The Fluid/Gravity Correspondence

Transcription

1 Review Construction Solution Summary Durham University PASI Summer School June 30, 2010

2 Review Construction Solution Summary Outline 1 Review fluid dynamics gravity 2 Iterative construction of bulk g MN and boundary T µν Structure of Einstein equations Implementation to 1st order 3 Analysis of 2nd order solution boundary stress tensor and transport coefficients bulk geometry and event horizon 4 Summary & Remarks

3 Review Construction Solution Summary fluid dynamics gravity Fluid dynamics Fluid dynamics is continuum effective description of any microscopic QFT valid when scales of variation are long compared to mean free path l mfp. The fluid description assumes that the system achieves local thermodynamic equilibrium. Regime of validity: long-wavelength approximation For local temperature of the fluid T and scale of variation of the dynamical degrees of freedom L, local equilibrium demands: L T 1 ɛ 1

4 Review Construction Solution Summary fluid dynamics gravity Fluid dynamics Dynamical degress of freedom: Local temperature T Fluid velocity u µ (normalized u µ u µ = 1) This leaves d functions, T (x µ ) and u ν (x µ ), which specify our fluid configuration. (x µ are coordinates on the boundary spacetime on which the fluid lives.) The conformal fluid stress tensor T µν Encode all the fluid information by stress tensor T µν, which is Traceless: T µ µ = 0 Conserved: µ T µν = 0 The conservation equation captures the dynamics.

5 Review Construction Solution Summary fluid dynamics gravity Form of conformal fluid stress tensor Form of stress tensor is determined by symmetries, order by order in derivative expansion; fluid properties specified by finite # of undetermined coefficients. Using Weyl-covariant formulation, in d dimensions T µν = P (γ µν + d u µ u ν ) 2 η σ µν [ ] + 2 η τ 1 u λ D λ σ µν τ 2 (ω µ λ σ λν + ω ν λ σ λµ ) + ξ σ [σ µ λσ λν Pµν d 1 σαβ σ αβ ] + ξ C C µανβ u α u β + ξ ω [ω µ λ ω λν + Pµν d 1 ωαβ ω αβ ] +... where Weyl-covariance demands that P T d, η T d 1, τ 1,2 T 2, ξ σ,c,ω T d 2

6 Review Construction Solution Summary fluid dynamics gravity Gravity in the bulk Consider any 2-derivative theory of 5-d gravity interacting with other fields with AdS 5 as a solution (e.g. II B SUGRA on AdS 5 S 5 ). Solution space has a universal sub-sector: pure gravity with negative cosmological constant E MN R MN 1 2 R g MN + Λ g MN = 0 (R AdS = 1 Λ = 6) We will focus on this sub-sector in long-wavelength limit. Apart from the AdS 5 solution, there is a 4-parameter family of solutions representing asymptotically-ads 5 boosted planar black holes. We will use these solutions to construct general dynamical spacetimes characterized by fluid-dynamical configurations.

7 Review Construction Solution Summary Structure 1st order 0 th order: boosted Schwarzschild-AdS black hole Start with the well-known stationary solution: (derivation) boosted Schwarzschild-AdS 5 black hole (w/ planar symmetry) ds 2 = 2 u µ dx µ dr + r 2 (η µν + [1 f (r/πt )] u µ u ν ) dx µ dx ν, with f (r) 1 1 r 4 It is parameterized by 4 parameters: temperature T and boosts u i. The bulk black hole is dual to a bdy perfect fluid with T µν = π 4 T 4 (η µν + 4 u µ u ν )

8 Review Construction Solution Summary Structure 1st order Deforming the 0 th order solution Now promote parameters u µ and T to functions depending on the boundary coordinates x µ : ds 2 = 2 u µ (x) dx µ dr + r 2 (η µν + [1 f (r/πt (x))] u µ (x) u ν (x)) dx µ dx ν Call such a metric g (0). Note: g (0) does NOT satisfy the equations of motion: E MN R MN 1 2 g MNR 6 g MN = 0 But starting from here we will construct an iterative solution.

9 Review Construction Solution Summary Structure 1st order A perturbation scheme for gravity Assume that the variation in local temperature and velocities are slow µ log T µ u O (ɛ), T T O (ɛ) In local patches the solution is like a boosted planar black hole. Basic idea The perturbative scheme is aimed at constructing a regular bulk solution, by patching together pieces of the uniform boosted black hole. Use ɛ as a book-keeping parameter (counting # of x µ derivatives), and expand: g = ɛ k g (k), T = k=0 ɛ k T (k), u = k=0 ɛ k u (k) k=0

10 Review Construction Solution Summary Structure 1st order A perturbation scheme for gravity At a given order in the ɛ-expansion we find equations for g (k). These are ultra-local in the field theory directions and take the schematic form: [ ] H g (0) (u µ (0), T (0) ) g (k) (x µ ) = s k H is a second order linear differential operator in r alone. s k are regular source terms which are built out of g (n) with n k 1.

11 Review Construction Solution Summary Structure 1st order A perturbation scheme for gravity Importantly the equations of motion split up into two kinds: Constraint equations: E rµ = 0, which implement stress-tensor conservation (at one lower order). Dynamical equations: E µν = 0 and E rr = 0 allow determination of g (k). We solve the dynamical equations subject to g (k) = particular(s k ) + homogeneous(h) regularity in the interior (coordinate choice for g) asymptotically AdS boundary conditions

12 Review Construction Solution Summary Structure 1st order Computation at first order To solve the equations to first order we need to ensure conservation of the perfect fluid stress tensor µ T µν (0) = 0 which needs to be solved only locally (at say x µ = 0). This can be used to eliminate derivatives of T (0) in terms of those of u (0) i. v (π T (0) ) 1 = 1 3 i u (0) i, i (π T (0) ) 1 = v u (0) i Then we solve Hg (1) = s 1 where the operators and sources are given as follows:

13 Review Construction Solution Summary Structure 1st order Computation at first order The operator H: Useful to decompose metric perturbations into SO(3) representations: scalars 1, vectors 3 and symmetric traceless tensors 5. For instance, we find: H 3 # = d ( ) 1 d dr r 3 dr # H 5 # = d (r 5 f (r) ddr ) dr # The source terms: These differ at various orders in perturbation theory. At first order: s1 3 = 3 r 2 v u (0) i s 5 1 = 6 r 2 σ (0) ij

14 Review Construction Solution Summary Structure 1st order Explicit solution to first order Bulk metric: ds 2 = 2 u µ dx µ dr + r 2 (η µν + [1 f (r/πt )] u µ u ν ) dx µ dx ν [ r + 2r πt F (r/πt ) σ µν u µu ν λ u λ 1 ] 2 uλ λ (u ν u µ ) dx µ dx ν, with x 2 + x + 1 F (r) = dx r x(x + 1) (x 2 + 1) = 1 [ ln 4 ( (1 + r) 2 (1 + r 2 ) r 4 ) ] 2 arctan(r) + π Boundary stress tensor: Note: T µν = π 4 T 4 (4 u µ u ν + η µν ) 2 π 3 T 3 σ µν. η s = 1 in agreement with well-known results. 4π

15 Review Construction Solution Summary T µν g MN 1 Review fluid dynamics gravity 2 Iterative construction of bulk g MN and boundary T µν Structure of Einstein equations Implementation to 1st order 3 Analysis of 2nd order solution boundary stress tensor and transport coefficients bulk geometry and event horizon 4 Summary & Remarks

16 Review Construction Solution Summary T µν g MN The d-dimensional conformal fluid from AdS d+1 Similar procedure can be implemented at any order. Generalizing to d dimensions and bdy metric γ µν [cf ], to second order: T µν = P (γ µν + d u µ u ν ) 2 η σ µν [ ] 2 η τ ω u λ D λ σ µν + ω λ µ σ λν + ω λ ν σ µλ + 2 η b [u λ D λ σ µν + σ λ µ σ λν σ αβ σ αβ ] d 1 P µν + C µανβ u α u β (1) with b d 4πT η = s 4π = 1 16πG AdS b d 1 and τ ω = b 1 ; P = 16πG AdS b d ; 1 y d 2 1 y(y d 1) dy

17 Review Construction Solution Summary T µν g MN The spacetime geometry dual to fluids The bulk solution thus constructed is tubewise approximated by a planar black hole! Bulk causal structure; in each tube metric approximates uniformly boosted Schwarzschild-AdS planar black hole.

18 Review Construction Solution Summary T µν g MN The event horizon The background has a regular event horizon. One can determine the event horizon locally using the fact that the solution settles down at late times to an uniformly boosted planar black hole. The horizon location can be determined within the perturbation scheme r = r H (x) = π T (x) + ɛ k r (k) (x) k=1 In fact, r (k) (x) is determined algebraically by demanding that the surface given by r = r H (x) be null. simpler analogy

19 Review Construction Solution Summary T µν g MN Cartoon of the event horizon Note: Horizon is null everywhere Late time approach to uniform planar black hole Horizon area increases

20 Review Construction Solution Summary T µν g MN The Entropy current Given a bulk geometry with a horizon we can determine the Bekenstein-Hawking entropy. Bulk construction of entropy: using area-form A of spatial slices of the event horizon in Planck units. Fluid entropy current The area-form A on event horizon can be pulled back to the boundary to define a fluid entropy current J µ S with non-negative divergence J S = η A µ J µ S 0 Entropy details Loganayagam

21 Review Construction Solution Summary T µν g MN Properties of Entropy current The bulk-boundary pull-back is facilitated by our coordinates: pull-back along radial ingoing geodesics (const r) x µ (H) x µ (bdy) Fluid entropy current consistent with second law and equations of motion naively has a 5 parameter ambiguity details Bulk construction of entropy current is also ambiguous: (i) ability to add total derivative terms without changing area (ii) pull-back is ambiguous to boundary diffeomorphisms. At second order this results in a two parameter ambiguity for Weyl covariant current with positive divergence.

22 Review Construction Solution Summary 1 Review fluid dynamics gravity 2 Iterative construction of bulk g MN and boundary T µν Structure of Einstein equations Implementation to 1st order 3 Analysis of 2nd order solution boundary stress tensor and transport coefficients bulk geometry and event horizon 4 Summary & Remarks

23 Review Construction Solution Summary Summary a map between conformal fluid configurations on R d 1,1 & regular asymp. AdS d+1 planar non-uniform black holes gain insight into generic behaviour of gravity Bulk spacetime solutions naturally uphold Cosmic Censorship imply a new variant of Uniqueness Theorem Long-wavelength regime of fluid dynamics allows this construction to any order in a boundary derivative expansion. This yields local determination of the event horizon. The solutions satisfy the Area increase theorem & corresponding entropy current satisfies the 2 nd law. Recovered the well-known value of viscosity: η s = 1 4π Predicted second order transport coefficients, characterising our CFT fluid.

24 Review Construction Solution Summary Generalizations other dimensions (cf. Van Raamsdonk; Haack, Yarom; Bhattacharyya, Loganayagam, Mandal, Minwalla, Sharma) fluids on curved manifolds (Cf. Bhattacharyya, Loganayagam, Minwalla, Nampuri, Trivedi, Wadia) include matter (richer dynamics, but at expense of losing universality) dilaton ( induces forcing) (cf. Bhattacharyya, Loganayagam, Minwalla, Nampuri, Trivedi, Wadia) Maxwell U(1) field ( extra conserved charge) (cf. Erdmenger, Haack, Kaminski, Yarom; Banerjee, Bhattacharya, Bhattacharyya, Dutta, Loganayagam, Surowka; Hur, Kim, Sin) 3 Maxwell fields + 3 scalars (cf. Torabian, Yee) magnetic and dyonic charges (cf. Hansen, Kraus; Caldarelli, Dias, Klemm) non-conformal fluids (cf. Kanitscheider, Skenderis; David, Mahato, Wadia) non-ralativistic fluids (cf. Rangamani, Ross, Son, Thompson; Bhattacharyya, Minwalla, Wadia)

25 Review Construction Solution Summary Puzzles & future directions Role of non-long-wavelength bulk semiclassical solutions More detailed bulk analysis: horizon topology, nature of curvature singularity, Cosmic Censorship striking difference between turbulence in 3+1 and 2+1 nonrelativistic fluids (eg. inverse cascade)? = qualitative difference in gravitational dynamics (eg. equilibration time of AdS 4 vs. AdS 5 BHs) Van Raamsdonk Relation to the black hole Membrane Paradigm Generalizations: extremal fluids (superfluids), confining theories (domain walls),... Finite N effects Gravity dual of turbulence comments

26 Review Construction Solution Summary

27 Boosted Schwarzschild-AdS black hole Static Schwarzschild-AdS black hole in planar limit: ( ds 2 = r 2 f (r) dt 2 + ) (dx i ) 2 + dr 2 r 2 f (r) i with f (r) = 1 r + 4. ( temperature T = r r 4 + /π.) To avoid coordinate singularity at the horizon r = r +, use ingoing coordinates: v = t + r where dr = dr r 2 f (r) : ds 2 = r 2 f (r) dv dv dr + r 2 i (dx i ) 2 Now covariantize by boosting: v u µ x µ, x i P iµ x µ. back

28 Choice of coordinates For dealing with regularity issues etc., it is simplest to work in an analog of ingoing Eddington-Finkelstein coordinates. ds 2 = 2 u µ (x) S(r, x) dr dx µ + χ µν (r, x) dx µ dx ν back The choice of coordinates is such that x µ = constant are ingoing null geodesics. It is well adapted to discuss features of horizon, such as entropy in the fluid language.

29 Event horizon in Vaidya-AdS Vaidya = spher. sym. black hole with ingoing null matter: ( ds 2 = 1 2 m(v) ) dv dv dr + r 2 dω 2 r suppose horizon is at r = r H (v) normal n = dr ṙ dv is null when r H (v) = 2 m(v) + 2 r H (v) ṙ H (v) Exact solution gives horizon non-locally in terms of m(v). But for m(v) slowly varying, ṁ(v) = O(ɛ), m m = O(ɛ 2 ), use ansatz r H = 2 m + a m ṁ + b m ṁ 2 + c m 2 m +... Iterative solution gives a = 8, b = 64, c = 32,... back

30 Expression for entropy current The gravitational entropy current: (4π η) 1 J µ S = [1 + b 2 ( A 1 σ αβ σ αβ + A 2 ω αβ ω αβ + A 3 R + b 2 [ B 1 D λ σ µλ + B 2 D λ ω µλ] + C 1 b l µ + C 2 b 2 u λ D λ l µ +... )] u µ with A 1 = π 16 + ln 2 4 ; A 2 = 1 8 ; A 3 = 1 8 B 1 = 1 4 ; B 2 = 1 2 C 1 = C 2 = 0 Entropy current

31 Divergence of entropy current Entropy current: (4π η) 1 J µ S [1 ( )] = + b 2 A 1 σ αβ σ αβ + A 2 ω αβ ω αβ + A 3 R u µ Divergence of entropy current: + b 2 [ B 1 D λ σ µλ + B 2 D λ ω µλ] + C 1 b l µ + C 2 b 2 u λ D λ l µ G (5) N b3 D µj µ S = b [ σ µν + b (2 A A ) ln 2 u λ D λ σ µν +4 b (A 2 + A 3 ) ω µα ω α ν + b (4 A )(σµα σ α ν ) + b C 2 D µ l ν ] 2 + (B 1 2A 3 ) b 2 D µd λ σ µλ + (C 1 + C 2 ) b 2 l µd λ σ µλ +... Non-negativity of divergence: D µ J µ S 0 (when σµν = 0) demands B 1 = 2 A 3, C 1 + C 2 = 0 back

32 Horizon physics described by fluid dynamics... Where does the fluid live? On the event horizon? (null hypersurface, defined globally...) On the dynamical horizon? (spacelike hypersurface) On the stretched horizon? (a la Membrane Paradigm) Membrane Paradigm Gourgoulhon & Jaramillo Thorne, Macdonald, Price Horizon interpreted as a fluid membrane with certain dissipative properties: (e.g. electrical conductivity, shear & bulk viscosity, etc.) On the spacetime boundary. (AdS/CFT) Fluid dynamics describes the full spacetime, not just horizon. back

33 Weyl covariant formalism Consider conformal transformations of metric and fluid variables: g µν = e 2φ(x) g µν, u µ = e φ(x) ũ µ, T = e φ(x) T (2) To construct Weyl-covariant derivative, use the gauge field A ν u λ λ u ν λu λ Then for tensor Q µ... ν... of weight w d 1 u ν = Ãν + ν φ. (3) D λ Q ν... µ... λ Q ν... µ... + w A λ Q ν... µ... + [ g λα A µ δ µ λ A α δ αa µ ] λ Q α... ν [g λν A α δλ α A ν δν α A λ ] Q α... µ is also Weyl-covariant with weight w. (4) Loganayagam

34 Kerr-AdS BH in fluid variables Weyl-covariant formalism allows to express rotating AdS black holes more elegantly and intuitively in fluid variables: ds 2 = 2u µ dx µ (dr + r A ν dx ν ) + [ r 2 g µν + u (µ S ν)λ u λ ω µ λ ω λν ] dx µ dx ν + r 2 u µ u ν b d det [r δ µ ν ω µ ν] dx µ dx ν (5) with A ν u λ λ u ν λu λ and Weyl-covariantized Schouten tensor S µν 1 d 2 d 1 u ν (6) ( R µν Rg µν 2(d 1) ) (7)