Off-shell hydrodynamics from holography

Journal of High Energy Physics, Feb 2016

Michael Crossley, Paolo Glorioso, Hong Liu, Yifan Wang

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Off-shell hydrodynamics from holography

HJE Off-shell hydrodynamics from holography Michael Crossley 0 1 Paolo Glorioso 0 1 Hong Liu 0 1 Yifan Wang 0 1 0 Cambridge , MA 02139 , U.S.A 1 Center for Theoretical Physics, Massachusetts Institute of Technology We outline a program for obtaining an action principle for dissipative fluid dynamics by considering the holographic Wilsonian renormalization group applied to systems with a gravity dual. As a first step, in this paper we restrict to systems with a non-dissipative horizon. By integrating out gapped degrees of freedom in the bulk gravitational system between an asymptotic boundary and a horizon, we are led to a formulation of hydrodynamics where the dynamical variables are not standard velocity and temperature fields, but the relative embedding of the boundary and horizon hypersurfaces. At zeroth order, this action reduces to that proposed by Dubovsky et al. as an off-shell formulation of ideal fluid dynamics. (AdS/CMT); Black Holes; Effective field theories 1 Introduction 2 Setup 3 Action for an ideal fluid 4 5 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 Generalization to higher orders Structure of derivative expansions to general orders Non-dissipative action at second order? Conclusion and discussions A Boundary term A.1 Boundary compatible with foliation A.2 Boundary incompatible with foliation B Explicit expressions of sources Isolating hydrodynamical degrees of freedom Saddle point evaluation Einstein gravity Solving the dynamical equations Effective action for τ and Xa Horizon limit Entropy current Hydrodynamical action and volume-preserving diffeomorphism More on the off-shell gravity solution 1 Introduction At distance and time scales much larger than the inverse temperature and any other microscopic dynamical scales, a quantum many-body system in local thermal equilibrium should be described by hydrodynamics. Except for ideal fluids, the current formulation of hydrodynamics has been on the level of equations of motion. There are, however, many physical situations where hydrodynamical fluctuations play an important role. An action principle is greatly desired. There are two main difficulties. One is to properly treat dissipation, and the other is to find the right set of dynamical degrees of freedom to formulate an action principle, as standard variables such as the velocity field appear not suitable. In principle it should be possible to derive hydrodynamics as a low energy effective field theory from a quantum field theory at a finite temperature via Wilsonian renormalization – 1 – patched together at a horizon hypersurface. Also labeled are stretched horizons Σ1, Σ2 discussed around (1.2). (b) A boundary theory Schwinger-Keldysh contour used to describe non-equilibrium physics. The two AdS regions map to the two horizontal legs of the Schwinger-Keldysh contour, while the analytic continuation around the horizon corresponds to the vertical leg which defines the initial thermal density matrix. group (RG) by integrating out all gapped modes, but in practice it has not been possible to do so. Such a formulation should lead to an action principle. For holographic systems, the holographic duality [ 1–3 ] provides a striking geometric description of the renormalization group flow in terms of the radial flow in the bulk geometry. In particular, the Wilsonian renormalization group flow of a boundary system can be described on the gravity side by integrating out part of the bulk spacetime along the radial direction [4, 5]. The proposal expresses the Wilsonian effective action in terms of a gravitational action defined at the boundary of the remaining spacetime region. The purpose of the current paper is to take a first step toward deriving an action for hydrodynamics using holographic Wilsonian RG.1 The basic idea is as follows: consider the gravity path integral Z[g¯1, g¯2] = spacetimes patched together at a dynamical horizon hypersurface, as shown in figure 1. Mc has two asymptotic boundaries ∂Mc 1,2 with boundary metrics g¯1, g¯2 respectively. The horizon is dynamical as its metric is integrated over. The two copies of AdS can be considered as corresponding to the two long horizontal legs of a Schwinger-Keldysh contour, with the continuation around the horizon corresponding to the vertical leg [13]. In (1.1) one integrates out all gapped degrees of freedom, and the resulting effective action for whatever gapless degrees of freedom remain is the desired action for hydrodynamics. For this purpose, it is convenient to introduce stretched horizons Σα, α = 1, 2 on each slice of the bulk manifold, which separate the bulk manifold into three different regions (see figure 1), i.e. 1The connections between holography and hydrodynamics has by now quite a history, starting with [6–9] and culminated in the fluid/gravity correspondence [10–12]. Z Mc = Z Σ1 ∂Mc1 + Z Σ2 boundary, the intersections of these geodesics with the boundary define X1a. The bulk path integral can be written as Z[g¯1, g¯2] = Z Dh¯1 Z where h¯1 and h¯2 are induced metric on the stretched horizons. Various factors in the integrand of (1.3) arise from the path integrals in three regions, e.g. ΨUV[h¯1, g¯1] = from that between ∂Mc1 and Σ1. integrates over all metrics G between ∂Mc1 and Σ1 with Dirichlet boundary conditions g¯1 and h¯1, and similarly with the others. The complex conjugate on Ψ∗UV[h¯2, g¯2] is due to that the bulk manifold in the region between Σ2 and ∂Mc2 has the opposite orientation Connections between hydrodynamics and Schwinger-Keldysh contour have been made recently in various contexts in [ 14–21 ]. In this paper we describe integrations over gapped degrees of freedom in the path integral (1.4). As anticipated earlier by Nickel and Son [22], in (1.4) the only gapless degrees of freedom are the relative embedding coordinates X1a(σμ) of the boundary Mc 1 and the stretched horizon hypersurface Σ1, see figure 2. Integrating out all other degrees of freedom we obtain an effective action IUV[X1a, g¯1, h¯1] for embeddings X1a, i.e. (1.4) becomes ΨUV[h¯1, g¯1] = Z DX1a eiIUV[X1a,g¯1,h¯1] . We develop techniques to compute IUV X1a, g¯1, h¯1 in expansion of boundary derivatives at full nonlinear level in a saddle point approximation. Plugging (1.5) into (1.3) and evaluating h¯1, h¯2 integrals one then obtains the full hydrodynamical action in terms of X1a and X2a, i.e. The evaluation of ΨIR h¯1, h¯2 requires developing new techniques for analytic continuations through the horizon. We will leave its discussion and the full evaluation of (1.7) elsewhere. Hydrodynamical actions based on doubled Xa degrees of freedom discussed here have also been discussed recently in [ 16–21 ]. We also show that at zeroth order in the derivative expansion, if one (i) takes h¯1 to the horizon, i.e. making Σ1 a null hypersurface, and (ii) requires h¯1 to be non-dissipative, i.e. the local area element is constant along the null geodesics which generate the horizon, h¯1 completely decouples from IUV[X1a, g¯1, h¯1], and IUV reduces to the conformal version of the ideal fluid action proposed by Dubovsky et al. [23, 24], i.e. IUV[X1a, g¯1, h¯1] = Iideal[ξ1, g¯1] Iideal[ξ, g¯] = −(d − 1) Z ddσ√ d −g¯ det α−1 2(d−1) , (α−1)ij = g¯μν ∂μξi∂ν ξj, and ξi(σμ) (with i = 1, 2, · · · , d − 1) are embeddings Xa(σμ) for a null hypersurface for which the time direction decouples. In particular, the volume-preserving diffeomorphisms which played a key role in the formulation of [23] arise here as residual freedom of horizon diffeomorphism. The entropy current also arises naturally as the Hodge dual of the pullback of the horizon area form to the boundary. It is tempting to ask whether conditions (i) and (ii) in the previous paragraph will also lead to a non-dissipative fluid action at higher orders. We find, however, that the 2nd order action is divergent unless one is restricted to shear-free flows. While it makes sense to make such restrictions in an equation of motion, imposing it at the level of path integrals for Xa appears problematic. We thus conclude that one must include dissipation in order to have a consistent formulation. We also note that the fact we find (1.9) when pushing Σ1 to the horizon does not necessarily imply that at zeroth order the full effective action (1.7) will be given by Ihydro = Iideal[ξ1, g¯1] − Iideal[ξ2, g¯2] (1.10) as the integrations over h¯1, h¯2 will generate new structures. At this stage the precise relation between (1.9) and the zeroth order of Ihydro is not yet fully clear to us. The plan of the paper is as follows. We will explain our holographic setup and the gravitational boundary value problem in details in section 2. In section 3, we perform the path integral (1.4) using saddle point approximation to obtain IUV defined in (1.5) at zeroth order in the boundary derivative expansion and relate it to the ideal fluid action (1.9). In section 4 we briefly comment on the generalization to higher orders in the derivative expansion. We conclude with a discussion of open questions and future directions in section 5. – 4 – (1.8) (1.9) have obtained similar results [25]. In this section, we describe in detail our setup for computing (1.4) to obtain IUV[Xa, g¯, h¯] Isolating hydrodynamical degrees of freedom In this subsection, we describe a series of formal manipulations of path integrals for gravity which allow us to isolate Xa as the “would-be” hydrodynamical degrees of freedom. There are standard difficulties in defining rigorously path integrals for gravity, which will not concern us as we will be only interested in the path integrals at a semi-classical level, i.e. in terms of saddle points and fluctuations around them. Consider a path integral of the form HJEP02(16)4 ¯ Z h g¯ Ψ[h¯, g¯] = where the integration is over all spacetime metrics ds2 = GMN (σ)dσM dσN = N 2dz2 + χμν (dσμ + N μdz)(dσν + N ν dz) between two hypersurfaces ΣUV and ΣIR at some constant-z slices and whose respective intrinsic geometries are specified by gμν and hμν , i.e. χμν ΣUV = g¯μν (σλ), χμν ΣIR = h¯μν (σλ) . In (2.1), one should integrate over all values of N and N μ without any boundary conditions for them on ΣUV and ΣIR. For this paper we will only be concerned with evaluating (2.1) to leading order in the saddle point approximation, thus will not be careful about the precise definition of integration measure, ghosts, and Jacobian factors for changes of variables. We will comment on these issues in section 5. Later we will take ΣUV to the boundary of an asymptotic AdS spacetime and ΣIR to an event horizon. We will for now keep them arbitrary for notational convenience. We will also for now keep the gravitational action S[G] general assuming only that it is diffeomorphism invariant and that the boundary conditions (2.3) give rise to a well defined variational problem. The variation of the action then has the form δS = without any boundary term. The equations of motion are thus EMN = 0 – 5 – (2.1) (2.2) (2.3) (2.4) (2.5) while diffeomorphism invariance of the action S[G] also leads to the Bianchi identities ∇M EMN = 0 . Given the diffeomorphism invariance of the bulk action S[G], Ψ[h¯, g¯] is invariant under independent coordinate transformations DiffIR × DiffUV of the two hypersurfaces, i.e. Ψ[h¯, g¯] = Ψ[ΛItRh¯ΛIR, ΛtUVg¯ΛUV] where ΛIR and ΛUV denote independent coordinate transformation matrices on ΣIR and HJEP02(16)4 Now consider transforming the metric (2.2) to the Gaussian normal coordinates (GNC) ds2 = du2 + γab(u, x)dxadxb ≡ G˜ABdξAdξB . Here we choose Gaussian normal coordinates for later convenience. The subsequent discussion applies with little changes to any set of “gauge fixed” coordinates. The metric components GMN can be expressed in terms of (γab, ξA) as GMN = G˜AB∂M ξA∂N ξB = ∂M u∂N u + γab∂M xa∂N xb . In choosing the Gaussian normal coordinates we have the freedom of fixing the values of u and xa at one end. For our later purpose it is convenient to choose a hybrid fixing u ΣUV = u0 = const, a x ΣIR = σμδμa. The values of u at ΣIR and xa at ΣUV are then determined dynamically, which we will parameterize as (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.15) ΣUV. ξA(σ) = (u, xa) in terms of which In terms of the foliation of (2.8), ΣUV and ΣIR can thus be written as and the boundary conditions (2.3) now become with ΣUV : u = u0, ΣIR : u = τ (xa) = τ˜(X−1(xa)) γab u=τ(xa) = hab, γab u=u0 = gab hab = h¯ab − ∂aτ ∂bτ, gab(X) = g¯μν(σ) ∂σμ ∂σν ∂Xa ∂Xb = J −1tg¯J −1 a ab , J μ ≡ ∂σμ ∂Xa . (2.14) Note that ξA = (u(σM ), xa(σM )) are dynamical variables and in going from (2.2) to (2.8), we have essentially traded GMN = (N, Nμ, χμν) for (u, xa, γab). The path integral can now be written as where γ is required to satisfy the boundary conditions (2.13). The coordinate invariance of the action implies that the action is independent of the bulk fluctuations of u and xa. Thus the path integrals over xa and u reduce to those over the boundary fluctuations (2.11) In the above integrals Xa always appears with g¯ through the induced metric g defined in (2.14). In addition to appearing in the IR boundary condition hab for γab integrals, τ also appears in the action S explicitly as the IR integration limit and boundary terms (which we will specify below). Xa and τ can be considered as the “Wilson line” degrees of freedom associated with N μ and N . Physically Xa(σμ) describes the relative embedding between the coordinates xa on ΣIR and the coordinates σμ on ΣUV, while τ (x) describes the proper distance between ΣUV and ΣIR. The path integrals (2.16) will be evaluated in stages: we first integrate over all possible γab with a fixed relative embedding Xa and proper distance τ to find and then integrate over τ (i.e. all possible proper distances) consistently integrated out to yield a local action I1[τ, h, g], which can be expanded in the number of boundary derivatives of τ, h and g, assuming they are slowly varying functions of boundary coordinates. In the boundary theory language γab should thus correspond to modes with a mass gap. τ depends only on boundary coordinates, and does not contain derivatives at leading order, and thus can also be consistently integrated out. By definition Xa always come with boundary derivatives as in (2.14), i.e. they correspond to gapless modes, and thus should be kept in the low energy theory. Integrating them out will lead to nonlocal expressions. Let us briefly consider the symmetries of I1[τ, g, h]. It is invariant under u-independent diffeomorphisms of x a → x′a(x) under which g, h transform simultaneously as tensors. These are large diffeomorphisms as they change the asymptotic behavior of AdS. I1 is also invariant under diffeomorphisms of σμ as it contains Xa and g¯ only through g, which is invariant due to the canceling of transformations in g¯ and Xa, g¯μν (σ) → g¯μ′ν (σ) = ∂σ′λ ∂σμ g¯λρ(σ′(σ)) ∂σ′ρ ∂σν Xa(σ) → X′a(σ) = Xa(σ′(σ)) . This implies that √ −g¯∇¯ ν 1 √ δI1 −g¯ δg¯μν = δXa δI1 ∂μXa – 7 – (2.16) (2.17) (2.18) (2.19) (2.20) HJEP02(16)4 where ∇¯ denotes the covariant derivative associated with g¯. Identifying √−g¯ δδg¯Iμ1ν as the 1 boundary stress tensor, we then see that the Xa equations of motion are equivalent to the conservation of the boundary stress tensor. Parallel statements can be made about IUV which comes from integrating out τ . Finally to conclude this subsection, let us be more explicit about the UV boundary condition. In an asymptotic AdS spacetime with ΣUV at a cutoff surface near the boundary, γab in (2.8) should have the behavior u→−∞ lim γab(u, xa) = e− 2Lu 2u γa(0b)(xa) + O(e L ) with L the AdS radius and γa(0b)(xa) finite. Thus we should replace the boundary condition (2.13) at u = u0 by HJEP02(16)4 lim u0→−∞ γab(u0, xa) = e− 2Lu0 gab(x) + O(e 2Lu0 ) , and g¯μν (σ) is the background metric for the boundary theory. Saddle point evaluation Now consider evaluating the path integrals (2.16)–(2.18) using the saddle point approximation. To elucidate the structure of equations of motion for τ and Xa, we consider (2.4) now with GMN considered as a function of γab and ξA via (2.9). Under variations of γab, we have √ −HEuB ∂∂ξzB ΣIR Euu − Eua ∂∂xτa ΣIR (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) which then (2.4) implies the equations of motion δGMN (x) = δγab∂M xa∂N xb EMN ∂M xa∂N xb = 0, ⇒ Eab = 0 where Eab = 0 is the ab-component of the equations of motion in coordinates (2.8). Below we will refer to these equations as “dynamical equations.” Under variations of ξA, we have δGMN = ∂∂G˜ξACB ∂M ξA∂N ξBδξC + 2G˜AC ∂M ξA∂N δξC . The bulk part of (2.4) then leads to the Bianchi identities in coordinates (2.8), which is as it should be since ξA(σ) is a coordinate transformation. But now there are boundary terms remaining δS = Z d x d √ −HEAB ∂z ∂ξB δξA −HEAB ∂z ∂ξB δξA which upon using (2.10)–(2.12) implies that −HEaB ∂∂ξzB ΣUV ⇒ with (2.28) corresponding to the equation of motion from varying τ while (2.29) corresponds to those from varying Xa. In deriving the second equations in both (2.28) and (2.29) we have assumed that √ −H and ∂∂uz at ΣIR and ΣUV are nonzero. It can be readily checked that (2.25) and (2.28)–(2.29) are equivalent to (2.5), and that the Bianchi identity ensures that (2.28) and (2.29) are satisfied everywhere once they are imposed at ΣIR and ΣUV, respectively. Following standard convention, below we will refer to (2.28) as the Hamiltonian constraint and (2.29) as the momentum constraints. Now recall from the general results on the holographic stress tensor [26] that the momentum constraints (2.29) in fact correspond to the conservation of the boundary stress where ∇a is the covariant derivative associated with gab and T ab is the stress tensor for the boundary theory with background metric gab in the state described by (2.8). Since g¯μν and gab are related by a coordinate transformation equation (2.30) is equivalent to ∇aT ab = 0 ∇¯ μT¯μν = 0 and tensor where (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) where ∇¯ μ is the covariant derivative associated with g¯μν and T¯μν is the stress tensor for the boundary theory with background metric g¯μν . This gives an alternative way to see that Xa equations of motion are equivalent to conservation of the boundary stress tensor. At the level of saddle point approximation, I1 as defined in (2.17) is obtained by solving (2.25) for γab, and IUV in (2.18) by solving (2.28) for τ (xa). In other words, IUV is computed by evaluating the gravity action with dynamical equations and the Hamiltonian constraint imposed, but not the momentum constraints. 2.3 Einstein gravity We now specialize to Einstein gravity, in which the gravity action S[G] in (2.1) can be written in the Gaussian normal coordinates (2.8) as are extrinsic curvatures for a constant-u hypersurface, and Sct is the standard AdS counterterm action at the ΣUV [26] Sct = Z u=u0→−∞ d x d √ −γ 2(1 − d) L + d − 2 L (d)R[γ] + · · · . – 9 – SIR is a boundary action at ΣIR which arises from the fact that ΣIR, given by u = τ (xa), is not compatible with the foliation of constant-u hypersurfaces, and can be written as (see appendix A for a derivation) with SIR = 2 Z where the indices are raised and lowered by the intrinsic metric h¯ab on ΣIR and D¯ is the covariant derivative associated with h¯ab. For convenience below we will use K to denote the matrix Kab and thus K = Tr K. Various components of the Einstein equations in Gaussian normal coordinates (2.8) can then be written as (2.35) (2.36) (2.37) (2.38) (2.39) −Eba = K′ − TrK′ + KTrK − Ric(d)[γ] − 2 −Euu = 1 2 TrK2 − 2 1 Tr2K + 2 1 R(d)[γ] − Λ = 0 −Eua = DaK − DbKba = 0 1 TrK2 + Tr2K + 2 1 R(d)[γ] − Λ = 0 with Da the covariant derivative associated with γab. As discussed in section 2.2, in order to not impose conservation of the stress tensor, i.e. leave hydrodynamical modes off-shell, at the saddle point level we should not impose the momentum constraint (2.39). We only need to solve the dynamical equations (2.37) for γab and a combination of (2.38)–(2.39) at ΣIR for τ (see (2.28)). From now on we will set the AdS radius L = 1. 3 Action for an ideal fluid In this section we first evaluate explicitly IUV[Xa, g¯, h¯] defined in (1.5) at zeroth order in the derivative expansion, assuming that Xa, g¯, h¯ are slowly varying functions. We then show that pushing h¯ to a horizon hypersurface and requiring it to be non-dissipative, we obtain the ideal fluid action of [23, 24]. 3.1 Solving the dynamical equations We will perform the γ integrals (2.17) using the saddle point approximation, i.e. it boils down to solving the dynamical Einstein equations (2.25) at zeroth order in boundary derivatives. At this order we can neglect boundary derivatives of τ (x), J aμ, and γab. The boundary conditions for γab become γ(u = τ ) = h = h¯, γ(u = u0 → −∞) = e−2u0 g, For notational simplicity here and below we will often use γ and g to denote the whole matrix γab and gab. Hopefully the context is sufficiently clear that they will not be confused with their respective determinants. which can be rewritten as K′ − TrK′ + KTrK − 2 1 TrK2 + Tr2K − Λ = 0 d − 1 d K′ + K2 1 2 where K is the traceless part of K From (3.4) Taking derivative on both sides of (3.3) leads to 1 2 d − 1 d K = K + K d 1, Tr K = 0 . (Tr K2)′ = −K Tr K2 . K′′ + KK′ = K Tr K2 . which is solved by where α1,2 are some constants. Inserting (3.9) into (3.4) we find Explicit expressions for the Einstein equations in Gaussian normal coordinates (2.8) are given in section 2.3. At zeroth order in boundary derivatives, the dynamical part of the Einstein equations (2.25) (more explicitly (2.37)) becomes (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) Eliminating Tr K2 between (3.6)–(3.7) and using (3.3) we then find an equation for K K′′ + 3K′K + K(K2 − d2) = 0 K = d α1α2e2du − 1 (1 + α1edu)(1 + α2edu) K = b0(α1 − α2)edu (1 + α1edu)(1 + α2edu) Tr b02 = d(d − 1) . with b0 a constant traceless matrix. Plugging (3.10) into (3.7) we find that b0 has to satisfy Combining (3.9) and (3.10) we then obtain K = 1 (1 + α1edu)(1 + α2edu) b0(α1 − α2)edu + (α1α2e2du − 1) . Now integrating (2.33) and imposing the boundary condition at UV (i.e u = u0 → −∞) we find that 2 γ = ge−2u (1 + α1edu)(1 + α2edu) d e2k(u)b0 k(u) = log 1 d where here and below we will always take α1 > α2. Introducing H ≡ g−1h¯, α1c ≡ α1edτ , α2c ≡ α2edτ from the IR boundary condition γ(u = τ ) = h¯ we then find √det H = e−dτ (1 + α1c)(1 + α2c), b0 = log Hˆ , Tr(log Hˆ )2 = 1 2k(τ ) where Hˆ denotes the unit determinant part of H and the last equation of (3.16) follows from (3.11). Requiring the metric γab to be regular and non-degenerate between u = −∞ and τ , we need 1 + α1c > 0, 1 + α2c > 0 . Given H and τ , we can use the first and last equations of (3.16) to determine α1,2 and then the second equation to find b0. Note at this stage τ is not constrained by H and thus can be chosen independent of h¯. More explicitly, Effective action for τ and Xa At zeroth order in boundary derivatives, the Einstein action (2.32) becomes and substituting (3.12) into (3.20) we have with the counterterm action given by I1[τ, h¯, g] = 2(d − 1) Z ddx√−g h−e−dτ + e−dΛ + α1α2edτ i + Sct Sct = −2(d − 1) We then find that with Z Λ→−∞ ddx√−γ = −2(d − 1) Z ddx√−g e−dΛ + α1 + α2 . I1[τ, h¯, g] = −2(d − 1) Z ddx√−g L1(H, τ ) L1(H, τ ) = e−dτ + α1 + α2 − α1α2edτ = −√det H + 4e− 21 dτ (det H) 14 cosh zc 2 − 2e−dτ where in the second line we have expressed the integration constants α1,2 in terms of boundary conditions via (3.18). We notice that at zeroth order, I1 depends on h¯ and g only through the combination H = g−1h¯. This follows from that I1 must be invariant under the diffeomorphisms of xa for which h¯ and g transform simultaneously, as noted in the paragraph before (2.19). At zeroth order Tr Hn for n = 1, 2, . . . , d are the only independent invariants. τ can now be integrated out by extremizing I1 which gives and thus Collecting everything together we thus find that2 IUV[Xa; h¯, g¯] = −2(d − 1) Z d x d √ −g √ det H cosh e−dτ0 = √ then from (3.28) α1c = −α2c = tanh zc α ≡ e−duh uh = τ − d 1 log tanh zc > τ . 2 One can readily check that the same result is obtained by solving instead the Hamiltonian constraint (2.28) at zeroth order. Also note that with τ = τ0 given by (3.25), equation (3.18) Equation (3.28) implies that after integrating out τ , α2 = −α1 is negative. We will now simply rename α1 as α. It is convenient to introduce (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) − det γ HJEP02(16)4 Now if we extrapolate the solution (3.13) beyond u = τ all the way to uh, then √ develops a simple zero at u = uh, which we will refer to as a “horizon.” Since the “horizon” lies outside the region where our Dirichlet problem is defined, γab does not have to be regular there, so this does not have to be the horizon in the standard sense. Now let us consider a sequence of h¯ whose time-like eigenvalue approaches zero. Equivalently, an eigenvalue of H (which we will denote as h0) goes to zero. At h0 = 0, h¯ describes a null hypersurface and ΣIR becomes a horizon for the metric between ΣIR and ΣUV. We thus define the h0 → 0 limit as the hydrodynamic limit. In this limit, we have det H → h0SdetH, 1 zc → − 2 log h0 + 1 2The overall minus sign has to do with our choice of orientation of bulk manifold. and the action (3.27) becomes where SdetH denotes the non-vanishing subdeterminant of H and can be written as where pd−1 is the standard polynomial which expresses the determinant of a non-singular (d − 1) × (d − 1) matrix in terms of its trace monomials. From (3.25) and (3.29) we thus find SdetH = pd−1 Tr H, Tr H2, · · · e−dτ0 → 4 1 (SdetH) 2(d−1) , d uh − τ → 0 = −(d − 1) Z Z d x d √ ddσ√ −g (SdetH) 2(d−1) −g¯ (SdetH) 2(d−1) . d d , i = 1, 2, · · · n, n ≡ d − 1 . (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40) Here we discuss the geometric meaning of SdetH and (3.35). Denote the null eigenvector of h¯ by ℓa, which give rises to a congruence of null geodesics which generate the null hypersurface. We can then choose a set of coordinates (v, ξi) on ΣIR with v the parameter along the null geodesics generated by ℓa and ξi remaining constant along geodesics. In this basis, we then write the metric on ΣIR as ds2ΣIR = h¯abdxadxb = σij(v, ξ)dξidξj, h¯ab = σij ∂xa ∂xb ∂ξi ∂ξj It then follows that where α−1 is defined as SdetH = det(αijσjk) = det σ det α−1 (α−1)ij ≡ αij = g¯μν EiμEjν , i E μ ≡ ∂xa ∂ξi J aμdσμ = ∂ξi ∂σμ . We thus find that √ SdetH can be written as horizon area density √σ normalized by the “area density” of the pull back of boundary metric g¯ to ΣIR. √ The physical meaning of SdetH can be better elucidated if instead we pull back all quantities to the boundary. We now show that it can be interpreted as a definition of (non-equilibrium) entropy density of the boundary system.3 For this purpose, consider the area form on the ΣIR which can be written as a = √ σ dξ1 ∧ dξ2 ∧ · · · ∧ dξn . Note that the horizon area √σ has no physical meaning itself in the boundary theory as its definition depends on a choice of local basis. It does become a physically meaningful quantity when we pull it back to the boundary via the relative embedding map J aμ introduced in (2.14). More explicitly, a = √ σE1 ∧ E2 ∧ · · · ∧ En, Ei = Eiμdσμ . From (3.40) we can define a current which is the Hodge dual of a on the boundary jμ = ǫμν1···νn aν1···νn = n! 1 ǫμν1···νn ǫi1···in Ei1 ν1 · · · Ein νn is natural to pull back the null vector ℓa to the boundary, giving where ǫμν1···νn is the full antisymmetric tensor for g¯ and ǫi1···in is that for σij. Similarly, it and we have chosen a convenient normalization for uμ. By construction, jμ is parallel to uμ and we can then write From (3.41), we find that has precisely the scaling of the local energy density as a function of entropy density for a conformal theory. From the perspective of evaluating the bulk action it can also be understood as follows: the bulk integration in (2.32) can be interpreted as giving the free energy while the Gibbons-Hawking term at the IR hypersurface becomes equal to entropy times temperature in the horizon limit. Their sum then gives the energy of the system. 3.5 Hydrodynamical action and volume-preserving diffeomorphism We now impose that the system is non-dissipative, which amounts to requiring that the entropy current (3.41) is conserved We will interpret uμ as the velocity field of the boundary theory, jμ (divided by 4GN ) as the entropy current, and s (divided by 4GN ) as the local entropy density. All these quantities are independent of choice of local coordinates on ΣIR. We also stress that their definitions do not depend on the derivative expansion and thus should apply to all orders. With this understanding the action (3.35) can be written as (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) (3.47) (3.48) (3.49) where form (3.39), or area density where ∇¯ is the covariant derivative for g¯. The above equation can also be written equivalently in various different ways in terms of horizon quantities. In terms of the horizon area uμ = (J −1)μaℓa, g¯μν uμuν = −1 jμ = suμ, s2 = −jμjμ . s = qdet(αijσjk) = √ SdetH . IUV = − Z ddσ√ −g¯ ǫ(s) . ǫ(s) = (d − 1)s d−1 d ∇¯ μjμ = 0. det σ = det σ(ξ) . i.e. the horizon area is independent of the horizon “time” v. Note that the form of the metric (3.36) is preserved with a v-independent coordinate transformation which can be used to set ξi → ξ′i(ξ) det σ = 1 → SdetH = det α−1. We thus see that with non-dissipative boundary condition at zeroth order the horizon metric completely decouples in the hydrodynamical action, and we find ddσ√ −g¯ det α−1 2(d−1) , d (α−1)ij = g¯μν ∂μξi∂ν ξj which is precisely that of [23, 24] applied to a conformal theory. After fixing (3.51), there are still residual volume-preserving diffeomorphisms in ξi, which played an important role in the formulation of [23, 24]. Here they arise out of a subgroup of horizon diffeomorphisms which leave “gauge fixing condition” (3.51) and the coordinate frame (3.36) invariant. If the non-dissipative horizon condition (3.49) can be consistently imposed to higher orders in derivative expansion, we should expect the resulting higher order action to respect the volume-preserving diffeomorphisms. As we will see in section 4, however, at second order we encounter divergences, which implies that (3.49) can no longer be consistently imposed for unconstrained integrations of Xa. 3.6 More on the off-shell gravity solution Here we elaborate a bit further on the off-shell gravity solution (3.13). First let us collect various earlier expressions. After integrating out τ , with (3.28) and (3.29) the off-shell metric (3.13) can be written as where we have introduced Also recall that e−dτ = √ Δ ≡ d(uh −τ ) = − log tanh zc . With a general regular h¯, the solution when extrapolated to the “horizon” uh is singular. This is perfectly okay as physically the behavior of the metric outside the region is of no concern to us. We will now show that if we do require the extrapolated metric to be also regular at the “horizon” u = uh, i.e. u = uh becomes a genuine horizon, then we recover the standard black brane solution. Of course this also implies that h¯ has to take a very specific form. γ = ge−2u 1 − α2e2du d2 e zz(uc) log Hˆ z(u) = log 1 + αedu 1 − αedu d 2 = −log tanh (uh − u) = tanh−1 αedu . (3.50) (3.51) (3.52) (3.53) (3.54) 2 (3.55) We now impose a “regularity” condition: γab has only one eigenvalue approaching zero as the horizon is approached with the other eigenvalues finite. Near the horizon, δ ≡ d(uh − u) → 0 with Denoting the eigenvalues of log Hˆ and g−1γ as ˆbμ and γμ respectively, from (3.53) we then z(u) → − log γμ → e−2uh(2δ) d δ 2 → +∞ . 2 δ − zc 2 ˆ bμ If we denote the time-like eigenvalue of g−1γ by γ0 and the rest by γi, the regularity condition amounts to that γ0 goes to zero while all γi finite. Requiring γi to be finite ˆbi = where the second equation follows from that log Hˆ is traceless. We thus find that the system has to be isotropic! Denoting the time-like eigenvector vector of Hˆ as ℓa, from (3.58) we can write Hˆ as where we have defined Plugging (3.59) into (3.53) we then find that Hˆ ab = (d − 1)bℓaℓb + b(δab + ℓaℓb) ℓa = gabℓb, ℓaℓa = −1 . γab = Cρ d4 gab + ρ−2ℓaℓb with C = e−2uh2 d4 , ρ(u) = cosh d(uh − u) 2 . This is precisely the black brane metric and ℓa is the null vector of the horizon hypersurface. Consider an arbitrary basis of vectors Eia which are orthogonal to ℓa, we can expand gab = −ℓaℓb + αijEiaEjb h¯ab = −h0ℓaℓb + σijEiaEjb then equation (3.59) implies that σikαkj = cδij where αij is the inverse of αij and c is some constant. In other words, regularity condition fixes h¯ in terms of g up to two constants c and h0. uh and τ can be expressed in terms of c and h0 as eduh = 4c1− d2 c − h0 , edτ = 4c1− d2 (√c + √ h0)2 . Given that ℓa is the null vector of the horizon and we can choose basis Eia in (3.63) to be that in (3.36) (with i index raise and lowered by α) and then αij of (3.63) then coincides that in (3.38), and thus the same notations. Similarly in the horizon limit h0 → 0, and σij of (3.64) is related to σij in (3.36) by raising and lowering using α. (3.56) (3.57) (3.58) (3.59) (3.60) (3.61) (3.62) (3.63) (3.64) (3.65) (3.66) In this section we discuss computation of IUV to higher orders in the derivative expansion. We first briefly outline the general structure of higher order calculations and then mention some results at second order. Structure of derivative expansions to general orders Assuming h¯, g¯ and Xa are slowly varying functions of boundary spacetime variables, we can expand γab, the extrinsic curvature K, and τ in the number of boundary derivatives, i.e. γ = γ0 + γ2 + · · · , K = K0 + K2 + · · · , τ = τ0 + τ2 + · · · . (4.1) HJEP02(16)4 where γ0, K0, τ0 (which we already worked out) contain zero boundary derivatives of g¯, h¯, J aμ = ∂μXa, whereas γ2, K2, τ2 contain two boundary derivatives, and so on. One can readily see that there is no first order contribution, as the equations for the saddle point (2.37) and (2.38) do not have first order terms, and neither does the action (2.32). The final hydrodynamical action (1.7) will receive first order contributions as the IR contribution ΨIR will contain first order terms, which will communicate via matching to ΨUV at the stretched horizons through equations for h¯1 and h¯2. Let us first look at the dynamical equations (2.37) which under decomposition (3.5) can be written as d − 1 d K′ + 1 2 K2 where R denotes the matrix of mixed-index Ricci tensor (d)Rab. Taking the u derivative on (4.2) and using (4.2)–(4.3) we find that K′′ + 3K′K + (K2 − d2)K = (d)RK + Plugging (4.1) into these equations we find at n-th order Kn′′ + 3K0Kn′ + (3K0′ + 3K02 − d2)Kn = Sn K′n + K0Kn + KnK0 = Pn where sources Sn and Pn contain only quantities which are already solved at lower orders. Note that Pn is a traceless matrix. Parallel to earlier zeroth order manipulations, the integration constants in K n will need to satisfy a constraint from (4.2) d − 1 d Kn′ + K0Kn + Tr K0Kn = Bn where Bn again contains only quantities solved at lower orders. Thus once we have solved the nonlinear equations at the zeroth order, higher order corrections can be obtained by solving linear equations. In particular, at each order the homogeneous part of the linear (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) equations are identical with only difference being the sources. This aspect is very similar to the structure of equations in the fluid/gravity approach [10]. For completeness we give explicit expressions of various sources Sn, Pn, Bn in appendix B. Similarly at n-th order the τ equation of motion (2.28) becomes TrK0Kn − K0Kn = Yn where the left hand side should be evaluated at zeroth order solution τ0 and Yn again depends only on lower order terms. For example at 2nd order it can be written as Y2 = − 2 1 (d)R2 + DaK0 − DbK0ba ∂τ0 ∂xa . Note that τn does not appear in (4.8) as ∂u(Euu)0 = 0. Non-dissipative action at second order? We have carried out the evaluation of IUV to second order. The full results are rather complicated and will be presented elsewhere. Here we will only mention results relevant for the following question: can we find boundary conditions for h¯ at the horizon which allow us to derive a non-dissipative hydrodynamical action to 2nd order in boundary derivatives? Mathematically this requires that in taking h¯ to be null, IUV[h¯, g¯, Xa] should have a well-defined limit and furthermore h¯ will either decouple (as in the zeroth order) or be determined in terms of g¯ and Xa, with Xa unconstrained. There are many reasons not to expect this to happen. After all, the holographic system we are working with has a nonzero shear viscosity, and things will eventually fall into horizon after waiting long enough time. Nevertheless it is instructive to work this out explicitly. Note that ideal fluid action of [23] has been generalized to second order in derivatives in [24, 28] based on volume-preserving diffeomorphisms. For simplicity we will take g¯μν = ημν . We find that in taking the horizon limit Δ ≡ d(uh − τ ) → 0, IUV develops various levels of divergences in terms of dependence on Δ: 1. The most divergent terms have the form tr log2 σˆ = 0 . (2) LUV ∼ ∂μj μ 1 Δ2 + 1 Δ ∂μjμ = 0 . (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (2) LUV ∼ tr log2 σˆ Δ3 1 log2 Δ + 1 log3 Δ + · · · where σˆ is traceless part of σij = αikσkj, and we have suppressed other finite constant coefficients. Interestingly all these divergences go away if we impose the regularity condition (3.65) at the horizon which is equivalent to 2. The next divergent terms are of the form where jμ is the entropy current (3.43). Thus they vanish if we impose the nondissipative condition where For this divergence to disappear, one then needs L (2) = − d1 23− d4 e(2−d)uh Σ2 log Δ + O(Δ0) Σμν = PμρPνσ∂(ρuσ) Pμν = ημν + uμuν . (4.14) (4.15) (4.16) HJEP02(16)4 (4.17) (4.18) (4.19) (4.20) 1 − d − 1 ∂ρuρPμν , Σ2 = 0 Eddington-Finkelstein coordinates. and (4.16), we obtain a simple result i.e. the system is shear free. Note that the divergence in (4.14) is proportional to Σ2, which is precisely the rate of increase of the horizon area.4 If we want to have unconstrained Xa, the shear-free condition (4.16) cannot be consistently imposed. Thus it appears not possible to generalize the non-dissipative horizon condition to obtain a second order non-dissipative action. As mentioned at the beginning of this subsection, this is hardly surprising. In particular, the specific divergence structure of (4.14) implies that we must take account of dissipation. We should note that in the full Schwinger-Keldysh program (1.7) outlined in the introduction, there is no need to impose any of the above conditions (4.11), (4.13) and (4.16). The divergences will cancel with those from ΨIR after we do a consistent matching at the stretched horizons. Also, the divergences mentioned above are not due to the use of Gaussian normal coordinates, which of course become singular themselves at the horizon. Similar divergences also occur in Eddington-Finkelstein coordinates. Being off-shell means that there are necessarily both in-falling and out-going modes at the horizon (which will further be magnified by nonlinear interactions) which will lead to divergences also in the Finally, for completeness, let us mention that if we do impose all of (4.11), (4.13) 3. Finally, we have the logarithmic divergence of the form ξi → ξi + δξi P μν ∂ν uh = uν ∂ν uμ. where (2) = 21− d4 e(2−d)uh θ2 − (d − 2)β2 − 2aμβμ , θ = ∂μuμ, βμ = Pμν ∂ν uh, aμ = uν ∂ν uμ. The second order Lagrangian (4.17) is subject to the ambiguity in the field redefinition which we fix by using the zeroth order equation of motion 4To see this explicitly, one need to study the Raychaudhuri equation associated with the null congruence ℓa on the horizon. In particular, one may need to put on shell the contraction of the Einstein equation with ℓa at this order. See [29] for details. to express βμ in terms of aμ. Eq. (4.17) can then be simplified to L (4.21) We should emphasize that due to various conditions imposed at the horizon, the nature of the above “action” is not clear at the moment. To derive a genuine off-shell action for one patch we should first compute the full action for both segments of the SchwingerKeldysh contour, and then integrate out modes of the other patch. At second order this “integrating-out” procedure likely does not make sense in the presence of dissipation. Even HJEP02(16)4 if this procedure makes sense after suppressing dissipation, it is not clear how our current prescription of imposing regularity and non-dissipative conditions relates to that. 5 Conclusion and discussions In this paper we outlined a program to obtain an action principle for dissipative hydrodynamics from holographic Wilsonian RG, and then developed techniques to compute IUV, as defined in (1.5), at full nonlinear level in the derivative expansion. The “Goldstone” degrees of freedom envisioned in [22] arise naturally from gravity path integrals, and the ideal fluid action of [23] emerges at zeroth order in derivative expansion when non-dissipative condition is imposed at the horizon. The volume-preserving diffeomorphisms of [23] appear here as a subgroup of horizon diffeomorphisms. We also found that a direct generalization of the nondissipative condition to higher orders does not appear compatible with the action principle. An immediate generalization of the results here is to compute ΨIR of (1.3) which will enable us to take into account of dissipations. In our discussion we have ignored possible corrections from Jacobian in the change of variables in going from (2.2) to (2.8), as well as higher order corrections in the saddle point approximation of gravity path integrals. Such corrections are suppressed at leading order in the large N limit of boundary systems. Nevertheless, they may be important for understanding the general structure of the hydrodynamical action, thus it would be good to work them out explicitly and explore their physical effects. It would be interesting to generalize the results to more general situations, such as charged fluids, fluids with more general equations of state (for example [30]), systems with anomalies (such as those considered in [31–33]), or higher derivative gravities. Acknowledgments We thank A. Adams, R. Loganayagam, G. Policastro, M. Rangamani, and D. T. Son for conversations, and M. Rangamani for collaboration at the initial stage. Work supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement DE-FG0205ER41360. We also thank the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during the completion of this work. Boundary term relation gives Σu with Now let us consider and apply the Stokes theorem to the last term of (A.1) Boundary compatible with foliation Consider a spacetime M with a boundary ∂M . Suppose ∂M is a slice of a foliation of M by hypersurfaces Σu. We denote the outward normal vector to Σu by nM . The Gauss-Codazzi R = (d)R + (K2 − KMN KMN ) − 2∇M (nM ∇N nN − nN ∇N nM ) where (d)R is the intrinsic scalar curvature of Σu and KMN is the extrinsic curvature for which then directly cancels the Gibbons-Hawking term. In this case we thus find that S = Z M dd+1x√−g h(d)R + (K2 − KMN KMN )i . A.2 Boundary incompatible with foliation Here we will consider an explicit example with a spacetime metric We further consider a foliation of the spacetime by hypersurfaces Σu specified by u = const. Denote the normal vector field to Σu by nM , which can be written as (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) K = ∇M nM . S = Z M dd+1x√−g (R − 2Λ) + Z ∂M d √ d x −h 2K −2 Z dd+1x √−g ∇M (nM ∇N nN − nN ∇N nM ) M = −2 = −2 Z Z ∂M ∂M d x d √ d x d √ −h nM (nM ∇N nN − nN ∇N nM ) −h ∇M nM R = (d)R + (K2 − KMN KMN ) − 2∇M (nM ∇N nN − nN ∇N nM ) The extrinsic curvature for Σu can be written as The Gauss-Codazzi relation gives ds2 = du2 + γab(u, xa)dxadxb . nM = (1, 0), nM = (1, 0) . 1 Kab = 2 ∂uγab, K = 1 γab∂uγab . 2 Note the relations and (A.12) can also be written as that hab = h¯ab + n12 ∂¯aτ ∂¯bτ, ∂¯aτ ≡ h¯ab∂bτ = n2∂aτ, −h = np−h¯, n = p1 − ∂¯aτ ∂aτ 1 K|∂M = nK + n Kab∂¯aτ ∂¯bτ − n 1 D¯ 2τ . Now combining the boundary term in (A.1) and the Gibbons-Hawking term we find where (d)R is the intrinsic scalar curvature of Σu. Now suppose the spacetime region M has a boundary ∂M which does not coincide with one of Σu. More explicitly, we specify ∂M by u = τ (xa) for some function τ (xa). The outward normal vector to ∂M can thus be written as ℓM = n(1, −∂aτ ), ℓ M = n(1, −∂aτ ), n = √ 1 1 + ∂aτ ∂aτ , ∂aτ ≡ γab∂bτ . The extrinsic curvature of ∂M is given by 1 1 K|∂M = √−γ ∂u(n√−γ) − √−γ ∂a(n√−γ∂aτ ) = nK + n3Kab∂aτ ∂bτ − nD2τ + n3∂aτ ∂bτ DaDbτ where we have used (A.8) and all indices and covariant derivatives are defined with respect to hab = γab(τ (x), xa). The induced metric on ∂M is given by h¯ab = hab + ∂aτ ∂bτ . ddxp−h¯ K|∂M − 2 ddxp−h¯ n1 Z ∂M d √ d x −h K h¯ab(∂¯τ )2 − ∂¯aτ ∂¯bτ Kab − D¯ 2τ . B Explicit expressions of sources Here we give explicit expressions of various sources introduced in section 4.1 Bn = d − 2 (d)Rn − 2 i=2 2d Pn = Rn − d 1 (d)Rn − X KiKn−i n−2 i=2 1 nX−2 Tr KiKn−i − d − 1 nX−2 KiKn−i 2d i=2 n−2 i=2 Sn = Jn − 3 X Ki′Kn−i − X KiKjKn−i−j n−2 i=0 Jn = X Ki(d)Rn−i + (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) (B.1) (B.2) (B.3) (B.4) Note that in the last term of (B.3), the sum should not include the term with i = j = 0 (which is denoted using a prime). Also note the relation Sn = d d − 1 Bn′ + 2K0Bn − Tr K0Pn . (B.5) Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [hep-th/9802150] [INSPIRE]. JHEP 06 (2011) 031 [arXiv:1010.1264] [INSPIRE]. [4] I. Heemskerk and J. Polchinski, Holographic and Wilsonian renormalization groups, [5] T. Faulkner, H. Liu and M. 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Michael Crossley, Paolo Glorioso, Hong Liu, Yifan Wang. Off-shell hydrodynamics from holography, Journal of High Energy Physics, 2016, 124, DOI: 10.1007/JHEP02(2016)124