#### 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.
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