Membrane paradigm of the static black bottle
aa r X i v : . [ g r- q c ] O c t Membrane paradigm of the static black bottle
Li Li and Towe Wang ∗ Department of Physics, East China Normal University,Shanghai 200241, China (Dated: November 7, 2018)In the membrane paradigm of black holes, it is usually assumed that the normal vector of the stretched horizonhas a vanishing acceleration. This assumption breaks down for black bottles, a class of solutions discoveredrecently in the asymptotically anti-de Sitter spacetime. In this paper, the membrane paradigm is generalized tothe stretched horizon with a nonvanishing acceleration of normal vector, and then it is applied to the static blackbottle. In this example, the membrane stress tensor and the fluid quantities are similar to those of black holes,while the fluid continuity equation and the Navier-Stokes equation are well satisfied in the near-horizon limit.
I. INTRODUCTION
In the past half century, the black hole thermodynamics has greatly deepened our understanding of gravity theory. However,the dynamics of gravity is not merely thermodynamics. The membrane paradigm of black holes has opened for us anotherwindow to gravity—the hydrodynamics. On the event horizon of a black hole, the gravitational equations resemble the low-dimensional fluid continuity equation and the Navier-Stokes equation. The fluid quantities, such as density, pressure, shear andexpansion, are encoded on the stretched horizon—a timelike hypersurface slightly outside the event horizon. Algebraically, theblack hole membrane paradigm can be derived from Einstein equations [1–5], or alternatively from an action with a surfaceterm [6]. The action-based method is convenient for studying more complicated examples. Although the membrane paradigm isnormally applied to black holes in the Einstein gravity, it is straightforward to extend it to other gravity theories [7–11] and to theFriedmann-Lemaˆıtre-Robertson-Walker spacetime [12]. Remarkably, turning to black branes in the asymptotically anti-de Sitter(AdS) spacetime, one can make use of the membrane paradigm and the AdS/CFT correspondence to calculate hydrodynamicquantities of the dual field theory, see [13–15] and references therein.Topologically, a black hole has a spherical horizon, while a black brane has a planar horizon. In AdS spacetime, there areblack hole-like solutions with various nontrivial topologies. In reference [16], Chen and Teo discovered a new class of solutionswith bottle-shaped horizons, which they named as black bottle. Contingent on the value of a rotation parameter, the black bottlecan be either static or rotating. In the present paper, we will focus on the static black bottle, whose horizon interpolates betweena spherical geometry and a hyperbolic one. We intend to build the membrane paradigm of the static black bottle. Unlike blackholes or black branes, on the stretched horizon, the black bottle has a nonvanishing acceleration of normal vector, violating theassumption of zero acceleration made in the literature. We will generalize some results in the literature to take this point intoconsideration.The paper is organized as follows. After a concise technical review of membrane paradigm in section II, we will reformulatethe static solution in spherical coordinates in section III. Then the membrane paradigm will be established in section IV for thestatic black bottle explicitly. We make a split of the spacetime in subsection IV A by specifying the timelike generatorand the spacelike normal of the stretched horizon. Most of the fluid quantities on the membrane will be worked out in subsectionIV B in spherical coordinates. In subsection IV C, the Eddington-Finkelstein (EF) coordinates will be introduced to evaluate thesurface gravity and inspect the limit that the stretched horizon tends to the true horizon. We will check the fluid equations insubsection IV D. The main results in section IV will be translated to the Pleba´nski-Demia´nski-like coordinates in section V. Thepaper concludes in section VI with a summary and discussion of the results. In appendix A, we will show that the membranestress tensor in the Einstein gravity receives no correction from the nonzero acceleration of normal vector.
II. REVIEW OF MEMBRANE PARADIGM
In this section, we will briefly review some geometric and algebraic details of the membrane paradigm of black holes in theEinstein gravity. ∗ Electronic address: [email protected]
The black hole event horizon H is a 3-dimensional null hypersurface with a null geodesic generator l a . At the event horizon,the geodesic equation is l b ∇ b l a = g H l a (1)where ∇ b is the covariant derivative with respect to metric g ab of the 4-dimensional spacetime. In a stationary spacetime, l a canbe taken as the null limit of a timelike Killing vector and then g H will be the surface gravity at H .The membrane paradigm is constructed on a timelike membrane S , which is outside H and very close to it. This membraneis dubbed the stretched horizon. The timelike generator u a and the spacelike normal n a of S are normalized to unity, n a u a = 0 , n a n a = 1 , u a u a = − . (2)The metric of S is given by h ab = g ab − n a n b . (3)Then we can define the extrinsic curvature of stretched horizon as K ab = h db ∇ d n a , (4)and the extrinsic curvature scalar K = g ab K ab . A 2-dimensional spacelike cross section of S normal to u a has the metric γ ab = h ab + u a u b . (5)In the membrane paradigm, we introduce a parameter α . When α → , the stretched horizon S will tend to the event horizon H , and αu a → l a , αn a → l a . (6)At the same time, the aforementioned 2-section of S will approach a 2-dimensional spacelike cross section of H , and K ab → α − k ab − α − g H u a u b . (7)Here k AB is the extrinsic curvature of the 2-dimensional spacelike section of H , k AB = γ aA γ bB ∇ b l a = 12 L l γ AB . (8)The notation L l is the Lie derivative along l a . It proves convenient to decompose k AB into a traceless part and a trace, k AB = σ AB + 12 θγ AB , (9)where σ AB is the shear of l a and θ is the expansion.In reference [6], assuming n a ∇ a n b = 0 , it was proven that the membrane stress tensor t ab = 18 πG ( Kh ab − K ab ) (10)on S in the Einstein gravity. As is demonstrated in appendix A, this result remains correct even if n a ∇ a n b = 0 . The abovestress tensor can be decomposed to a form like a viscous fluid, t ab = 1 α ρu a u b + 1 α γ aA γ bB ( pγ AB − ησ AB − ζθγ AB ) + π A ( γ aA u b + γ bA u a ) . (11)Here we have inserted the renormalization parameter α , hence ρ , p , σ AB , θ and π A correspond to fluid quantities on H .Taking the limit α → , we expect the fluid quantities satisfy the (2 + 1) -dimensional fluid continuity equation and theNavier-Stokes equation, L l ρ + θρ = − pθ + 2 ησ AB σ AB + ζθ + T ab l a l b , (12) γ eA L l π e + π A θ = − p || A + 2( ησ BA ) || B + ( ζθ ) || A − T ab l b γ aA , (13)where T ab is the bulk stress tensor, and || A is the 2-covariant derivative with respect to the metric γ AB .Some fluid quantities and relations can be known without going into a specific spacetime. Substituting equations (7), (9) intoequation (10), one gets t ab = − πGα θu a u b − πGα σ ab + 18 πGα (cid:18) θ + g H (cid:19) γ ab . (14)Identifying it with equation (11), we can read off ρ = − θ πG , p = g H πG , π A = 0 ,η = 116 πG , ζ = − πG . (15)In subsection IV B, we will obtain the other fluid quantities from the details of a static black bottle. III. BLACK BOTTLE IN SPHERICAL COORDINATES
In reference [16], the term “black bottle” was coined to refer to a new class of solutions with bottle-shaped event horizons,among which the static solution has the form ds = ℓ (1 − b )( x − y ) (cid:20) Q ( y ) dt − dy Q ( y ) + dx P ( x ) + P ( x ) dφ (cid:21) ,P ( x ) = 1 + x − x − x ,Q ( y ) = b + y − y − y . (16)Here ℓ is related to the cosmological constant by ℓ = − / Λ . The ranges of parameters and coordinates are b < − / , y < x, − < x ≤ +1 , ≤ φ < π. (17)For our purpose in this paper, it would be helpful to rewrite the above metric in the system of spherical coordinates. This canbe achieved by the following transformation of coordinates t = τℓ p − b ) , x − y = ℓ √ − b √ r , x = cos ϑ, φ = ϕ . (18)For simplicity, we will use the abbreviated notation L = ℓ √ − b/ √ . Then it is straightforward to write down the line elementof a static black bottle ds = − f ( r, ϑ ) dτ + 1 f ( r, ϑ ) (cid:18) dr − r sin ϑL dϑ (cid:19) + 2 r ϑ dϑ + r sin ϑ ϑ ) dϕ ,f ( r, ϑ ) = − L r + 12 (1 + 3 cos ϑ ) + r L (1 − ϑ − ϑ ) − r L ( b + cos ϑ − cos ϑ − cos ϑ ) (19)as well as the ranges of parameters and coordinates b < − / , r > , ≤ ϑ < π, ≤ ϕ < π. (20)It is remarkable that the function f ( r, ϑ ) can also be written as f ( r, ϑ ) = − r L Q ( y ) ,Q ( y ) = b + (cid:18) cos ϑ − Lr (cid:19) − (cid:18) cos ϑ − Lr (cid:19) − (cid:18) cos ϑ − Lr (cid:19) . (21)In the case b < − / , Q ( y ) has only one real root y . As demonstrated in reference [16], this root corresponds to a Killinghorizon in the spacetime. Translated into spherical coordinates, the horizon is located at the hypersurface cos ϑ − Lr = y . (22)Substituting this equation into metric (19), one can plot the same image of black bottle as figure 4 in reference [16]. Notice thatthe shape of black bottle is controlled by parameter y or alternatively b , while its area is determined by y and L together, A BB = 4 πL y − . (23)It is easy to check that the above value is consistent with equation (27) in reference [16].It is interesting to observe that ∂ r Q = ( L/r ) ∂ y Q , ∂ ϑ Q = − sin ϑ∂ y Q . They can be combined to yield a useful relation ∂ ϑ f = r sin ϑL (cid:18) r f − ∂ r f (cid:19) . (24)In this paper, we will not use the explicit expression of f ( r, ϑ ) . Instead, we will make use of equation (24) implicitly. IV. MEMBRANE PARADIGM OF BLACK BOTTLEA. The split
In this section, we will apply the general setup of section II to a static black bottle. As we will see, although the black bottlehas a horizon of unusual topology, its membrane paradigm can be explicitly built as usual. To this end, the first and key step ismaking a split of spacetime, which can be done most conveniently in spherical coordinates.In the spirit of section II, we wish to rearrange the line element of a black bottle as g ab dx a dx b = − u a u b dx a dx b + n a n b dx a dx b + γ ab dx a dx b , (25)in which u a is proportional to a timelike Killing vector, and n a is normal to the stretched horizon. The most efficient way to dothis is comparing the above line element with equation (19), which suggests that n a dx a = f − / (cid:18) dr − r sin ϑL dϑ (cid:19) , u a dx a = − f / dτ,γ AB dx A dx B = 2 r ϑ dϑ + r sin ϑ ϑ ) dϕ . (26)Accordingly, the metric of stretched horizon is h ab dx a dx b = − f ( r, ϑ ) dτ + 2 r ϑ dϑ + r sin ϑ ϑ ) dϕ . (27)The one-forms n a dx a , u a dx a are dual to vectors n a ∂ a = f / ∂ r , u a ∂ a = f − / ∂ τ , (28)meeting the orthonormal condition (2).We can get a clearer picture of the membrane paradigm from the explicit results above. In the membrane paradigm, thestretched horizon, located slightly outside the event horizon, plays the role of a membrane. The low-dimensional fluid lives onthis membrane. From equation (26), we notice that n a is the normal vector of hypersurface cos ϑ − L/r = constant , includingthe event horizon (22). This implies that the stretched horizon of a static black bottle is cos ϑ − Lr = y + ǫ, (29)where ǫ is a small positive constant.For a static black bottle, the null generator of event horizon can be chosen as the timelike Killing vector l a ∂ a = ∂ τ (30)so that the non-affine coefficient g H in formula (1) is the surface gravity. In practice, this formula can be implemented in the EFcoordinates, but not in the spherical coordinates. We will return to this formula in subsection IV C and confirm that vector (30)is a generator of the event horizon. In addition, the limit (6) can be better understood in the EF coordinates. Nevertheless, fromequations (28) and (30), we can guess the renormalization parameter α = f / . (31)It is important to point out that the acceleration a b of the normal vector is not zero, a b = n a ∇ a n b = − r sin ϑL ( dϑ ) b , (32)violating the assumption a b = 0 made in the literature. We take this point into consideration in this paper. In appendix A, wedemonstrate the membrane stress tensor (10) remains correct when a c = 0 . B. Fluid quantities
Having set up the geometry of the static black bottle, the following steps are straightforward. In order to get thefluid quantities on the event horizon, we should calculate the stress tensor (10) on the stretched horizon, and then decompose itto the form (11).By definition of the 3-dimensional extrinsic curvature (4), we find the nonzero components K ττ = − f / ∂ r f, K ϑϑ = 2 r ϑ f / , K ϕϕ = r sin ϑ ϑ ) f / . (33)One may check that K ab = 1 r f / γ ab − f − / ( ∂ r f ) u a u b . (34)The trace of the extrinsic curvature is K = 12 r f − / (4 f + r∂ r f ) . (35)Substituting them into equation (10), we can get the stress tensor of the fluid on the stretched horizon. The nonvanishingcomponents are t ττ = − πGr f / ,t ϑϑ = r πG (1 + cos ϑ ) f − / (2 f + r∂ r f ) ,t ϕϕ = r sin ϑ πG (1 + cos ϑ ) f − / (2 f + r∂ r f ) , (36)which are equivalent to t ab = 116 πGr f − / (2 f + r∂ r f ) γ ab − πGr f / u a u b . (37)Comparing it with equation (11) and inserting the renormalization parameter (31), we arrive at some fluid quantities or theircombinations ρ = − πGr f, π A = 0 , ησ AB = 0 ,p − ζθ = 116 πGr (2 f + r∂ r f ) . (38)Combining equations (15) and (38), we can conclude that the fluid quantities on the event horizon are Energy density : ρ = − πGr f, Pressure : p = 116 πG ∂ r f, Momentum density : π A = 0 , Shear : σ AB = 0 , Expansion : θ = 2 r f, Shear viscosity : η = 116 πG , Bulk viscosity : ζ = − πG . (39)The shear σ AB and the expansion θ can be calculated directly with the 2-dimensional extrinsic curvature (8), which turns outto be zero, k AB = 0 . (40)According to equation (9), this means σ AB = 0 , θ = 0 . (41)In the near-horizon limit α → , it is consistent with equation (39). C. EF coordinates and surface gravity
When building the membrane paradigm in details, the line element (19) in spherical coordinates is very useful but not enough.In particular, it is unsuitable for computing the surface gravity g H or studying the near-horizon limit of n a . These difficultiescan be overcome by introducing the Eddington-Finkelstein (EF) coordinates.The ingoing EF coordinates ( v, r, ϑ, ϕ ) are related to the standard spherical coordinates ( τ, r, ϑ, ϕ ) via dv = dτ + f − (cid:18) dr − r sin ϑL dϑ (cid:19) (42)for the static black bottle solution (19). In the EF coordinates, the line element of a static black bottle takes the form ds = − f dv + 2 dv (cid:18) dr − r sin ϑL dϑ (cid:19) + 2 r ϑ dϑ + r sin ϑ ϑ ) dϕ . (43)Rewritten in the EF coordinates, the vector (30) becomes l a ∂ a = ∂ v . (44)It is not hard to check that l a is a Killing vector and is null at the event horizon. To make sure that it is the generator of eventhorizon, we should also check if it is normal to the event horizon. This can be done in the EF coordinates as follows. Fromequation (22), we can see the normal vector of event horizon has the dual form proportional to dr − r sin ϑL dϑ. (45)At the same time, the dual form of vector (44) is l a dx a = − f dv + dr − r sin ϑL dϑ. (46)Obviously, on the event horizon f = 0 , this vector is the normal vector. The surface gravity at the event horizon can be safelyderived with formula (1). In the EF coordinates (43), it gives a definite answer g H = 12 ∂ r f. (47)This answer is exactly what we have expected in equations (15), (39).For consistency, we should scrutinize equations (6) in the near-horizon limit α → , taking α = f / . Translated into EFcoordinates, αu a ∂ a = ∂ v , αn a ∂ a = ∂ v + f ∂ r . (48)Both of them tend to vector (44) in the near-horizon limit satisfactorily. D. Fluid equations
In principle, the continuity equation (12) and the Navier-Stokes equation (13) can be derived from the Gauss-Codazzi equa-tions and the Einstein equation, by considering the fact that n a ∇ a n b = 0 . With all fluid quantities in hand, here we will take ashortcut. We will insert the fluid quantities into equations (12), (13) to check them directly.In the membrane paradigm with a cosmological constant [12], we relegate the cosmological term to the bulk stress tensor T ab = − Λ8 πG g ab = 3(1 − b )16 πGL g ab . (49)Projected to the null-null and null-transverse directions respectively, it becomes T ab l a l b = − − b )16 πGL f, T ab l b γ aA = 0 . (50)For the rest terms in equations (12), (13), starting from the fluid quantities in equation (39), lengthy but straightforward compu-tations lead to the result L l ρ + θρ + pθ − ησ AB σ AB − ζθ = − πGr f (cid:18) r f − ∂ r f (cid:19) , (51) γ eA L l π e + π A θ + p || A − ησ BA ) || B − ( ζθ ) || A = sin ϑ πGL f ( dϑ ) A . (52)We should warn that the above result cannot be gained without the help of equation (24). In view of equations (50), (51), (52), itis clear that the continuity equation (12) and the Navier-Stokes equation (13) are well satisfied in the near-horizon limit α → . V. TRANSLATED TO PLEBA ´NSKI-DEMIA ´NSKI-LIKE COORDINATES
In the previous section, we have constructed the membrane paradigm of static black bottles in the spherical coordinates (19).All of the results can be accurately translated to the Pleba´nski-Demia´nski-like coordinates (16) via relation (18). For futurereference, let us accomplish the translation of equations (26), (28), (30),(31), (39), (47). These equations specify the geometricstructure and the fluid quantities of the membrane.Firstly, in the Pleba´nski-Demia´nski-like coordinates (16), we can rewrite equations (26), (28), (30) as n a dx a = √ Lx − y [ − Q ( y )] − / dy, u a dx a = − √ Lx − y [ − Q ( y )] / dt,γ AB dx A dx B = 2 L ( x − y ) (cid:20) dx P ( x ) + P ( x ) dφ (cid:21) ,n a ∂ a = x − y √ L [ − Q ( y )] / ∂ y , u a ∂ a = x − y √ L [ − Q ( y )] − / ∂ t , l a ∂ a = 12 L ∂ t . (53)Secondly, in the Pleba´nski-Demia´nski-like coordinates, equations (31), (39), (47) take the form α = 1 √ x − y ) [ − Q ( y )] / , ρ = 18 πGL ( x − y ) Q ( y ) ,p = − πGL (cid:20) x − y Q ( y ) + ∂ y Q ( y ) (cid:21) ,π A = 0 , σ AB = 0 , θ = − L ( x − y ) Q ( y ) ,η = 116 πG , ζ = − πG , g H = − L (cid:20) x − y Q ( y ) + ∂ y Q ( y ) (cid:21) . (54)What is more, the job in subsection IV C can be done by introducing the Pleba´nski-Demia´nski-like counterpart of EF coordinates.This should be straightforward and thus we leave it to the readers. VI. CONCLUSION
We have shown that the membrane paradigm of black holes can be applied to the static black bottle successfully and explicitly,although the normal vector of the stretched horizon violates the zero acceleration assumption in the literature. The success relieson two “miracles”: first, the membrane stress tensor is not affected by the acceleration; and second, the fluid equations continueto hold. We have demonstrated the first miracle very generally in appendix A, and presented the second miracle for the staticblack bottle specifically in subsection IV D.In the membrane paradigm, the stretched horizon is interpreted as a fluid with certain dissipative properties, while the fluidquantities meet the continuity equation (12) and the Navier-Stokes equation (13) in the near true horizon limit. That is to say, themembrane paradigm has established a relation between the black hole horizons and the hydrodynamics. Unfortunately, hithertothe relation has been confirmed only for black branes in asymptotically AdS spacetime. For black holes in asymptotically AdSspacetime, this is difficult because their horizons lack translational invariance. The black bottles suffer the same difficulty. How-ever, if in the far future one discovers the dual field theory of black bottles in asymptotically AdS spacetime, its hydrodynamicslimit should reproduce some of the results in this paper.
ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China (Grant No. 91536218), and in part by theScience and Technology Commission of Shanghai Municipality (Grant No. 11DZ2260700). T. W. is indebted to Shi-Ying Caifor encouragement and support.
Appendix A: On the robustness of (10)
The assumption a b = 0 has been made in reference [6] and other works. In this appendix, we will demonstrate that themembrane stress tensor (10) is not modified if we get rid of this assumption. After examining the proof of equation (3.24) inreference [6], i.e. equation (10) in our main text, we find the acceleration a b of normal vector appears exclusively in equation(A1) of reference [6]. Therefore, to get the corrections of a b to stress tensor (10), we have to only revise equation (A1) ofreference [6].In equation (A1) of reference [6], the terms involving a b are Z d x √− h (cid:2)(cid:0) n c a b + n b a c (cid:1) δh bc − h bc n a δh ab a c − a b n a δh ab (cid:3) . (A1)Using the symmetry δh ab = δh ba , we can put it in the form Z d x √− h (cid:0) n a a b δh ac − h bc n a δh ab a c (cid:1) = Z d x √− h (cid:0) g bc − h bc (cid:1) n a a c δh ab = Z d x √− hn b n c n a a c δh ab . (A2)By the identify n c ∇ a n c = 0 , it is easy to see n c a c = 0 and thus the last line vanishes. Consequently, in the general case a b = 0 ,the terms involving a b does not contribute to the membrane stress tensor (10). [1] T. Damour, Phys. Rev. D , 3598 (1978).[2] T. Damour, Ph.D. thesis, University of Paris VI, 1979.[3] T. Damour, in Proceedings of the Second Marcel Grossman Meeting on General Relativity , edited by R. Ruffini (North-Holland, Amster-dam, 1982), p. 587.[4] R. H. Price and K. S. Thorne, Phys. Rev. D , 915 (1986). [5] K. S. Thorne, R. H. Price and D. A. Macdonald, “ Black Holes: The Membrane Paradigm ,” Yale University Press, 1986.[6] M. Parikh and F. Wilczek, Phys. Rev. D , 064011 (1998) [gr-qc/9712077].[7] S. Chatterjee, M. Parikh and S. Sarkar, Class. Quant. Grav. , 035014 (2012) [arXiv:1012.6040 [hep-th]].[8] T. Jacobson, A. Mohd and S. Sarkar, arXiv:1107.1260 [gr-qc].[9] S. Kolekar and D. Kothawala, JHEP , 006 (2012) [arXiv:1111.1242 [gr-qc]].[10] W. Fischler and S. Kundu, JHEP , 112 (2016) [arXiv:1512.01238 [hep-th]].[11] T. Y. Zhao and T. Wang, JCAP , no. 06, 019 (2016) [arXiv:1512.01919 [gr-qc]].[12] T. Wang, Class. Quant. Grav. , no. 19, 195006 (2015) [arXiv:1411.6445 [gr-qc]].[13] P. Kovtun, D. T. Son and A. O. Starinets, JHEP , 064 (2003) [hep-th/0309213].[14] A. O. Starinets, Phys. Lett. B , 442 (2009) [arXiv:0806.3797 [hep-th]].[15] N. Iqbal and H. Liu, Phys. Rev. D , 025023 (2009) [arXiv:0809.3808 [hep-th]].[16] Y. Chen and E. Teo, Phys. Rev. D93