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JHEP01(2019)044

Published for SISSA by Springer

Received: October 20, 2018

Revised: December 12, 2018

Accepted: December 25, 2018

Published: January 4, 2019

Fluid description of gravity on a timelike cut-oﬀ

surface: beyond Navier-Stokes equation

Shounak De and Bibhas Ranjan Majhi

Department of Physics, Indian Institute of Technology Guwahati,

Guwahati 781039, Assam, India

E-mail: shounakde@iitg.ac.in, bibhas.majhi@iitg.ac.in

Abstract: Over the past few decades, a host of theoretical evidence has surfaced that

suggest a connection between theories of gravity and the Navier-Stokes (NS) equation of

ﬂuid dynamics. It emerges out that a theory of gravity can be treated as some kind of ﬂuid

on a particular surface. Motivated by the work carried out by Bredberg et al. [6], our paper

focuses on including certain modes to the vacuum solution which are consistent with the

so called hydrodynamic scaling and discuss the consequences, one of which appear in the

form of Damour Navier Stokes (DNS) equation with the incompressibility condition. We

also present an alternative route to the results by considering the metric as a perturbative

expansion in the hydrodynamic scaling parameter  and with a speciﬁc gauge choice, thus

modifying the metric. It is observed that the inclusion of certain modes in the metric

corresponds to the solution of Einstein’s equations in presence of a particular type of matter

in the spacetime. This analysis reveals that gravity has both the NS and DNS description

not only on a null surface, but also on a timelike surface. So far we are aware of, this

analysis is the ﬁrst attempt to illuminate the possibility of presenting the gravity dual of

DNS equation on a timelike surface. In addition, an equivalence between the hydrodynamic

expansion and the near-horizon expansion has also been studied in the present context.

Keywords: Classical Theories of Gravity, Black Holes, Gauge-gravity correspondence

ArXiv ePrint: 1810.07017

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP01(2019)044

JHEP01(2019)044

the usual NS equation. A corresponding action formulation has been discussed extensively

by Parikh and Wilczek in [4]. The same has also been done in [5] for DNS.

In a more recent work by Bredberg et al. [6], which forms the basis of our work, a

general solution of the vacuum Einstein equations with certain prescribed boundary data

is presented in (p + 2) dimensions (p corresponds to all angular coordinates while 2 stands

for one time and one radial coordinate) using the well-known hydrodynamic expansion of

the incompressible NS equation in (p+1) dimensions (here p is, like earlier, collection of all

angular coordinates and 1 refers to only the timelike coordinate). In this paper, they seek

a relation between the (p + 2)-dimensional Einstein and (p + 1)-dimensional NS equation

by the construction of a metric in accordance with a scaling symmetry the incompressible

NS equation possess. The metric construction is an expansion in the hydrodynamic scaling

parameter  and is parameterized by the velocity ﬁeld v

, τ) and pressure P (x

, τ) that

solves the full nonlinear incompressible NS equation. It may be worth mentioning that

the metric in [6] is constructed so as to strictly obtain incompressible NS dynamics by

conserving the corresponding Brown-York stress tensor on a timelike hypersurface r = r

where r

lies outside the black hole horizon r

(also, an earlier seminal work [7] is done in

the AdS/CFT context). The choice of the metric thus lies on the following facts. It has

been obtained perturbatively in the hydrodynamic expansion parameter around Rindler

space-time and up to certain gauge conditions and reparameterizations, satisﬁes certain

boundary conditions and solves the vacuum Einstein’s equations. The explicit meaning of

these statements will be made clear in the next paragraph. This cut-oﬀ surface approach has

been applied in various cases [8]. For example, it was extended for higher curvature gravity

theories [9–13] as well as for the AdS [14, 15] and dS [16] gravity theories (for other theories,

like black branes, see [17]). Also a corresponding relativistic situation has been discussed

extensively in [18]. Symmetries of the vacuum Einstein equations have been exploited to

develop a formalism for solution generating transformations of the corresponding NS ﬂuid

duals in [19]. The ﬂuid description on the Kerr horizon has also been explored extensively

in [20] (see [21] for the isolated horizon case).

Our motivation is to analyse the metric construction more deeply and see how far we

can generalise the form of the metric which abide by the same scaling laws and thereby

investigate the corresponding consequences. On analysing the vacuum metric we realized

that certain additional modes can still be incorporated in the metric and in accordance

with the relevant hydrodynamic scaling. For example, terms like (∂

+ ∂

)dx

and

are perfectly allowed modes of the order of 

and hence can be added to the

metric of [6]. In our work we try to ﬁgure out the implications of retaining these scaling

consistent modes which serve as modiﬁcations to the original vacuum solution presented

in [6]. One such consequence due to the presence of the term ∂

(

)

is that the modiﬁed

metric generates ﬂuid dynamics described by the equation which contains an additional

term proportional to ∂

, while retaining the incompressibility condition. Interestingly,

such a structure is similar to that obtained in [22] where the DNS equation is expressed

in terms of coordinates adapted to a null surface. As discussed in length in the work

by Padmanabhan [22], the key structural diﬀerence of the DNS equation in comparison

to the NS equation is the presence of a ∂

kind of term. We obtain a seemingly close

– 2 –

JHEP01(2019)044

structure which has only the kinematic viscosity (thereby calling it a “restricted” DNS

equation) when one keeps a term of the type (∂

+ ∂

)dx

in the original metric

and approaching in the same route as in [6]. We show later in the subsequent sections

that these additional mode(s) result from deﬁning an appropriate bulk matter and the

corresponding generalized metric solves the Einstein ﬁeld equations with an appropriate

energy-momentum tensor.

This is a new scenario quite in contrary to the approach by Damour [2] which was

set in context to null surface dynamics alone, whereas the present one is constructed on

a timelike surface. Such an investigation fulﬁls an important purpose. In literature, the

NS equation has both types of gravity duality: one on the timelike surface and another on

the null surface. On the contrary, so far we are aware of, the DNS equation has a gravity

dual constructed on a null surface. So the present analysis can be regarded as the ﬁrst

attempt to have a gravity description of the DNS equation on a timelike surface, along

with a particular constraint.

We also describe an alternative framework in which the geometry described in [6], is

now viewed as a perturbative expansion of the leading order metric and leading to NS or

DNS dynamics depending on the manner in which the metric is modiﬁed at a particular

order. In this case, the extrinsic curvature and the surface stress-tensor is also taken to have

a linear order correction. The perturbative expansion follows a speciﬁc choice of ﬁxed gauge

conditions. This gauge condition conﬁrms the invariance of the structure of the seed metric

on the cut-oﬀ surface. In both the frameworks, the equivalence between the hydrodynamic

expansion and the near-horizon expansion approaches is explored. Implications of these

results are ﬁnally discussed at the end.

The organization of the paper is as follows. In section 2, we brieﬂy review the well

known scaling symmetry of the incompressible NS equation which serves as the basis of our

main goal. In the next section, we present a generalized metric where the additional scaling

consistent tensor modes are sourced by an appropriately deﬁned bulk matter characterized

by an energy-momentum tensor and the metric is shown to solve the complete Einstein

ﬁeld equations, i.e., in presence of matter, upto the order the bulk solution is presented.

The constraint equations on the cut-oﬀ boundary surface is shown to reduce to the incom-

pressible DNS-like ﬂuid equations. In section 4, we present an alternative framework to

represent the gravity dual of this constrained DNS ﬂuid, with the metric now modiﬁed by a

speciﬁc choice of gauge conditions and the extrinsic curvature along with the Brown-York

stress tensor taken to have linear order corrections only. The equivalence between the hy-

drodynamic expansion and the near-horizon expansion approaches, similar in spirit to [6]

is explored in section 5. Next, we present some discussions that could possibly illuminate

our understanding of certain key aspects. Finally, we present our conclusions in section 7.

We present the detailed proof of the scale invariance of the restricted DNS-like equation

in appendix A. In appendix B, we show the explicit computation of the Ricci tensor com-

ponents for the metrics presented in the paper, but whose coeﬃcients have been taken to

be pre-factored with suitable multiplicative constants, indicating as to how various modes

contribute to the Ricci tensor at various orders.

The notations used throughout the paper are clariﬁed as follows: all uppercase Latin

letters (i.e., A, B, C, etc.) denote the bulk spacetime co-ordinate indices while the lower-

– 3 –

JHEP01(2019)044

case Latin letters (i.e., a, b, c etc.) are used only for denoting the transverse coordinates.

The Greek letters (i.e., α, µ, ν etc.) denote the coordinates on the boundary cutoﬀ hyper-

surface Σ

2 Setup: scaling laws

In this section we shall brieﬂy review the scaling symmetry of the incompressible NS equa-

tion, which forms the basis of the work [6] and also lays the foundation of our analysis.

Now, if the amplitudes of the solution space (v

, P ) of the incompressible NS equation is

scaled down by the parameter :



, τ) = v

(x

, 

τ) ;



, τ) = 

P (x

, 

τ) , (2.1)

then the NS equation remains invariant under the above scaling transformation, thus gen-

erating a family of solutions parameterized by  from the original solution space. Any

deviations from ideality lead to certain typical corrections which vanish under the scaling

law (2.1), as shown in [6]. A term proportional to ∂

arises in our analysis later in the

paper, as an addition to the incompressible NS equation. This resulting equation is also

shown to obey the same scaling laws (2.1) (the explicit proof is shown in appendix A). The

hydrodynamic scaling parameter  serves as the expansion parameter for the metric in [6].

The hydrodynamic scaling of the spatial and time derivatives along with the pair (v

, P )

follows as:

∼ O() , P ∼ O(

) , ∂

∼ O() , ∂

∼ O(

) . (2.2)

In our analysis in the subsequent sections, we focus on including certain additional modes

consistent with the above hydrodynamic scaling sourced by an appropriately deﬁned bulk

matter tensor and working with a modiﬁed metric than in [6]. We then strive to discuss the

consequences of this generalized metric with these scaling consistent modes in a formalism

similar to [6] and also discuss it in a diﬀerent framework. We shall see that it not only

includes the NS equation, but also gives birth to a restricted type of DNS equation which

also preserves its structure under the scaling (2.1) of the ﬂuid parameters.

3 Metric and incompressible DNS

We present a metric keeping some more possible terms consistent with the scaling argument

as discussed above. The leading order base metric is taken to be ﬂat and is in (ingoing)

Rindler form in Eddington-Finkelstein coordinates. The metric retains its original structure

as in [6], albeit for the two additional scaling consistent terms that now appear at O(

)

which are pre-factored with constants ˜a

and ˜a

. Of course, these terms vanish on the

timelike hypersurface r = r

as well, such that the induced metric is ﬂat. We therefore

– 4 –

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