The fluid/gravity correspondence: Lectures notes from the 2008 Summer School on Particles, Fields, and Strings
PPreprint typeset in JHEP style - HYPER VERSION
The fluid/gravity correspondence:
Lectures notes from the 2008 Summer School on Particles, Fields, and Strings, UBC, Canada
Nicola Ambrosetti
University of Bern, [email protected]
James Charbonneau
University of British Columbia, [email protected]
Silke Weinfurtner
University of British Columbia, [email protected]
Abstract:
This is a paper compiled by students of the 2008 Summer School on Particles,Fields, and Strings held at the University of British Columbia on lectures given by VeronikaHubeny as understood and interpreted by the authors. We start with an introduction tothe AdS/CFT duality. More specifically, we discuss the correspondence between relativis-tic, conformal hydrodynamics and Einstein’s theory of gravity. Within our framework theEinstein equations are an effective description for the string theory in the bulk of AdS spacetime and the hydrodynamic fluid equations represent the conformal field theory nearthermal equilibrium on the boundary. In particular we present a new technique for calcu-lating properties in fluid dynamics using the stress-energy tensor induced on the boundary,by the gravitational field in the bulk, and comparing it with the form of the stress-energytensor from hydrodynamics. A detailed treatment can be found in [JHEP 02 (2008) 045]and [arXiv:0803.2526]. Keywords:
AdS/CFT, hydrodynamics, fluid/gravity correspondence. a r X i v : . [ g r- q c ] O c t ontents
1. Introduction 12. Review of the AdS/CFT correspondence 2
3. Review of fluid dynamics 54. Schwarzschild AdS × S black hole (black branes) 9 × S
5. Inducing a stress-energy tensor with a metric 136. Introducing dynamics and non-uniformity 15
7. Conclusions 18
1. Introduction
The motivation for studying the fluid/gravity correspondence is to gain insight into stringtheory and strongly coupled gauge theories. There is a symbiotic relationship betweenthese theories and the insight gained is rooted in the fact that they are dual to each othervia the AdS/CFT correspondence. Their respective coupling constants can be matchedto scale inversely to each other, meaning that doing perturbative calculations in the weakcoupling limit of one theory can give us non-perturbative results in the strong couplinglimit of the other. This is provided we have the dictionary between the two theories thatgive us a map between objects in both theories.One of the fundamental questions to be answered in quantum gravity is what is thefundamental nature of spacetime? Somehow spacetime is an emergent property, but whatdoes it emerge from? Unfortunately, because the dictionary is incomplete, it is often easierto do straight gravitational calculations rather than use conformal field theory (CFT) atweak coupling to learn about gravity. For now the question of the fundamental nature ofspacetime is too ambitious. There are many simpler questions we can ask. Which CFTconfigurations have gravity duals? What types of curvature singularities are allowed?For strongly coupled gauge theories we want to use the gauge/gravity correspondenceto explore the universal properties of matter. The aim is to eventually do calculations in– 1 –he strongly coupled regimes of quantum chromodynamics (QCD), relevant to describe forexample the quark-gluon plasma produced in relativistic heavy ions collisions. Using grav-ity at weak coupling to calculate hydrodynamic properties has been successful in yieldingthe entropy to shear viscosity ratio, see [1]. Furthermore the authors conjectured this ratioto satisfy a universal lower bound, such that ηs ≥ π , (1.1)for any CFT with a gravity dual.The key to doing this calculation was the observation that the long distance dynamicsof any interacting quantum field theory near thermal equilibrium is well described by arelativistic fluid equation. In doing these calculations the real correspondence is betweenthe fluid dynamics, an effective description of CFT, and gravity, an effective description ofstring theory.In this review we will discuss recent techniques for calculating hydrodynamic propertiesusing the fluid/gravity correspondence [2, 3]. The original methods for calculating dynamicproperties of gauge fields using Minkowski correlators is reviewed in [4].
2. Review of the AdS/CFT correspondence
The AdS/CFT correspondence we are interested in is the classic example between type IIBstring theory on AdS × S , a 10 dimensional theory of gravity, and N = 4 supersymmetricYang–Mills (SYM), a 4 dimensional gauge theory, see for example [5]. The difference indimension seems like a problem until one realizes that the extra dimensions on the gravityside become particle degrees of freedom on the gauge side.In terms of the correspondence the gauge theory lives on the 4 dimensional boundary,located at r = ∞ , of the 5 dimensional AdS space. The dual gravity theory lives in thebulk of the AdS space where r < ∞ , see Figure 1. The fundamental ideas behind theconjectured correspondence are outlined in Table 1. Details of the dictionary between CFToperators on the boundary and field configurations in the bulk will be discussed later.Before using the correspondence for our calculations we must find an effective descrip-tion for the theories that are dual to each other. Next we find static solutions to theseeffective theories from which it is possible to calculate static properties such as the entropyof strongly coupled SYM. The real goal is to calculate dynamic properties by perturbingthe static solution in one theory and seeing what that looks like in the other theory. To doso we need to know what the perturbations in each theory look like. Since introducing why the duality exists is better left for another review, we will focus onthe parameter matching that arises from the two theories being dual. This is the simplestpart of of the AdS/CFT dictionary. A more detailed discussion can be found in [6], wherea precise matching of fields and operators in the two theories is listed.On the field theory side there are two parameters — the number of colours N (i.e., therank of the gauge group SU ( N )) and the gauge coupling g . When the number of colours is– 2 – igure 1: An illustration of where N = 4 SYM lives on AdS space. The 4 dimensions of thegauge theory ( t, x ) live on the boundary of the AdS space at r = ∞ . Table 1:
Summary of the corresponding elements that appear in our duality. bulk boundaryAdS/CFT type IIB string theory onasymptotically AdS × S N = 4 SYM on S × R or with a Poincar´e patch R × R effective description Einstein equation withcosmological constant relativistic fluid dynamics known static solutions black hole or black branein AdS static configuration of aperfect fluid perturbation non-uniformly evolvingblack branes dissipative fluid flowlarge, known as the planar limit, perturbation theory is controlled by the ’t Hooft coupling λ = g N . On the string theory side the parameters are the string coupling g s , the stringlength l s = √ α (cid:48) , and the radius L of the AdS space, which is proportional to N / , where N is the number of branes. The connection between the two originates from the doublenature of D-branes: the gravitational and the gauge theoretical.We will start by considering the gravity side. A single Dp-brane has a tension thatscales as M ∼ /g s , thus it is a nonperturbative object, and in the weak coupling limit canbe treated as a rigid object. Vice-versa, at strong coupling Dp-branes become light and alarge number, of order N ∼ /g s , are required to have a sizable gravitational effect. TheDp-brane is also charged under a ( p + 1)-form potential with a charge Q equal to its tension– 3 –esulting in the BPS bound being saturated. In a supersymmetric theory saturation ofthe bound means that half of the supersymmetry is preserved and the other half is broken,but also that the configuration is stable. In our case this implies that we can stack anarbitrary number of Dp-branes and the configuration will remain stable. The mass of ablack hole scales as N ∼ /g s so stacking N branes creates a background with a blackbrane equivalent of a black hole. The solution to the equations of motion of type IIBsupergravity for a D3-brane is [7, 8], ds = H − / ( r ) (cid:104) − dt + (cid:80) i =1 ( dx i ) (cid:105) + H / ( r ) (cid:2) dr + r d Ω (cid:3) (2.1) F (5) = Q ( ε + ∗ ε ) and Φ = constant , (2.2)where H ( r ) = 1 + L /r , the event horizon is at r = r + = 0 (compare with equation (4.1)where r + (cid:54) = 0), the volume form for the coordinates ( t, x i , r ) is given by ε , and the Hodgestar operator is denoted ∗ . The Ramond–Ramond 5-form, F (5) , is self dual and couplesto the D3-brane, and the dilaton field Φ is constant. Since g s = e Φ we are free to chooseany value for the string coupling. A more general, non-extremal, r + (cid:54) = 0, version of themetric (2.1) is considered in section 4. It is possible to rewrite strings in terms of gravityparameters 16 πG = (2 π ) g s l s and by associating this black brane (black hole) with N D3-branes we get L l s = 4 πN . (2.3)Another relationship between parameters comes from the gauge theory created bythe open strings attached to Dp-branes. The massless spectrum of the open strings onthe Dp-brane is that of the maximally supersymmetric ( p + 1) dimensional gauge theorywith gauge group SU ( N ) for N stacked branes. In the case of D3-branes this is fourdimensional N = 4 SYM. The effective action of the Dp-brane is the Dirac–Born–Infeld(DBI) action that, when expanded at first order in α (cid:48) , yields the usual Yang–Mills action.By identification of the coefficient of the gauge kinetic term we get the gauge coupling interms of string theory parameters, g = g s l s (4 πl s ) p − (2.4) ⇒ g = 4 πg s , for p = 3 . (2.5)We are interested in N = 4 SYM that is created by D3-branes, so p = 3, leaving a simplerelationship between the gauge and string coupling constants.We can rewrite these parameters as L α (cid:48) ∼ (cid:112) g s N ∼ √ λ . (2.6) The BPS bound is in general M ≥ Q . If we take the near horizon limit, r ∼ r + (cid:28) L , the geometry becomes that of AdS × S spacetime withradius L . From viewpoint of AdS the 5-form flux can be identified with the cosmological constant and canbe neglected in the following discussion. – 4 –he goal of this lecture series is to establish a correspondence between the gravitationallimit of Type IIB string theory on AdS xS space, and the hydrodynamic limit of the non-gravitational supersymmetric N = 4 Yang–Mills gauge theory defined on the conformal4-dimensional boundary of AdS . For the gravitational description of string theory to bevalid we require for the two dimensionless string parameters that Ll s = L √ α (cid:48) (cid:29) g s (cid:28) , (2.7)where the ratio between the curvature scale for the string background L and the stringlength l s to be large (to suppress stringy effects), and simultaneously we assume the stringcoupling to be small (to further suppress quantum effects). The equivalent of the gravita-tional limit in terms of fundamental parameters for the conformal field theory, the ’t Hooftcoupling λ and the Yang–Mills coupling g , is given by the following correspondence, (cid:26) L √ α (cid:48) , g s (cid:27) (cid:10) (cid:8) λ = g N, g (cid:9) . (2.8)Therefore the suppression of stringy and quantum effects on the boundary requires that λ (cid:29) λN (cid:28) , (2.9)both λ → ∞ and N → ∞ . This is the ’t Hooft limit with λ → ∞ .Further, to obtain a hydrodynamical description for the boundary theory, we considerthe local energy density of the conformal field theory such that we are able to thermody-namically associate a local notion of temperature T and mean free path l mfp ∼ /T . Thescale for the field fluctuations, R , has to be long wavelength in time and space. In otherwords, the scale of variations has to be large compared to the mean free path l mfp (cid:28) R . Interms of the derivative expansion of the stress-energy tensor we want the first order termto be small compared to the zeroth order term,1 st order0 st order ∼ η σ µν ρ u µ u ν ∼ ηρ R ≡ l mfp R ∼ R T (cid:28) , (2.10)where we have assumed that u µ ∼ O (1), and σ µν ∼ /R .Using the parameter matching to write these in terms of AdS parameters we get, R (cid:29) l mfp ⇒ r + (cid:29) L . (2.11)So we see that the regime where the fluid is valid corresponds to a theory with large AdSblack holes.
3. Review of fluid dynamics
We are relatively familiar with the effective description of string theory on AdS back-grounds, that is classical (super)gravity, but less so with the effective description of SYM.When first writing the fluid equations of motion we assume an ideal fluid - one that has no– 5 –iscosity and no thermal conduction. This means there is no energy dissipation and thatthe entropy of the fluid is constant. The effects we are interested in later are fundamen-tally dissipative so we will have to add corrections to these initially simplified equations ofmotion. Thus we start with an adiabatic fluid and later we relax this constraint allowingthe fluid parameters to vary slowly.The standard description of fluid dynamics is given by the continuity and Euler equa-tions. The differential form of the continuity equation is obtained by realising that anychange in the amount of fluid in a volume V with a density ρ must be accompanied by afluid traveling at velocity v through the boundary of that volume, δV , ∂∂t (cid:90) V ρ dV = − (cid:90) δV ρ v · dA (3.1)= − (cid:90) V ∇ · ( ρ v ) dV (3.2)where we have used Stokes theorem to rewrite the right hand side in terms of a volumeintegral. Since this is valid for any volume V the equation of motion can be read off to be, ∂ρ∂t + ρ ∇ · v + v · ∇ ρ = 0 . (3.3)The Euler equation comes from Newton’s law and relates the pressure P of the fluid to itsvelocity −∇ P = ρ (cid:18) ∂ v ∂t + ( v · ∇ ) v (cid:19) . (3.4)For our purposes it is more convenient to write these equations in a covariant mannerby expressing the characteristics of the fluid in terms of a symmetric stress-energy tensor T µν . Here T represents the energy density, T ii the pressures in each direction, T ij theshear stresses, and T i the momentum density. Within this formalism the equations ofmotion can easily be written as ∇ µ T µν = 0 . (3.5)The stress-energy tensor describes a fluid of density ρ ( x µ ), pressure P ( x µ ), and fluid veloc-ity u ν ( x µ ), which is normalised to u µ u µ = −
1. The stress-energy tensor of any ideal fluidis not permitted to contain derivatives and thus must of the form of T µν = (scalar) u µ u ν + (scalar) g µν . (3.6)If we further define a projection operator, P µν = g µν + u µ u ν , (3.7)that has the property P µν u µ = 0 . (3.8)– 6 –e can make use of dimensionality arguments to find the explicit expression for the stress-energy tensor T µν = ρ u µ u ν + P P µν . (3.9) Exercise: . Show that the relativistic versions of the ideal fluid equations can be obtainedby replacing T µν = ρ u µ u ν + P P µν into the equation of motion ∇ µ T µν = 0 . Ultimately, we are interested in calculating dissipative properties of conformal fluids.The stress-energy tensor up to zeroth order, being void of dissipation, captures none ofthese. We need to look at the first order dissipative corrections to the stress-energy tensor.We want to construct the most general n − derivative dissipative correction order byorder. It should be proportional to a single derivative of the fluid velocity ∇ µ u ν . Thistensor can be decomposed into irreducible representations, separating into componentsparallel or orthogonal to u µ ∇ µ u ν = − a µ u ν + σ µν + ω µν + 13 θP µν , (3.10)where the trace is fully contained in the last term, θ = ∇ µ u µ , and the first three terms areleft traceless. The first term contains the acceleration, a µ = u ν ∇ ν u µ , that is orthogonal tothe projection tensor P µν . The second term is the shear, σ µν = ∇ ( µ u ν ) + u ( µ a ν ) − θP µν , (3.11)which is symmetric and traceless by construction, and is orthogonal to u µ . The vorticity, ω µν = ∇ [ µ u ν ] + u [ µ a ν ] , (3.12)is also orthogonal to u µ and is antisymmetric by construction.Now that each irreducible representation has been identified with a physical quantitywe can find the constants of proportionality to make it a correction to the stress-energytensor. We find T µν dissip = − ζθP µν − ησ µν + 2 q ( µ u ν ) , (3.13)where ζ is the bulk viscosity, η is the shear viscosity, and the last term is constructed fromthe vorticity and represents the heat dissipation q = − κP µν ( δ µ T + a ν T ).The appearance of a the dissipation term can also be quantified by looking at thederivative of the entropy current J ( s ) µ = su µ , where s is the entropy density. At zerothorder we assume an adiabatic fluid which means the entropy is constant. To obtain thiswe set all derivatives of u µ to zero. Clearly then the entropy current is conserved, ∇ µ J ( s ) µ = 0 . (3.14)– 7 –hen dissipative terms are introduced there are now non-zero derivatives of u µ and theentropy current is no longer conserved, T ∇ µ J ( s ) µ = q µ q ν κT + ζθ + 2 ησ µν σ µν (3.15) ≥ . (3.16)We can see that because everything is squared, the correction is always positive, and theentropy always increases with dissipation.For the AdS/CFT correspondence it is necessary to constrain T µν to represent con-formal fluid equations. Before we proceed we would like to briefly recall the definition forconformal invariance. Conformal invariance . Consider a rescaling of the metric field tensor by some conformalfactor, g µν = e φ ˜ g µν , involving a scalar field φ . Further consider a field ψ satisfying the fieldequations H [ ψ, g µν ] = 0 , depending on ψ and g µν . H is said to be conformally invariant,if and only if we can find a conformal weight s of ψ , where s ∈ R , such that ˜ ψ and ˜ g µν ,where ψ = e s φ ˜ ψ and g µν = e φ ˜ g µν , also satisfy the field equations, H [ ˜ ψ, ˜ g µν ] = 0 .For example, the relativistic wave equation for massless scalar field minimally coupledto the gravitational field g µν , ∂ µ (cid:0) √− g g µν ∂ ν ψ (cid:1) = 0 , (3.17) is conformally invariant in d = 2 , where s = 0 . However, we are able to extend the equationof motion for ψ such that the fields couple conformally to the metric tensor, ∂ µ (cid:0) √− g g µν ∂ ν ψ (cid:1) + d −
24 ( d − R ψ = 0 , (3.18) for d > and with s = 1 − d/ . In order to have the conformal invariance (i.e., invariance under rescaling of T µν = e sφ ˜ T µν and g µν = e φ ˜ g µν ) of the relativistic Navier–Stokes equation, ∇ µ T µν = 0, it isnecessary to demand T µµ = 0 and s = − d − d = 4 thus s = − . (3.19)Thus conformal invariance implies Weyl symmetry, that is the trace of T µν must vanish,and the equation of state for a perfect fluid in d − dimensions reduces to P = ρd − . (3.20)Altogether, to first order, in terms of T ( x µ ) and u ν ( x µ ) we get T µν = ( πT ) ( η µν + 4 u µ u ν ) − πT ) σ µν . (3.21)With this equation we are able to read off the entropy to shear viscosity defined earlier on ηs = π T π T = 14 π . (3.22)– 8 –his can easily be seen as the shear viscosity η is given by the factor in front of the sheartensor σ µν , compare with equation (3.13), and the entropy is defined by s = ∂T /∂T where T = ( πT ) . Please note that the variable s has previously been used in a differentcontext where it expressed the conformal weight.
4. Schwarzschild AdS × S black hole (black branes) We will study the gravity side of the correspondence provided by the underlying stringtheory. A static solution for Einstein equations with negative cosmological constant in thebulk for N near extremal D3 branes is given by ds = H − / ( r ) (cid:34) − f ( r ) dt + (cid:88) i =1 ( dx i ) (cid:35) + H / ( r ) (cid:2) f − ( r ) dr + r d Ω (cid:3) , (4.1)where we have introduced the following parameters H ( r ) = 1 + L r and f ( r ) = 1 − r r . (4.2)Notice that for r + = 0 the extremal D3 brane, see equation (2.1), has its horizon at r = 0,see Figure 2, and at the near horizon limit, where r ∼ r + (cid:28) L and thus H ( r ) = L /r , themetric simplifies to ds = r L (cid:34) − (cid:16) − r + r (cid:17) dt + (cid:88) i =1 ( dx i ) (cid:35) + L r (cid:18) − r r (cid:19) − dr + L d Ω . (4.3) Figure 2:
The figure illustrates the various limits. For r ∼ r + (cid:28) L we are in the near horizonlimit, while for r/L → ∞ we are asymptotically approaching flat spacetime. It is also interesting to compare the near-extremal D3-brane with the global SchwarzschildAdS black hole, ds = − h ( r ) dt + dr h ( r ) + r d Ω + L d Ω , (4.4)– 9 –here h ( r ) = r L + 1 − r r ≡ r L + 1 − r r (cid:18) r L + 1 (cid:19) . (4.5)In the planar limit, r + (cid:29) L , thus h ( r ) → r L − r r and r → r L , (4.6)and the 3-sphere decompactifies, r d Ω → r dx i dx i L . (4.7)Substituting this in we get the our near horizon metric. So the planar limit r + (cid:29) L ofthe global Schw-AdS metric gives the near horizon limit of the string theory metric for N near extremal black branes, see Figure 3. Figure 3:
Planar limit, r ∼ r + (cid:29) L . × S Before continuing we would like to summarize the key features of the gravitational field inthe bulk. There exists a scaling symmetry in the planar limit, t → λ t, x i → λ x i , and r → rλ ⇒ r + → r + λ , (4.8)– 10 –uch that ds → ds with r + rescaled. Therefore it can be shown that the Hawkingtemperature [9, 10], given by T BH = h (cid:48) ( r + )4 π = 2 r + L π r + L , T BH = r + π L → T BH λ , (4.9)is also part of the scaling symmetry.For the causal structure, see the Penrose diagram in Figure 4(b), we are interested inthe asymptotic behaviour, the existence of horizons and curvature singularities. As r → ∞ we get for the line element ds → r L (cid:0) − dt + d(cid:126)x (cid:1) + L r dr + L d Ω , (4.10)corresponding to an asymptotic AdS xS spacetime. There exists a Killing-vector field ∂ t with norm given by g tt , and at r = r + the norm of Killing-vector field vanishes, thus thereexists a Killing horizon at r + . Finally, it can be shown by studying the behaviour of thediffeomorphism invariant quantity R abcd R abcd , the square of the Riemann tensor, that at r = 0 we are dealing with a curvature singularity, where R abcd R abcd → ∞ .It is also insightful to directly compare the causal structure for a Schwarzschild blackhole in two different spacetimes — asymptotic Minkowski/flat and AdS — using Penrosediagrams .In Figure 4(a) we plotted the worldline for an observer in an asymptotically flat space-time that starts at past timelike infinity I − . Once the observer crosses the horizon at r = r + no light rays can be transmitted to future null infinity I + (i.e., to any otherobserver outside the black hole) and the infalling observer inevitably falls towards thespacetime singularity at r = 0. Consequently no information can leave the black hole.This is the opposite of a white hole where no information can enter. The white hole isobtained from mathematically extending the spacetime solution. To make the spacetimediagram more accessible we indicate future and past null infinity as I ± , future and pasttimelike infinty as I ± , and spacelike infinity I . These indicate respectively where lightlike,timelike, and spacelike worldlines start and end.The global structure for a Schwarzschild black hole in an asymptotic AdS spacetimeis shown in Figure 4(b). Here I ± = I and spatial infinity is now an extended line ratherthan a single point. A consequence of the extended spacelike infinity is that a single lightray can reach infinity, bounce back, and return to its origin in a finite time t , see greenlines in Figure 4(b).From now on we will drop the term, L d Ω , in the metric on S and use ds = r L (cid:34) − (cid:18) − r r (cid:19) dt + (cid:88) i =1 ( dx i ) (cid:35) + L r (cid:18) − r r (cid:19) − dr . (4.11) Penrose diagrams are plots of conformally transformed spacetimes. The conformal factor has beenchosen such that the entire (infinite) spacetime is mapped onto a finite region. These diagrams are espe-cially useful to study causal relations between any two points in the “original” spacetime, as conformaltransformations maintain the light-cone structure. – 11 – a) Schwarzschild black hole in asymptotically flat spacetime.(b) Schwarzschild black hole in asymptotically AdS spacetime.
Figure 4:
The figures illustrate the Penrose diagrams for black holes in spacetimes with differentasymptotic behaviour. Here light rays propagate along 45 ◦ lines, see yellow dashed lines as emittedfrom an observer falling into the black hole. .The extra degrees of freedom corresponding to fluctuations around the S are dual tooperators on the CFT side that we are not currently interested in. For the metric (4.11)the boundary lies at r = ∞ , the event horizon lies at r = r + , there is a curvature singularity– 12 –t r = 0, and the black hole temperature, which corresponds to the thermal temperatureof the gauge theory, is T BH = r + πL , thus all the key features are preserved. One way to avoid the coordinate singularity at r = r + is to use “ingoing” Eddington-Finkelstein coordinates, ( V, x i , r, · · · ), where V = t + r ∗ and dr ∗ = dr r L (cid:16) − r r (cid:17) . (4.12)Under the new coordinates ( V, x i , r ) the metric of the planar Schw-AdS black hole (4.11)can be rewritten as ds = − r L (cid:18) − r r (cid:19) dV + 2 dV dr + r L (cid:88) i =1 ( dx i ) . (4.13)We now find that g µν and g µν remain finite ∀ r > x µ = 0, and r = constant.Covariantizing x µ = ( V, x i ) with respect to the boundary directions we get V = − u µ x µ ⇒ dV = − u µ dx µ ⇒ dV = u µ u ν dx µ dx ν (4.14) x i = P iµ x µ ⇒ dx i dx i = P µi P iν dx µ dx ν = P µν dx µ dx ν , (4.15)where u µ = (1 , , , , u µ = ( − , , , black hole we have ds = − r L (cid:18) − r r (cid:19) u µ x ν dx µ dx ν − u µ dx µ dr + r L P µν dx µ dx ν = − u µ dx µ dr + r L (cid:18) η µν + r r u µ u ν (cid:19) dx µ dx ν . (4.16)The metric (4.16) is called as the boosted uniform black hole.
5. Inducing a stress-energy tensor with a metric
Now we consider the boundary stress-energy tensor T µν for the asymptotical AdS geometry,where we will use the unboosted metric, ds = r L (cid:34) − (cid:18) − r r (cid:19) dt + (cid:88) i =1 ( dx i ) (cid:35) + L r (cid:20) − r r (cid:21) − dr . (5.1)We have already established what the stress-energy tensor looks like in terms of dynamicalfluid variables and that it lives on the boundary of the AdS space. To do calculationsusing the correspondence we are interested in what the gravitational stress-energy tensoron the boundary looks like. Knowing this we are able to calculate the stress-energy tensor.– 13 –he correct prescription for determining the stress-energy tensor on the boundarygiven a metric in any dimension is found in [11]. The method requires one to consider theboundary surface ∂M n of the n dimensional bulk M n . To do this we want to foliate thespacetime such that the slices are parallel to the boundary and calculate the stress-energytensor for this foliation, see for example [12]. In particular we are interested in its behaviouron the boundary. We would like to point out that this prescription gives a unique stress-energy tensor on the boundary implying that the stress-energy tensor induced for a givenmetric is unique [11].The first thing we do is find the stress-energy tensor in the bulk. The gravitationalaction with cosmological constant Λ in n + 1 dimensions is, S = − πG (cid:90) M d n +1 x √− g ( R − − πG (cid:90) ∂M d n x √− γK + 18 πG S ct ( γ µν ) . (5.2)The first term is the bulk action whose solutions δS M n +1 = 0 for AdS n +1 with curvature L gives Λ = − n ( n − L , the cosmological constant. The second term is a surface term thatcontains the extrinsic curvature K µν = − γ µσ ∇ σ n ν , which is a Lie derivative of γ µν pointingin the direction of n µ . The action also contains counter terms to cancel the divergences dueto the infinite volume of AdS and obtain a finite stress-energy tensor T µν . The form ofthese counter terms depend on the number of dimensions. For the AdS case the counterterms are L ct = − L √− γ (1 − L R ( γ )).Here we are concerned only with the AdS result. The general result in other dimen-sions can be found in [11]. The boundary stress-energy tensor is T µν = 2 √− γ δS cl δγ µν = 18 πG (cid:20) K µν − Kγ µν − L γ µν − L G µν (cid:21) . (5.3)This is obtained in the standard manner by varying the action with respect to the boundarymetric γ µν . Note that the last two terms came from the counter term in the Lagrangian.The problem is then reduced to finding the extrinsic curvature. We must choose acurve to slice along. Since we are looking at how slices approach the boundary the naturalchoice is along the radial coordinate of the AdS space, d ˜ r = √ g rr dr , (5.4)for any constant t and x i . With this slicing the metric decomposes into, ds = N dr + γ µν ( dx µ + V µ dr )( dx ν + V ν dr ) (5.5)= N dr + γ µν dx µ dx ν , (5.6)where γ µν is the induced metric given by, γ µν = g αβ e αµ e βν where e αµ = (cid:18) ∂x α ∂ ˜ x µ (cid:19) , (5.7)and µ, ν run from 0 to 3, α, β run from 0 to 4. The initial metric (5.1) has no off-diagonalelements such that V µ = 0. The extrinsic curvature is then given as, K µν = − N n σ ∂ σ γ µν + D µ V ν + D ν V µ (5.8)= − N n σ ∂ σ γ µν . (5.9)– 14 –e are left then with finding the normal vector n µ and the form of the induced metric γ µν .The vector normal to the hypersurface is given by n α = N ∂ α ˜ r (5.10)= N √ g rr δ αr . (5.11)Raising the index gives n α = n β g αβ = N √ g rr δ rα (5.12)= N (cid:115) r L (cid:18) − r r (cid:19) δ rα . (5.13)Due to the simple diagonal structure and the exclusive dependence of the metric on r wecan set ˜ t = t and ˜ x i = x i , thus all e αµ = 1 . (5.14)The induced metric is given by dγ = r L (cid:34) − (cid:18) − r r (cid:19) dt + (cid:88) i =1 ( dx i ) (cid:35) , (5.15)where r = r (˜ r ).The Einstein tensor G µν ( γ ) vanishes at the boundary and using the previous resultswe can find T µν , see equation (5.3), for a metric g αβ corresponding to planar Schw-AdS .We first ensure that the divergent terms cancel out and then it can been shown that, T tt = 316 πG Lr r + O (cid:18) r (cid:19) (5.16) T ii = 116 πG r r + ... . (5.17)Altogether the stress-energy tensor on the boundary is T µν∂M = lim r →∞ r T µν = 116 πG (cid:2) ( πT ) ( η µν + 4 u µ u ν ) (cid:3) , (5.18)where we can rewrite η µν +4 u µ u ν = P µν +3 u µ u ν . We now have the form of the stress-energytensor on the gravity side that is induced on the boundary.
6. Introducing dynamics and non-uniformity
Without loss of generality we can set L = 1 , f ( r ) = 1 − r and b = 1 πT = 1 r + , (6.1)– 15 – able 2: Summary for the relevant parameters in the bulk (5 dimensional metric tensor) and onthe boundary (4 dimensional stress-energy tensor) and the corresponding equations of motion theEinstein equations in the bulk and the relativisitic hydrodynamic equations for the boundary.
Bulk Boundary E MN = R MN − Rg MN + Λ g MN = 0 ∂ µ T µν = 0 ds = − u µ dx µ dr + r [ η µν + (1 − f ( br )) u µ u ν ] dx µ dx ν T µν = b (4 u µ u ν + η µν )such that f ( b r ) = 1 − r r . (6.2)At zeroth order the boosted black brane metric corresponds to the stress-energy tensorof a perfect fluid. A summary of the current picture can be found in Table 2, with thestress-energy tensor induced from the metric as explained in the previous section.We then look for solutions of Einstein’s equations where the parameters { b, u i } arepromoted to slowly varying functions of the boundary coordinates x µ . Henceforth we shallcall the metric with the parameters promoted to functions g (0) , g | b → b ( x ) ,u i → u i ( x ) = g (0) [ b ( x ) , u i ( x )] . (6.3)But g (0) is no longer a solution of the Einstein equations for arbitrary b ( x ), u i ( x ). Never-theless it has two nice features; it is manifestly regular (i.e., non-singular) for every positive r and for slowly varying { b ( x ) , u i ( x ) } we expect it to be a good approximation to the truesolution since locally in x µ it can be “tubewise” well approximated by a boosted blackbrane.For a fluid in local thermal equilibrium with typical fluctuations R much larger thanthe scale of inverse temperature we can expand the solution in a series expansion in bR ∼ T R ≡ (cid:15) (cid:28)
1. Having order (cid:15) n then corresponds to n boundary derivatives. Inserting theansatz g (0) into Einstein’s equations we get E MN (cid:104) g (0) (cid:105) = O ( (cid:15) ) (cid:54) = 0 , (6.4)where O ( (cid:15) ) correspond to terms with derivatives and the linear term has vanished becauseit is a solution of the equations. The approach is to expand the metric in (cid:15)g MN = g (0) MN + (cid:15) g (1) MN + (cid:15) g (2) MN + . . . , (6.5)where the g ( i ) MN depend on { b ( x ) , u i ( x ) } and are correction terms such that g MN solves E MN = 0 to a given order in (cid:15) . This will be possible only if { b ( x ) , u i ( x ) } satisfy certainequations of motion (namely ∂ µ T µν = 0) that are corrected order by order in (cid:15) . Conse-quently we must also correct { b ( x ) , u i ( x ) } order by order in (cid:15) , so we expand them as b = b (0) + (cid:15) b (1) + . . . (6.6) u i = u (0) i + (cid:15) u (1) i + . . . (6.7)– 16 –ith constant { b ( i ) , u ( j ) i } . Now we can solve for g ( i ) iteratively. Let us imagine that we have solved the perturbationtheory to the ( n − th order (i.e., we have determined g ( m ) for m ≤ n −
1) and we havedetermined the functions u ( m ) i and b ( m ) for m ≤ n −
2. Inserting the expansion (6.5) intothe Einstein equation as given in Table 2, and extracting the coefficient of (cid:15) n , we obtainan equation of the form H (cid:104) g (0) (cid:16) u (0) i , b (0) (cid:17)(cid:105) g ( n ) ( x µ , r ) = S n . (6.8)Here H is a linear differential operator of second order in the variable r alone and containsno boundary derivatives. As g ( n ) is already of order (cid:15) n , and since every boundary derivativeappears with an additional power of (cid:15) , H is an ultralocal operator (no derivatives in x µ ).Hence H is a differential operator only in the variable r and does not depend on thevariables x µ .The precise form of this operator at a point x µ depends only on the values of u (0) i and b (0) at x µ but not on the derivatives of these functions at that point. Furthermore,the operator H is independent of n ; it is the same at all orders in (cid:15) . The difficulty insolving Einstein’s equations does not come from H but from the increasing complexity ofthe “source term” S n that is an expression of n th order in boundary derivatives of u (0) i and b (0) , as well as of ( n − k ) th order in u ( k ) i , b ( k ) for all k ≤ n − · / g ( n ) and only constrain the form of expression of b and u i . We will call these constraintequations and they turn out to be equivalent to ∂ µ T µν ( n − = 0 , (6.9)where T µν ( n − is the boundary stress-energy tensor dual to the metric g up to O ( (cid:15) n − ). Ofthe other 11 equations, one is redundant and the 10 dynamical equations left are used todetermine g ( n ) to second order in r . We now make a gauge choice by imposing that themetric be of the form ds = − u µ ( x ) S ( x, r ) dx µ dr + χ µν ( x, r ) dx µ dx ν , (6.10)such that every constant x µ trajectory corresponds to an ingoing null geodesic in r . Theresidual SO (3) symmetry in the spacial boundary directions is what allows us to reducethe second order differential equations in r to first order ones. The dynamical equationscan then be recast into a set of first order decoupled equations that we can solve byintegrating the source and thus guaranteeing regularity of the solution. The ambiguities inthe integration constant can by removed by choosing the Landau gauge, u µ T µν dissip = 0 , (6.11)where T µν dissip includes all higher dissipative orders.– 17 – .2 Results Implementing this procedure to first order the metric is found to be ds = − u µ dx µ dr + r [ η µν + (1 − f ( br )) u µ u ν ] dx µ dx ν (6.12)+2 r (cid:20) brF ( br ) σ µν + 13 u µ u ν θ − u ρ ∂ ρ ( u µ u ν ) (cid:21) dx µ dx ν , (6.13)with F ( r ) ≡ (cid:90) ∞ r x + x + 1 x ( x + 1)( x + 1) dx = 14 (cid:20) ln (cid:18) (1 + r ) (1 + r ) r (cid:19) − − r + π (cid:21) , (6.14)and the stress-energy tensor T µν = 1 b (4 u µ u ν + η µν ) − b σ µν . (6.15)The first line of the metric is just the zeroth order solution and the second line is the firstorder correction. Similarly the first term of the stress-energy tensor is the usual perfectfluid result and the second term is the dissipative correction.The second order calculation can be carried out in the same way and the result can befound in [2]. T µν (2) has five independent (Lorentz and Weyl) covariant terms (of the form σσ , ωω , aa , . . . ) from which we can read off second order fluid parameters like the relaxationtime, which form a signature of conformal fluids with a gravity dual.In [3] an analysis of the event horizon of the solution yielded the entropy current onthe boundary, which was shown to be never decreasing.
7. Conclusions
The fluid/gravity correspondence provides us with an exceptionally powerful tool for calcu-lations and provides an interesting connection between two seemingly disconnected fields.It is analogous to a giant “Laplace transform” where, when confronted with a difficultproblem, we can switch to a “space” where the calculations can be carried out, then theresults are translated back into the language of the original problem. For example, inthis paper we have calculated the stress-energy tensor of a gauge theory, with dissipativecorrections, by rephrasing the problem in the language of gravity, in a regime where thecalculation is tractable, then translating the result back into the language of gauge theory.In this review we have discussed a new technique that facilitates both the calculationand parameter matching. By calculating the stress-energy tensor on the boundary inducedfrom the bulk, and then comparing that with the stress-energy of fluid dynamics, we canquickly calculate hydrodynamic properties. Because this process is iterative, dissipativecorrections can be evaluated with relatively little effort.This new correspondence can be used to describe the quark-gluon plasma created inexperiments of relativistic heavy ions collisions [13, 14, 15, 16]. This has been done forthe shear viscosity to entropy ratio with encouraging results, although it remains to beunderstood to what extent these calculations apply to real QCD and how to estimate– 18 –heoretical errors [17]. This does not constitute experimental evidence in favour of stringtheory, but merely shows that string theory can give new insights into other fields of physics.We would like to end these lecture notes with a brief discussion on the role of fluidhydrodynamics in quantum gravity. Over the last twenty years a broad class of hy-drodynamic systems have been investigated, referred to as analogue models/ emergentspacetimes, whose linear excitations experience an effective metric tensor, see for exam-ple [18, 19, 20, 21]. The gravity analogy is only valid in the linear regime, beyond thisits exhibits model-dependent dynamics. Any attempt to relate these toy models withEinstein gravity has only been partially successful, see for example Sakharov’s inducedgravity [22, 23].The correspondence outlined exhibits a surprising relation between fluid dynamicsand general relativity at a fully dynamical level. The relevance of this duality, from aconceptional viewpoint, for alternative approaches towards quantum gravity has yet to beinvestigated. In general we would like to close our interpretation of Veronika Hubeny’slecture series by stressing the importance of dualities such as the one studied here, as theyare of interdisciplinary nature and open a window for an alternative approach towardsquantum gravity.
Acknowledgments
We would like to thank Veronika Hubeny for presenting these lectures and for her com-ments and suggestions on these notes. Also, we wish to thank the organizers of the 2008Summer School on Particles, Fields, and Strings held at the University of British Columbia,particularly Gordon W. Semenoff for initiating these proceedings. SW would like to thankBill Unruh for many discussions and comments on the fluid/gravity duality.
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