ ν (12) • The dissipative hydrodynamics of a CFT is determined by only one kinetic coefﬁcient - the shear viscosity η. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Relativistic Turbulence: Experimental Predictions (with Fouxon)

• In relativistic hydrodynamics we consider the hydrodynamics equation with a random force term

ν ∂ Tµν = fµ (13)

and derive the exact scaling relation ǫr hT (0, t)T (r, t)i = i (14) 0j ij d

where hT0j (0, t)fj (0, t)i≡ ǫ. • In the non-relativistic limit we get the Kolmogorov exact relation hvj (0, t)vi (r, t)vj (r, t)i = ǫri /d (15)

where hvj (0, t)fj (0, t)i≡ ǫ Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Relativistic Turbulence

• The Reynolds number is

TL R ∼ (16) e η/s

When Re is large we expect turbulence. • Is there a universal structure in relativistic turbulence? • Does it show up in RHIC and LHC? For gold collisions at RHIC, the characteristic scale L is the radius of a gold nucleus L ∼ 6 Fermi, the temperature is the QCD scale η 1 T ∼ 200 MeV, and s ∼ 4π is a characteristic value of strongly coupled gauge theories. With these one gets Re > 100. Is it an experimental realization of the steady state relativistic turbulence (Romatschke:2007). Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook η The Shear Viscosity to Entropy Density Ratio s

Ideal Viscous Tµν = Tµν + ηTµν µ ∂ Tµν = 0

$/s 2 1/ ( log (

1/4 4

(

Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Elliptic Flow

• Hydrodynamic simulations at low shear viscosity to entropy ratio are consistent with RHIC Data. • The elliptic ﬂow parameter is the second Fourier coefﬁcient v2 = hCos(2φ)i of the azimuthal momentum distribution dN/dφ dN ∼ 1 + 2v Cos(2φ) dφ 2 Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Elliptic Flow (Luzum, Romatschke:2008) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Relativistic Turbulence in 1+1 Dimensions (to appear with X.Liu)

• The scaling behavior of all structure functions take the form

cpq = p + q, p + q ≤ 1; (18)

cpq = 1, p + q ≥ 1. (19)

• 1+1 dimensional relativistic turbulence is strongly intermittent, at short scales the contribution from singular objects, the shock waves, takes over the contribution from the smooth component as p + q increases across 1. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Non-Relativistic Limit of CFT Hydrodynamics

• The hydrodynamics of relativistic conformal ﬁeld theories is intrinsically relativistic as is the microscopic dynamics. • However, the limit of non-relativistic macroscopic motions of a CFT hydrodynamics leads to the non-relativistic incompressible Euler and Navier-Stokes equations for ideal and dissipative hydrodynamics of the CFT, respectively (with Fouxon; Bhattacharyya, Minwalla, Wadia: 2008). • The non-relativistic slow motions limit: v ≪ c where v i is the three-velocity of the ﬂuid, uµ = (γ, γv i /c) and γ = [1 − v 2/c2]−1/2.

~ 2 2 c F(λ) T = T0 1 + P/c + o(1/c) , ν ≡ (20) h i 4kBT0 • For example, for strongly coupled CFTs described by an AdS gravity dual F = 1/π. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Holographic Principle

• The microscopic degrees of freedom of gravity in a volume of space V are encoded on a boundary of the region A. • In the AdS/CFT examples, the theory on the boundary is a local quantum gauge ﬁeld theory. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Energy Scale • The additional coordinate is an energy scale, thus we unify in the gravity description space and time and energy.

IR Energy Scale UV (t,x) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook UV-IR • The size of the object (for instance instanton) in the boundary gauge ﬁeld theory is determined by its location in the energy scale direction.

IR UV Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Gravitational Dual Description

• Relativistic CFT hydrodynamics provides a universal description of the large scale dynamics of the CFT. The AdS/CFT correspondence suggests that the large-time dynamics of gravity provides a dual description of the CFT hydrodynamics. • The four-dimensional CFT hydrodynamics equations are the same as the equations describing the evolution of large scale perturbations of the ﬁve-dimensional black brane. • Since we can obtain the Euler and NS equations in the non-relativistic limit of CFT hydrodynamics, the AdS/CFT correspondence implies that Euler and NS equations have a dual gravitational description. The dual description is obtained by taking the non-relativistic limit of the geometry dual to the relativistic CFT hydrodynamics. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Gravitational Dual Description

• Consider the ﬁve-dimensional Einstein equations with negative cosmological constant

Rmn + 4gmn = 0, R = −20 (21)

• These equations have a particular "thermal equilibrium" solution - the boosted black brane

2 µ 2 µ ν 2 µ ν ds = −2uµdx dr − r f [br]uµuνdx dx + r Pµνdx dx (22) where 1 f (r)= 1 − , Pµν = uµuν + ηµν (23) r 4 and the constant T = 1/πb is the temperature. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Gravitational Dual Description

• One looks for a solution of the Einstein equation by the method of variation of constants using the ansatz (Bhattacharyya, Hubeny, Minwalla, Rangamani)

gmn = (g0)mn + δgmn (24)

m n α µ (g0)mndy dy = −2uµ(x )dx dr − 2 α α α µ ν 2 α µ ν r f [b(x )r]uµ(x )uν(x )dx dx + r Pµν (x )dx dx (25)

• y = (x µ, r). As in the Boltzmann equation, the condition of constructibility of the series solution produces equations µ α α α for u (x ) and T (x )= 1/πb(x ). The series for gmn is the series in the Knudsen number of the boundary CFT hydrodynamics. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Horizon Dynamics

• The way the black brane horizon geometry encodes the boundary fluid dynamics is reminiscent of the Membrane Paradigm in classical general relativity, according to which any black hole has a fictitious fluid living on its horizon. • The real fluid whose dynamics we wish to study is at the boundary of the space. It is natural to ask to what extent it can be identified under the duality map with the membrane paradigm fluid. • We analyze the dynamics of the event horizon of a black brane in asymptotically AdS space-time and show that it is described by the Navier-Stokes equations. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Horizon Dynamics

• The analysis that we perform also holds for any non-singular null hypersurface when a large scale hydrodynamic limit exists. Thus it applies, for instance, to the Rindler acceleration horizon. • The connection between the horizon hypersurface dynamics and the NS equation is analogous to the connection between the Burgers and the KPZ equations. The Burgers equation provides a simpliﬁed model for turbulence, while the KPZ equation describes a local growth of an interface using a height function. The height gradient obeys the Burgers equation. • Our result shows that real turbulence may also be seen as resulting from a physically natural surface dynamics. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Membrane Dynamics: KPZ and Burgers

∂t v + v∂x v − ν∂xx v = f , v = ∂x h (26) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Deﬁnition of Coordinates

(Formalism developed by Damour in his Ph.D. thesis: 1979)

• We use the convention 8πG = c = ~ = kB = 1. • Consider a (d + 2)-dimensional bulk space-time M with coordinates X A, A = 0, ..., d + 1 with a Lorentzian metric gAB. • Let H be a (d + 1)-dimensional null hypersurface (notion akin to horizon) characterized by the null normal vector n which components nA obey

A B n · n = gABn n = 0 (27) Note this condition implies that for a null hypersurface the normal vector is also a tangent vector. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Normal to the Horizon

• We deﬁne the hypersurface in the bulk space-time by x d+1 ≡ r = const, and denote the other coordinates as x µ = (t, x i ), i = 1, ..., d. • The coordinate t parameterizes a slicing of space-time by spatial hypersurfaces and x i are coordinates on sections of the horizon with constant t. • In this coordinate system

nr = 0, nt = 1, ni = v i (28) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Geometrization of the Fluid Variables • The ﬂuid pressure and velocity in the geometrical picture

V(x)

P(x) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Slow Motion Limit

• In the case of black branes in AdS, the event horizon is located in the bulk space-time at r = πT0, where T0 corresponds to the Hawking temperature. The horizon coordinates (t, x i ) can be identiﬁed with time and space Eddington-Finkelstein coordinates in the AdS boundary. • We consider the slow motion limit where v i is a small perturbation. In order to keep track of the different terms 2 i we impose the scaling ∂t ∼ ε , v ∼ ∂i ∼ ε, where ε is a small parameter corresponding to c−1. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Induced Metric

• The ﬁrst fundamental form (induces metric) reads

2 i i j j dsH = hij (dx − v dt)(dx − v dt) (29)

where hij is the metric on sections St of the horizon H at constant t. • For the equilibrium black brane, at zeroth order

(0) 2 hij = (πT0) δij (30)

The details of the subleading term, which is of order ε2, will not be needed for our analysis. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Extrinsic Curvature

• The second fundamental form of the horizon hypersurface ν is the extrinsic curvature Kµ deﬁned by

A ν A ∇µn = Kµ eν (31)

A where we use the horizon basis eµ • Together, the ﬁrst and second fundamental forms provide a complete description the embedding of the null hypersurface in the bulk space-time. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Expansion and Shear

• Consider Lie transport of hij along the null normal vector n, which is given by the Lie derivative Ln 1 1 L h = ∂ h + (πT )2(D v + D v ) (32) n ij 2 t ij 2 0 i j j i

Di is the covariant derivative with respect to the metric hij . • This can be split into its trace part (the expansion θ) and trace free part (the shear σij )

1 θ = hij ∂ h + D v j (33) 2 t ij j σij = Lnhij − θhij/d (34) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Expansion and Shear

Expansion

Shear Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Expansion and Shear

• For the black brane we get to leading order the O(ε2) expressions

θ = ∂i vi (35) 1 σ = (πT )2(∂ v + ∂ v − 2∂ v δ /d) (36) ij 2 0 i j j i k k ij Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Pressure

A • Using n and ei as a tangent basis, the components of the horizon extrinsic curvature are

n Kn = κ(x) (37) n Ki = Ωi (38) i i i Kj = σj + θδj /d (39)

B A A • κ(x) is the surface gravity deﬁned by n ∇Bn = κ(x)n . It can be parameterized as

κ(x)= 2πT0(1 + P(x)) (40)

where P(x) scales as ε2. We will identify P(x) as the ﬂuid pressure. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Momentum

• Ωi is deﬁned by A Ωi = m ∇i nA (41) A where m nA = 1. For the black brane at leading order we have Ωi = 2πT0vi (42) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Focusing Equation

• The dynamics of the horizon geometry perturbations are governed by the Einstein equations. Consider ﬁrst the contraction of the Einstein equations with nAnB. The black brane is a solution to the Einstein equations with negative cosmological constant

matt RAB + (d + 1)gAB = TAB (43)

• There is no contribution from the cosmological constant term proportional to the metric, and we get the focusing equation

A 2 AB matt A B −n ∇Aθ + κ(x)θ − θ /d − σABσ − TAB n n = 0 (44) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Incompressibility Condition

matt • Consider ﬁrst the case TAB = 0. Plugging our previous results for the expansion, shear, and surface gravity, we ﬁnd at leading order the incompressibility condition

∂i vi = 0 (45) Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Navier-Stokes Equations

• Consider next the contraction of the Einstein equations A B with n ei . Again the cosmological constant term does not B A contribute because by construction ei and n are orthogonal. From the remaining terms one ﬁnds

1 L Ω = −∂ κ(x)+ D σj − ∂ θ − nAeBT matt (46) n i i j i d i i AB where j j LnΩi = (∂t + θ)Ωi + v Dj Ωi +Ωj Di v (47) matt • When TAB = 0 we ﬁnd that the leading order terms are at order ε3 and give the NS equation (1) without a force term −1 and with a kinematic viscosity ν = (4πT0) . Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Area Theorem

∂ (A/A )= − ∂ v 2/2 d d x (48) t 0 Z t d where A0 is the zeroth order area density (πT0) . Thus, as the kinetic energy of the ﬂuid on the boundary decreases in time due to viscous dissipation, the horizon area grows. This is consistent with the classical area increase theorem of General Relativity. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook The Open Question

How does the black hole horizon geometry encodes turbulence? • Space-averaged equal-time correlators of velocity differences Sn(r) in the inertial range are characterized by critical exponents. In the geometrical picture, Sn(r) correspond to the space-averaged equal-time correlators of differences of normals to the horizon. • Does the horizon surface develop a multifractal structure that encodes the anomalous exponents of turbulence in terms of the spectrum of fractal dimensions ? • Is there an appropriate generalization of quasi-normal modes that captures the turbulent structure ? Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Black Holes in Asymptotically Flat Space-Time

• Consider for example black holes in asymptotically flat spaces with horizons of a spherical topology. • By dimensional analysis the correlation length of a fluid will −1 scale as lc ∼ T0 , where T0 is the Hawking temperature. −1 • In the asymptotically flat cases T0 ∼ r0, where r0 the horizon radius. Since the horizon is now also compact, the L can be no greater than ∼ r0. The dimensionless Knudsen number Kn ≡ lc/L is of order unity, implying that the derivative expansion used above is not valid and that hydrodynamics is not the appropriate effective description. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Hydrodynamics and the QCD Axial Anomaly in HIC (with B. Keren-Zur)

• It has been recently revealed that the hydrodynamics description exhibits an interesting effect when a global symmetry current of the microscopic theory is anomalous. This has been ﬁrst discovered in the context of the the gauge/gravity correspondence (Erdmenger et.al, Banerjee et.al.:2008). • The Chern-Simons term in the gravity action, which corresponds to having an anomalous global symmetry current in the dual gauge theory, has been shown to modify the hydrodynamic current by a term proportional to the vorticity of the ﬂuid

1 ωµ ≡ ǫµνλρu ∂ u (49) 2 ν λ ρ Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Hydrodynamics and the QCD Axial Anomaly in HIC

• At ﬁrst sight the additional vorticity term seemed in contradiction with the second law of thermodynamics (Landau). • This, however, has been resolved by a redeﬁnition of the entropy current (Son,Surowka:2009). • (with Y. Neiman to appear) A non-abelian generalization. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Hydrodynamics and the QCD Axial Anomaly in HIC

• The basic idea is that the the axial charge density, in a locally uniform ﬂow of massless fermions, is a measure of the alignment between the fermion spins. • When the QCD ﬂuid freezes out and the quarks bind to form hadrons, aligned spins result in spin-excited hadrons. The ratio between spin-excited and low spin hadron production and its angular distribution may therefore be used as a measurement of the axial charge distribution. • Due to the short lifetime of high-spin hadrons such as the ρ mesons and ∆ baryons, we propose to focus on narrow resonances such as Ω−. • We predict the qualitative angular distribution and centrality dependence of the axial charge. Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Hydrodynamics and the QCD Axial Anomaly in HIC • Our main proposal is that for off-central collisions we expect enhancement of Ω− production along the rotation axis of the collision Turbulence and Black Holes Fluid Dynamics and Turbulence Relativistic CFT Hydrodynamics Black Hole Horizon Dynamics Outlook Outlook

• Non-relativistic and relativistic turbulence are still poorly understood, despite the fact that turbulence is rather generic in nature. • Gravity and a geometrical formulation may help. • Many applications in a wide variety of ﬁelds including atmospheric science, heavy ion collisions, astrophysics ...