Compressible forced viscous fluid from product Einstein manifolds
aa r X i v : . [ h e p - t h ] J a n Compressible forced viscous fluid fromproduct Einstein manifolds
Xin Hao ∗ , Bin Wu † and Liu Zhao ‡ School of Physics, Nankai University, Tianjin 300071, ChinaJanuary 19, 2015
Abstract
We consider the fluctuation modes around a hypersurface Σ c in a ( d + 2)-dimensional product Einstein manifold, with Σ c taken either near the horizon orat some finite cutoff from the horizon. By mapping the equations that governsthe lowest nontrivial order of the fluctuation modes into a system of partial differ-ential equations on a flat Newtonian spacetime, a system of compressible, forcedviscous fluid is realized. This result generalizes the non bulk/boundary holo-graphic duality constructed by us recently to the case of a different backgroundgeometry.Keywords: Gravity/Fluid correspondence, holography, flat space, compressiblefluidPACS numbers: 04.20.-q, 04.50.Gh During the last two decades our understanding on the properties of relativistic gravi-tation has been strengthened considerably. The major new concept which centralizesthe studies on gravitation is holographic duality [1, 2] originated from the area lawof black hole entropy and has been realized in a number of different physical con-figurations, mostly with gravity in the bulk and some other physical system on theboundary. Among the various realizations of holographic duality, the most important ∗ email : [email protected] † email : [email protected] ‡ email : [email protected], correspondence author. d +2)-dimensional productEinstein manifold M = M × M , where dim( M ) = 2 , dim( M ) = d . Following aprocess which is parallel to that made in [40], we will show that similar fluid dual canbe realized in a flat Newtonian spacetime with one less dimension. Moreover, we willshow that the near horizon limit is not essential in the construction – we can realizesimilar flat space fluid system by looking at the fluctuation modes around a finitecutoff surface in the background geometry as well. Since the technics used is extremelysimilar to that of [40], we shall be as brief as possible in the main text and just presentthe brief route leading to the results. 2 Product Einstein manifolds in ( d + 2) -dimensionsand hypersurface geometry The ( d + 2) dimensional product Einstein manifold M which we consider is equippedwith the line elementd s M = g µν d x µ d x ν = − f ( r )d u + 2d u d r + e Φ( x ) δ ij d x i d x j , (1)where ( u, r ) and ( x i ) are coordinates on M and M respectively, f ( r ) = 1 − ωr − d r , (2)and Φ( x ) obeys the following system of equations: δ jk ∂ j ∂ k Φ + d − (cid:18) ∂ i Φ + δ jk ∂ j Φ ∂ k Φ − ( ∂ i Φ) (cid:19) + 4 d Λ e Φ = 0 , (no summation over i ) (3)( d − (cid:18) ∂ i ∂ j Φ − ∂ i Φ ∂ j Φ (cid:19) = 0 ( i = j ) . (4)For generic d , we present a special solution to the equations (3) and (4) in the appendix.More solutions might exist because these equations are nonlinear. Given any solutionto eqs. (3) and (4), the metric g µν of the total manifold M obeys the vacuum Einsteinequation G µν = − Λ g µν . (5)We assume that ω > <
0, so that f ( r ) always haszeros. Let the (biggest, if Λ <
0) zero of f ( r ) be r h , which represents the radius ofa horizon. Consider a ( d + 1)-dimensional timelike hypersurface Σ c located at r = r c .Evidently, we need r c > r h when Λ < r c < r h when Λ ≥ c is timelike. The induced line element on this hypersurface is given byd s c = γ ab d x a d x b = − f ( r c )d u + e Φ δ ij d x i d x j = − (d x ) + e Φ δ ij d x i d x j = − λ d τ + e Φ δ ij d x i d x j , (6)where τ = ( λ √ f c ) u , x = √ f c u , f c is a shorthand for f ( r c ). Similar notations havebeen used in [40]. One can easily promote the hypersurface tensor γ ab to a bulk tensor γ µν and define the extrinsic curvature of Σ c as K µν = L n γ µν , with n µ being a unitvector field which is normal to Σ c at r = r c . In the coordinates ( u, r, x i ), n µ hascomponents n µ = (cid:18) √ f , p f , , · · · , (cid:19) . c yields the momentum and Hamiltonian constraints, D a ( K ab − γ ab K ) = 0 , (7)ˆ R + K ab K ab − K = 2Λ , (8)where ˆ R is the Ricci scalar of Σ c , D a is the covariant derivative that is compatiblewith γ ab . In terms of the Brown-York tensor t ab = γ ab K − K ab , the momentum andHamiltonian constraints can be rewritten as D a t ab = 0 , (9)ˆ R + t ab t ba − t d = 2Λ . (10)Explicit values of the extrinsic curvature and the Brown-York tensor will be made useof in the following context, so we present these values by direct calculations, K ττ = f ′ c √ f c , K τi = 0 ,K ij = 0 , K = K aa = f ′ c √ f c . (11)These in turn lead to the background Brown-York tensor t τ ( B ) τ = 0 , t τ ( B ) i = 0 ,t i ( B ) j = f ′ c √ f c δ ij , t ( B ) = t a ( B ) a = d f ′ c √ f c . (12)We have intentionally added a superscript ( B ) to indicate that these are the valueswith respect to the background geometry. When considering fluctuation modes, thesevalues must be supplemented with fluctuation modifications. In this section we shall consider the case with Σ c placed very closed to the horizon,i.e. r c − r h = ǫ α λ , where ǫ = 1 if Λ < ǫ = − ≥
0, and λ → α is introduced to eliminate someun-necessary complexity when taking the near horizon limit. We will focus ourselves onthe fluctuation modes around Σ c and pay particular attention to the Petrov I boundarycondition C ( l ) i ( l ) j | Σ c = l µ ( m i ) ν l ρ ( m j ) σ C µνρσ | Σ c = 0 , (13)where C µνρσ represents the bulk Weyl tensor, and l µ = 1 √ (cid:18) √ f ( ∂ u ) µ − n µ (cid:19) , k µ = 1 √ (cid:18) √ f ( ∂ u ) µ + n µ (cid:19) , m i ) µ = e − Φ ( ∂ i ) µ (14)constitute a set of Newman-Penrose basis vector field which obeys l = k = 0 , , ( k, l ) = 1 , , ( l, m i ) = ( k, m i ) = 0 , , ( m i , m j ) = δ ij . (15)When restricted on Σ c , we can write l µ | Σ c = 1 √ (cid:0) ( ∂ ) µ − n µ (cid:1) , k µ | Σ c = 1 √ (cid:0) ( ∂ ) µ + n µ (cid:1) . (16)Therefore the boundary condition can also be cast in the form C i j + C ij ( n ) + C ji ( n ) + C i ( n ) j ( n ) = 0 , (17)with C abcd = γ µa γ νb γ σc γ ρd C µνσρ = ˆ R abcd + K ad K bc − K ac K bd − d ( d + 1) γ a [ c γ d ] b ,C abc ( n ) = γ µa γ νb γ σc n ρ C µνσρ = D a K bc − D b K ac ,C a ( n ) b ( n ) = γ µa n ν γ σc n ρ C µνσρ = − ˆ R ab + KK ab − K ac K cb + 2Λ( d + 1) γ ab , (18)where ˆ R abcd and ˆ R ab are the Riemann and Ricci tensors of Σ c . Inserting (18) into(17) and inverting the relationship between K ab and t ab , the boundary condition (17)becomes 2 λ t τi t τj + t d γ ij + 2Λ d γ ij − ( t ττ − λD τ ) (cid:18) td γ ij − t ij (cid:19) − λ D ( i t τj ) − t ik t kj − ˆ R ij = 0 . (19)Note that the appearance of λ in this equation comes purely from the rescaling of thecoordinate x → τ /λ .To see the effects of the fluctuation modes let us expand the metric, Ricci tensorand the Brown-York tensor on Σ c in the following form, γ ab = γ ( B ) ab + ∞ X n =1 γ ( n ) ab λ n , (20)ˆ R ab = ˆ R ( B ) ab + ∞ X n =1 λ n ˆ R ( n ) ab , (21) t ab = t a ( B ) b + ∞ X n =1 λ n t a ( n ) b , (22)where the leading terms on the right hand side represent the background values. Inthe near horizon limit, f c can be rearranged in the form f c = f ′ h · ( ǫ α λ ) + 12 f ′′ h · ( α λ ) , ′ h = − (cid:18) ω + 4Λ r h d (cid:19) , f ′′ h = − d . which follows from the Taylor series expansion around r h . Consequently, the back-ground values given in (12) will also develop λ dependences. In the end, we have t ττ = 0 + λt τ (1) τ + · · · ,t τi = 0 + λt τ (1) i + · · · ,t ij = ǫ (cid:16) p ǫf ′ h αλ − αλ d p ǫf ′ h (cid:17) δ ij + λt i (1) j + · · · ,t = dǫ (cid:16) p ǫf ′ h αλ − αλ d p ǫf ′ h (cid:17) + λt (1) + · · · . (23)What we need to do is to expand (9), (10) and (19) into power series in λ and lookat the first nontrivial order contributions. For this purpose we need to supplement(23) with expansions of ˆ R ab and D ( i t τj ) . To evaluate the latter we need to expand theChristoffel connection ˆΓ abc . Omitting the details we present the results as follows:ˆΓ abc = Γ a ( B ) bc + O ( λ ) , ˆ R ab = ˆ R ( B ) ab + O ( λ ) , (24)where the background valuesˆΓ τ ( B ) ab = ˆΓ a ( B ) τb = 0 , ˆΓ k ( B ) ij = 12 (cid:0) δ ki ∂ j Φ + δ kj ∂ i Φ − δ ij ∂ k Φ (cid:1) , ˆ R ( B ) τa = 0 , ˆ R ( B ) ij = 2Λ d e Φ δ ij are all λ -independent. Therefore, we have D ( k t τj ) = ∂ ( k (cid:0) λt τ (1) j ) + O ( λ ) (cid:1) − (cid:0) ˆΓ l ( B )( kj ) + O ( λ ) (cid:1)(cid:0) λt τ (1) l + O ( λ ) (cid:1) ≡ λζ (1) kj + λ ζ (2) kj + O ( λ ) , (25)where ζ (1) kj = ∂ ( k t τ (1) j ) − ∂ ( k Φ t τ (1) j ) + 12 δ kj δ lm ∂ l Φ t τ (1) m , (26)and ζ (2) kj is some complicated expression which depends on γ (1) kj and t τ (2) j . We do notneed to use the explicit form of ζ (2) kj in this paper.Substituting eqs. (23) and (25) into (19), we get in the first nontrivial order O ( λ )the following identity, t i (1) j = αǫ p ǫf ′ h · γ ik (0) (cid:0) t τ (1) k t τ (1) j − ζ (1) kj (cid:1) + 1 d t (1) δ ij , (27)6here γ ik (0) = e − Φ δ ik . Notice that the dependences on α and ǫ can be get rid of bychoosing α = ǫ p ǫf ′ c , which we will take from now on. Similarly, expanding the τ component of (9) into power series in λ yields, at order O ( λ − ), δ ij (cid:18) ∂ i + d − ∂ i Φ (cid:19) t τ (1) j = 0 , (28)and expanding the spatial components of the same equation yields, at the order λ ,the following equation, ∂ τ t τ (1) i − (cid:0) t (1) − t τ (1) τ (cid:1) ∂ i Φ + (cid:18) ∂ j + d ∂ j Φ (cid:19) t j (1) i = 0 . (29)The lowest order contribution to the Hamiltonian constraint is at order O ( λ ), whichyields t τ (1) τ = − γ ij (0) t τ (1) i t τ (1) j . (30)The form of the equations (27), (28), (29) and (30) is exactly the same as those appearedin [40] – though with different values for each quantity entering the equations – whichhad led to a flat space fluid system after introducing some appropriate holographicdictionary. Therefore, we can follow the same line of argument as made in [40] andintroduce the holographic dictionary ρ = e d Φ , µ = e d − Φ , ν = µρ = e − Φ , (31)and t τ (1) i = v i ν , t (1) d = p µ , (32)where ρ, µ, v i , p are to be interpreted as the density, viscosity, velocity field and thepressure of the dual fluid. Finally, eq. (28) becomes the continuity equation ∂ j ( ρv j ) = 0 , (33)and eqs. (27) and (29) combined together give rise to the standard equation ρ ( ∂ τ v i + v j ∂ j v i ) = − ∂ i p + ∂ j d ij + f i (34)for the velocity field of the fluid, where d ij = µ (cid:18) ∂ j v i + ∂ i v j − d δ ij ∂ k v k (cid:19) (35)represents the deviatoric stress, which is symmetric traceless and depends only on thederivatives of the velocity field, hence vanishes in the hydrostatic equilibrium limit,and f i = ∂ j Φ (cid:18) d ij + d − pδ ij (cid:19) + 2 d v j v j ∂ i ρ − d ( v j ∂ j ρ ) v i (36)7epresents an extra force. Since the first two terms − ∂ i p + ∂ j d ij on the right hand sideof (34) correspond to the ordinary surface forces, the last term which cannot be castin the form of the first two terms must represent a body force. The equations (33) and(34) constitute a system of equations governing the motion of a compressible, forced,stationary and viscous fluid moving in the ( d + 1)-dimensional Newtonian spacetime R × E d . The whole construction of the last section is very similar to that made in [40] for blackhole background. Now we would like to ask a different question: Can we construct a flatspace fluid dual without making use of the near horizon Petrov I boundary condition?To answer this question, let us go over the whole process of the construction. Itis clear the every formulae until (22) still holds if Σ c is not placed near the horizon.However, from eq. (23) and onwards, things begin to change a little. Concretely, (23)must be replaced by t ττ = 0 + λt τ (1) τ + · · · ,t τi = 0 + λt τ (1) i + · · · ,t i j = f ′ c √ f c δ ij + λt i (1) j + · · · ,t = d f ′ c √ f c + λt (1) + · · · , (37)where f c , f ′ c etc must be kept un-expanded. If we replace (23) with (37) and carry onthe rest process as made in the last section, then we will encounter some trouble. Unlikewhat has been derived in the near horizon limit, at finite cutoff the first nontrivial orderof the boundary condition becomes t τ (1) k t τ (1) j − ζ (1) kj = 0 . (38)insert (38) into (28), we could only come to the conclusion that at order O ( λ ) themomentum constraints will not allow fluctuation in the t τi component, i.e. t τ (1) i = 0 . (39)If we expand the boundary condition to the next order, then the following relationwould follow, t i (1) j = t (1) d δ ij − √ f c f ′ c (cid:0) R i (1) j + 2 ζ i (2) j (cid:1) , (40)where ζ i (2) j = γ ik (1) ζ (1) kj + γ ik (0) ζ (2) kj . d ∂ i t (1) = √ f c f ′ c (cid:18) ∂ j + d ∂ j Φ (cid:19)(cid:16) R j (1) i + 2 ζ j (2) i (cid:17) = √ f c f ′ c e − d Φ ∂ j h e d Φ (cid:0) R j (1) i + 2 ζ j (2) i (cid:1)i , (41)but it is exceedingly hard to explain (41) as a partial differential equation that describesthe fluid motion.To avoid the above problem and realize the fluid dual at finite cutoff, we employ atranslation of Φ as follows: Φ → Φ − ln λ. (42)Such a translation does not alter the order of derivatives of Φ but does change theorder of e Φ . After the translation (42), the spatial components of the induced metric(6) will be rescaled: γ ( B ) ab d x a d x b → ˜ γ ( B ) ab d x a d x b = − λ d τ + e Φ λ δ ij d x i d x j , (43)and eq. (3) becomes δ jk ∂ j ∂ k Φ + d − (cid:18) ∂ i Φ + δ jk ∂ j Φ ∂ k Φ − ( ∂ i Φ) (cid:19) + 4Λ d e Φ λ = 0 . (44)As a result the Ricci tensor of Σ c will be inversely proportional to λ ,ˆ R ( B ) ij → ˜ R ( B ) ij = 2Λ d e Φ λ δ ij . (45)The background Brown-York tensor t a ( B ) b , the Christoffel connection ˆΓ a ( B ) bc and ζ (1) ij areall kept invariant. So we can list the new boundary condition as2 λ ˜ γ ik t τk t τj + t d δ ij + 2Λ d δ ij − ( t ττ − λD τ ) (cid:18) td δ ij − t i j (cid:19) − λ ˜ γ ik D ( k t τj ) − t ik t kj − ˆ R ij = 0 . (46)The form of this equation looks identical to (19), however with a different backgroundmetric ˜ γ ij which is λ dependent. Using (46) in place of (19) and inserting (25) and(37), we get the following new equation at the order O ( λ ), t i (1) j = √ f c f ′ c · γ ik (0) (cid:0) t τ (1) k t τ (1) j − ζ (1) kj (cid:1) + 1 d t (1) δ ij − √ f c f ′ c ˆ R i (1) j . (47)9bviously the momentum constraints are invariant under the translation (42) forΦ, so there will not be any new terms in equation (28) and (29). Substituting (47) into(29), and introducing the following holographic dictionary ρ = f ′ c √ f c e d Φ , µ = e d − Φ , ν = µρ = √ f c f ′ c e − Φ ,t τ (1) i = v i ν , t (1) d = p µ , (48)eqs. (28) and (29) will again become the continuity equation and the Navier-Stokesequation ∂ j ( ρv j ) = 0 , (49) ρ ( ∂ τ v i + v j ∂ j v i ) = − ∂ i p + ∂ j d ij + f i , (50)where d ij takes the same form as in (35) (of course with different µ and v i ), and f i = ∂ j Φ (cid:18) d ij + d − pδ ij (cid:19) + 2 d v j v j ∂ i ρ − d ( v j ∂ j ρ ) v i + 2 √ f c f ′ c ν∂ j ( ρ ˆ R j (1) i ) . (51)Comparing to the case of the last section, we see that an extra force term √ f c f ′ c ν∂ j ( ρ ˆ R j (1) i )appears in f i at finite cutoff. The result of this work indicates that a compressible, forced, viscous fluid in flat New-tonian spacetime can not only be realized as the holographic dual of fluctuation modesaround a black hole background, but also be realized as the dual of fluctuating modesaround a product Einstein manifold. The construction relies on taking a timelike hy-persurface Σ c – which is placed either near the horizon or at some finite cutoff – andmapping it into an Euclidean space which lies in its conformal class. It is surprisingthat the final fluid equations, including the form of the extra force term f i , are basicallythe same as that arising from fluctuations around a black hole background (though inthe case of a finite cutoff a novel force term appears). It remains to understand thenature of the extra force term from the point of view of fluid dynamics.Experienced readers on Gravity/Fluid correspondence might feel that the fluid equa-tions in flat space are only those for a curved space incompressible fluid in disguise. Thisis true in some sense. However, writing the fluid equations in flat Newtonian space-time is still a nontrivial step forward, because the holographic dictionary we adopt isquite different for that leading to a curved space incompressible fluid. In practice, anyphysical system which is dual to another one in some way can be regarded as its dualin disguise. However this view point does not disvalue the duality relationship betweenthe two systems. 10s a technical addendum let us mention why we stick to the approach which makesuse of the Petrov I boundary condition rather than adopting the boost-rescaling ap-proach which is also widely used in Gravity/Fluid correspondence [18–29]. The reasonlies in that the background geometry which we use is not necessarily flat, and boost-ing requires the existence of a flat boundary. For this reason, we do not expect theboost-rescaling approach to be applicable in our situations.The product manifold taken in this work is of the simplest kind, i.e. the totalmanifold is the product of two submanifolds. We can of course consider the case whenthe total manifold is the product of several submanifolds, e.g. the submanifold M itself has a product structure, with line element of the formd s M = e Φ( x ) δ ij d x i d x j + e Ψ( y ) δ mn d y m d y n + · · · . We expect that flat space fluid may still be constructed out of the fluctuating modesaround such backgrounds, and the resulting fluid might become anisotropic. We willhave more to say along the lines of research of [40] and the present work.
Appendix
In this appendix we discuss some properties regarding the system of equations (3) and(4) that determine the function Φ( x ).For the particular case of d = 2, these equations degenerate into a single equation δ jk ∂ j ∂ k Φ + 2Λ e Φ = 0 , which can be easily recognized to be the Euclideanized Liouville equation for Λ = 0 orthe Laplacian equation for Λ = 0. The fact that the metric function Φ obeys Liouvilleor Laplacian equation in the case d = 2 is first observed in [41] while consideringblack hole metrics. In the case of product Einstein manifold such equations still hold.Existence of infinitely many solutions to such equations are well known.For d >
2, these equations are much more complicated. However, there is an explicitsolution which we reproduce below. LetΦ = − χ. (52)Then, eq. (4) becomes ∂ i ∂ j χ = 0 , i = j. Therefore, χ must take the form χ = X i χ i ( x i ) , x i . Moreover, eq. (3) can be rewritten as X k ( d − ∂ k χ ) − χ∂ k χχ − ( d − χ∂ i χχ + 4Λ dχ = 0 . (53)Suppose χ = a + b X i x i , eq. (53) becomes an algebraic equationΛ d − ab ( d −
1) = 0for the coefficients a and b . When Λ = 0, both a and b cannot vanish. So, we canfreely choose a = 1 and then get b = Λ d ( d − . This yields the solution χ ( x ) = 1 + Λ d ( d − X k x k , i.e. e Φ = 1 χ = 1 (cid:18) Λ d ( d − P k x k (cid:19) . If Λ = 0, then either a or b must be zero. Taking a = 0 and let b be arbitrary, we get χ ( x ) = b X k x k , i.e. e Φ = 1 χ = 1 b (cid:18) P k x k (cid:19) .e Φ becomes singular in this case at x i = 0. Alternatively, we can take b = 0 andnormalize a = 1, which corresponds to χ = 1, i.e. Φ = 0. In this particular case thefluid we obtain becomes incompressible but still lives in flat space, and the force f i given in (36) (and also the first two terms in (51)) becomes vanishing. Since eqs. (3)and (4) are essentially nonlinear, there may be other solutions to these equations. References [1] G. ’t Hooft, “Dimensional reduction in quantum gravity”, Salamfest 1993:0284-296,[arXiv:gr-qc/9310026].
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