Kerr/Fluid Duality and Singularity of Solutions to the Fluid Equation

An equation for a viscous incompressible fluid on a spheroidal surface which is dual to the perturbation around the near-near horizon extreme Kerr (n-NHEK) black hole is derived. It is also shown that an expansion scalar $\theta$ of a congruence of null geodesics on the null horizon of the perturbed n-NHEK spacetime, which is dual to a viscous incompressible fluid, is not positive semi-definite, even if initial conditions on the velocity are smooth. Unless initial conditions are elaborated, caustics of null congruence will occur on the horizon in the future. A similar result is obtained for a perturbed Schwarzschild black hole spacetime which is dual to a viscous incompressible fluid on $S^2$. An initial condition that $\theta$ be positive semi-definite at any point on $S^2$ is a necessary condition for the existence of smooth solutions to incompressible Navier-Stokes (NS) equation on $S^2$.


Introduction
Fluid/Gravity correspondence has a long history starting from [1] and [2]. In the context of AdS/CFT this subject has been extensively studied. [3] [4] In the case of asymptotically flat spacetimes a prescription was developed in [5] and the incompressible Navier-Stokes (NS) equation on a plane which is dual to Rindler space [6] and that on a sphere which is dual to Schwarzschild black hole [7] [8] are derived.
Furthermore Petrov type I condition on the hypersurface geometry reduces the Einstein equation to the NS equation. [9] To our knowledge, however, an equation of fluid mechanics which is dual to a more realistic black hole, a rotating one in an asymptotically flat spacetime has not been obtained until now.
In this paper we will derive an extension of an incompressible Navier-Stokes (NS) equation to that which is dual to the Kerr black hole solution in four-dimensional asymptotically flat spacetime. For simplicity, we will consider the near-near horizon extreme Kerr (near-NHEK) limit [10] which is introduced in the context of Kerr/CFT correspondence [11] [12]. This problem is interesting from the point of view of symmetry. Kerr black hole has only axial symmetry and the horizon of the Kerr Black hole is not a round sphere, but a distorted, spheroidal surface. This reduces the symmetry of the fluid equation, and it makes it difficult to infer the form of the fluid equation solely from symmetry considerations.
Another subject of this paper is the expansion scalar θ of the null congruence of the perturbed event horizon of the near-NHEK spacetime. In this paper this perturbed event horizon H is defined by the condition that H coincides with the stationary event horizon in the absence of gravitational perturbation, and H is null.
This definition was also adopted for example in [17]. This H is generated by null geodesics. In the case of the spacetime we study in this paper such a null surface H is uniquely determined by the λ expansion which is introduced in the derivation of the incompressible NS equation. We will compute θ to the leading non-trivial order of the λ expansion and find that it is not necessarily positive semi-definite. If θ is negative at some point the congruence of null geodesics will end with caustics in finite time due to the focusing equation. At the caustics we have θ = −∞ and the velocity v i of the fluid becomes singular, since θ is a polynomial of v i and its derivatives. This means that the solution to the incompressible NS equation has singularities, even if it is smooth in the past. To avoid such a singularity we need to impose an initial condition that θ be non-negative anywhere on H at some time τ = τ 0 . The calculation of the expansion scalar is carried out for both near-NHEK and Schwarzschild black hole. This result will be relevant to the problem [18] of the existence and smoothness of solutions to the incompressible NS equation in spaces with topology different from a plane. 1 By the assumption of cosmic censorship [13] it is considered that with a smooth initial condition there are no singularites on and outside the event horizon in the future. It is known that the entropy of a black hole is related to the area of the event horizon. The area of the event horizon and the entropy will not decrease. [14] In the non-stationary case, there are some proposals for definition of the horizons. [25] In the previous paragraph we defined the event horizon of a perturbed spacetime as a null surface H which agrees with the event horizon of the unperturbed one when the perturbation is switched off. Then, if θ were negative at some point on H, H would not be a true event horizon. The true event horizon will be outside H. However, even if this is the case, the above argument for the singularity of the solution to the fluid equation still applies, since the argument is valid on some null surface H. To the contrary, when θ is non-negative all over H, H might as well be regarded as an event horizon. This paper is organized as follows. In sec.2 procedure for constructing fluid mechanics dual to asymptotically flat spacetime [6][7] is reviewed. Then the limit of near horizon extreme Kerr (NHEK) spacetime and the near-NHEK spacetime is explained. In sec. 3 an incompressible NS equation on spheroidal surface which is dual to near-NHEK is derived. This equation contains several extra terms. In sec. 4 the expansion scalar θ for the congruence of the null geodesics on the perturbed event horizon H is computed. Discussions are given in sec. 5. In appendix A quantities related to the un-perturved horizon, a spheroidal S 2 is collected. In appendix B the perturbation part of the metric which is dual to the fluid is presented. In appendix C we report on some results for fluid equation which is dual to a perturbed Schwarzschild black hole and which was obtained by using the same prescription as for near-NHEK spacetime.

Fluid/Gravity correspondence in asymptotically flat spacetime
In this section we will recall some ingredients for deriving NS equation from Einstein gravity. We will restrict discussions to asymptotically-flat spacetimes. We start with some static or stationary black hole solution. An idea is to intorduce a cutoff surface Σ c at some fixed radius r = r c outside a black hole. We then impose some boundary condition for gravitational perturbations on Σ c and look for solutions which are non-singular at the horizon. Einstein equation has a solution space which is much larger than that of NS equation. By choosing suitable boundary conditions on Σ c it is possible to make some degrees of freedom remain which describe fluids on the horizon.
In the case of flat Minkowski space, we choose in-going 'Rindler coordinates', [6] and introduce a dimensionless parameter λ and change coordinatesr,t →, t witĥ Upon introducing metric perturbations, we demand that the induced metric on the cut-off surface Σ c be flat. The cutoff surface approaches the horizon by taking a limit λ → 0. Within a power series expansion around λ = 0 we determine the metric components of the perturbations in terms of the velocity field v i (t, x) and pressure P (t, x) in such a way that the full metric including the perturbations satisfies Einstein equation as far as v i and P satisfy NS equation. In taking the limit λ → 0 the coordinate frame is infinitely accelerated and this allows us to probe the near-horizon dynamics. In this way the NS equation and the incompressibility condition are obtained.
Here ν is the viscosity parameter. The above equations have scaling symmetry.
Here ǫ is a small constant parameter.
In the case of Schwarzschild black hole, the incompressible NS equation was obtained by a similar procedure. [7] Here, R ij is Ricci tensor on a round sphere and ∇ i is a covariant derivative associated with the metric on the sphere. One notable difference is that it is not possible to impose a boundary condition such that the metric on Σ c be that of a direct product of time and a round sphere of radius r c , but what can be done at most is that the induced metric on Σ c is conformal to such a fixed metric.
In In the case of Kerr black hole, however, the spacetime does not have sufficient symmetries to determine the form of the fluid equation uniquely. Actually, rotation symmetry is reduced to only ∂ ϕ . The horizon is not a round sphere, but a deformed, spheroidal surface. This suggests that the equation for the fluid on the spherical horizon also has only a rotation symmetry ∂ ϕ . To the best of our knowledge, a form of NS equation or its generalization, which is dual to such a curved spacetime without sufficient isometry has not been studied. Hence we cannot predict a priori the exact form of the NS equation soley from the symmetry argument due to the lack of symmetries. In this paper we will show that this fluid equation can be determined uniquely by requiring that the equation can be written in terms of covariant derivatives associated with the metric on the horizon.
Another problem associated with the Kerr spacetime is the fact that it is more complicated than that of Schwarzschild. It is known that the Kerr spacetime can be simplified by considering near horizon limit of the extremal Kerr or that of near extreme Kerr, and the correspondence with conformal field theories has been studied.
In this paper we will treat this limiting case of Kerr spacetime.
We wish to remove the apparent singularity of (2.8) at the horizon, and we change coordinates to in-going Kerr coordinates. They are analogous to the Eddington-Finkelstein coordinates for Schwarzschild black hole which are appropriate for discussing perturbations around the horizon. The necessary coordinate transformations and the resulting metric are, respectively, dr = dr, dt = dt + (r 2 + a 2 ) dr ∆(r) , dθ = dθ, dϕ = dφ + a dr ∆(r) (2.13) and 14) The extreme Kerr has r + = r − , T H = 0. In this case the upper bound for J is saturated: J = M 2 . We can blow up the near horizon region by the following followed by a limit λ → 0. The resulting metric is given by where Γ(θ) = 1 + cos 2 θ 2 , Λ(θ) = 2 sin θ 1 + cos 2 θ , (2.17) and the associated geometry is the near horizon extremal Kerr (NHEK). [15] [11] While NHEK has vanishing Hawking temperature, its generalization with nonvanishing temperature, a near-NHEK spacetime was constructed. [10] This is achieved by taking the limit λ → 0, while keeping the associated temperature finite and non-zero. This means that parameter a is adjusted according to Now we make the following coordinate transformations to the metric (2.14) instead of the transformation (2.15).

Scale transfomation
We will perform an extra rescaling of the coordinates r and t (2.20) to (2.14) by with the same parameter λ as in the previous subsection. As λ → 0, the coordinate frame is infinitely accelerated, and the near-horizon dynamics can be probed. The metric after the transformation (3.1) and expansion in power series of λ is given by and it turns out that a term with a power λ −2 appears in front of dτ 2 . This λ −2 order term can be removed by means of a shift in ϕ.
After this shift we obtain We also present some components of the inverse metric up to O(λ 1 ).
The leading order of g ρρ is also λ −1 like that of g τ τ .
We define Σ c as a hypersurface located at ρ = 1. By taking a limit λ → 0, Σ c approaches the event horizon r = r + , which is a null surface. This is a distorted 2-sphere S 2 , whose metric γ ij is given in (A.1). We attempted to impose a boundary condition on Σ c such that the induced metric on Σ c is (possibly conformal to) that of a semi-direct sum of time and the distorted sphere. It turns out, however, that this is not possible.

Derivation of the Navier-Stokes equation
We consider perturbation around the Kerr background, and we wish to find a fluid equation which is an extention of NS equation with possibly some appropriate extra terms. The metric with perturbations will be denoted as where the perturbation part of the metric is expanded in a power series of λ and ρ: The vacuum Einstein equation is R µν = 0. Here and henceforth, Greek indices are reserved for four-dimensional spacetime (ρ, τ, θ, ϕ) and Latin ones for the distorted 2-sphere (θ, ϕ). R µν represents the Ricci tensor in four dimensions, while R ij that in two dimensions. Each component of the Ricci tensor is also expanded in a power series of λ and ρ in the same fashion as that of the metric µν refers to the Ricci tensor at order λ n and R (n,m) µν at order λ n ρ m . We found that the integer n in the summation of (3.8) should start from −1. All R To relate g (n) µν to fluid variables v i and P such that if v i and P satisfy fluid equations, then g (n) µν satisfy Einstein equations, we impose the following requirements.
1. We choose a gauge g ρρ = 0 not only for the background field but also for the full gravitational field including perturbations.

The leading order of the perturbations g (n)
µν is n = 0. This is expected because the leading behavior of g τ τ is O(λ −1 ) after the shift (3.3).
3. The leading and subleading perturbations g (0) µν and g (1) µν are cubic polynomials of ρ. Actually, it turns out they can even be quadratic polynomials. 4. We require that the metric should be smooth on the boundary Σ c , and that the fluid equation be expressed covariantly in terms of covariant derivatives with respect to the metric on the distorted 2-sphere (A.1). It turns out this requirement is strong enough to restrict the form of the fluid equation. We do not require that the induced metric on Σ c to coincide with or be conformal to that of a semi-direct sum of time and the distorted sphere. 5. We do not require that the perturbation part of the metric be written in terms of covariant derivatives associated with the distorted sphere.
Through analysis of the Einstein equations we found that by setting where v i is the velocity field and c 0 is a constant, it is possible to identify R as the incompressibility condition.
Here ∇ i is covariant derivative associated with the metric on the distorted S 2 . By solving other R (n,m) µν = 0 it was also found that R where P (τ, θ, ϕ) is pressure, c 1 and c 2 are constants, ǫ ij is a unit anti-symmetric tensor defined in (A.3). The term R ij v j also appeared in the fluid equation dual to Schwartzschild black hole. [7] The appearance of ǫ ij in the last line may be surprising.
However, a rotation about an axis breaks parity. 2 The term proportional to ∇ ϕ v i in (3.11) is also consistent with the isometry ∂ ϕ of the background. Actually, to the order we are considering there also appears in NS equation another new term proportional to ǫ ij ∂ τ v j , which is consistent with the axial symmetry. This is, however, a result of the analysis to the leading and sub-leading orders. By analysis to higher orders, we find that this term is excluded at the end. Hence the structure of the Lagrange derivative of NS equation is not modified.
The perturbative part of metric tensor which corresponds to this fluid equation is presented in appendix B. In the NS equation we can take T R → 0 limit. However, such a limit cannot be taken in the metric presented in appendix B. In (0,0) τ i = 0 after substitution of (3.9) and ansatz 3 where α (n) (θ) and β (n) (θ) are unknown functions of θ, takes a complicated structure.
Especially, although NS equation must be a vector equation, terms such as To remove extra terms which cannot be combined into a vector quantity, many conditions must be imposed on α (n) (θ) and β (n) (θ). Some of them are differential equations and solving them gives the integration constants c n and determines α (n) (θ), β (n) (θ) except for

Stationary solution to Navier-Stokes equation on the distorted 2-sphere
If one fixes the constants of integration as NS equation is obtained with the conventional weight where ν is a viscosity parameter.
The solution for the pressure P is given by where c 3 is an integration constant. Choosing c 3 = −2M 2 ξ 2 reduces (3.18) to

Higher order equations
In appendix B some components of g We have found that the equation takes a form where G is some known function. To solve the equation (3.21), we require that G be written in a form of total divergence. We accomplished this by tuning the form of g (0,0) τ τ and g (1,0) ij appropriately. We set the following ansatz and incompressibility condition (3.9) and the NS equation (3.11) makes G take the form G g (0,0) τ τ , g Now the equation (3.21) can be written as This result is quite different from that in [7] for the study of dual fluid of Schwarzschild black hole. In that case, time derivatives of v 2 and P appear in the equation, R When the constants are fixed as in (3.13), V i are given by

Singularity of the Solution to the Fluid Equation
In this section we will consider a fluctuating event horizon H. Its null normal is denoted as k µ . This normal vector defines a family of null geodesics on H, and by extending it off H appropriately, we will obtain an expansion scalar θ of null geodesics. This represents a logarithmic derivative of local area on H, and if this is a true event horizon, the second law of black hole thermodynamics [14] asserts that θ is not negative on the future horizon.
In the case of the non-stationary black hole there are some proposals for definition of the event horizon [17]. In this paper we define the event horizon of a perturbed

Congruence of null geodesics
Let the horizon H be a hypersurface Φ(x) = 0. The normal vector derived from Φ, is null (k 2 = 0) on H, but in general not null off H. To compute the expansion scalar θ associated with a congruence of null geodesics, it is convenient to extend the null normal vector k µ off the horizon H. 6 This procedure will be briefly explained. We introduce another normal vector N µ which satisfies the following conditions off H and definek Here α is a function defined by Now the vectork µ is null in the neighborhood of H and defines a congruence of null geodesics on and off H. The vectork µ is, however, hypersurface orthogonal only on H. A projection matrix onto the subspace spanned byk µ and N µ are defined by Now the following tensor is introduced. 7 It can be shown that although the null vectork µ is not hypersurface orthogonal off H, the rotation tensor ω µν vanishes on H.
The expansion scalar obeys the Raychaudhuri's equation.

Sign of expansion scalar θ
When the perturbations in near-NHEK are absent, the event horizon is located at ρ = 0. When the perturbations are switched on, the horizon is deformed. [17] Φ(ρ, τ, θ, ϕ) ≡ ρ − λ F 1 (τ, θ, ϕ) − λ 2 F 2 (τ, θ, ϕ) − λ 3 F 3 (τ, θ, ϕ) + · · · = 0, (4.12) where F a are unknown functions to be determined by a null normal condition, whose solution defines a new surface H. If F a are chosen such that k · k vanishes, then H becomes a null surface. 8 Generally, this procedure would yield a set of partial differential equations for F a and could not be solved explicitly.
By using the procedure given in the previous subsection, θ is computed to O(λ 1 ).
After using Einstein equation we have θ = θ (0) + λ 1 θ (1) + O(λ 2 ) with θ (0) = 0 owing to incompressibility condition, and Although several terms are positive semi-definite, some are not. It must still be taken into account that in the above result the incompressibility condition (3.9) is not completely solved. In two spatial dimensions, however, the velocity of incompressible fluid can be expressed in terms of a scalar potential f (τ, θ, ϕ): We checked that even when these equations are substituted into (4.14), the result is still not positive semi-definite. The result (4.14) does not depend on the ansatz (3.12) and the choice of c 1 and c 2 .
The above result implies that in order to avoid the singularity of the expansion θ in the future, an initial condition on the velocity field v i must be imposed at some time τ = τ 0 such that θ (1) (τ 0 , θ, ϕ) ≥ 0 for all region of θ and ϕ as well as condition of smoothness. Then the velocity field v i will remain non-singular afterwards. If θ (1) (τ 0 , θ, ϕ) ≥ 0 is not obeyed, however, caustics will occur on H in the future. We also performed similar analysis on the expansion scalar for the Schwarzschild black hole and obtained a same result. (See appendix C.) The above result is relevant to the existence and smoothness problem of the incompressible NS equation [18][19] [7]. Although this problem is not solved in three spatial dimensions, it was studied in two dimensions (especially for R 2 and T 2 ) [20].
Indeed, we can show that the expansion scalar is positive semi-definite in the case of a two-dimensional planar horizon in Rindler space (2.1) 9 .
However, solution of this problem in the case of a sphere is not known up to now.
Study of Fluid/Gravity correspondence along this line may shed some light on the solution of this problem for space with topologies other than plane. The above result is obtained only for O(λ 1 ). It is interesting to investigate whether higherorder analysis imposes further restrictions on the initial conditions for v i to avoid caustics.

Discussions
In this paper we derived an equation for viscous incompressible fluid which is dual to perturbations of near-NHEK black hole. This equation contains terms proportional to ∇ ϕ v i and ǫ ij v j , respectively. The structure of this hydrodynamic equation is interesting by itself and deserves further study. In this paper we considered only the near-NHEK geometry. This is simply because in the case of general Kerr geometry we did not succeed in finding appropriate expressions of the metric perturbation in terms of quantities in fluid dynamics such that if v i and P satisfy the viscous NS equations, Einstein equation is also satisfied. It is interesting to study whether this near horizon limit is essential, or there exist dual fluids for any angular momentum parameter a (≤ J). In this work we could not make metric perturbations to satisfy some boundary conditions on a cut-off surface Σ c except for smoothness. As for future studies an extension of the present work to the duality of fluid with higher dimensional rotating black hole and a study of turbulence based on the new fluid equation may be attempted. It is also interesting to study the possibility of systematic higher-order correction to the duality found in this paper.

A Tensors on the distorted 2-sphere
The background metric on the distorted 2-sphere in the near-NHEK limit is A covariant derivative associated with γ ij are denoted as ∇ i . The epsilon tensor and the Ricci tensor are The Ricci scaler and the derivative are The non-zero Christoffel symbols are

B Fluctuation of the metric tensor
We found the following results on the perturbations of the metric tensor, by an expansion in a power series of λ and ρ: α (i) (θ) and β (i) (θ) are undetermined functions of θ and V i is defined by (3.28) and (3.29). The terms which do not appear above can be arbitrary functions up to the order we consider.

C Metric for the perturbation of Schwarzschild black hole and expansion scalar
We also derive NS equation for the perturbation to the Schwarzschild background. The boundary condition on Σ c is only that the functions v i and P are smooth. Here we do not require that the perturbation part of the metric be written covariantly in terms of covariant derivatives associated with the 2-sphere.
The background metric in a power series of λ is where M is a mass of the Schwarzschild black hole.
The perturbation of the metric is expressed as The non-vanishing coefficient functions are given as follows. and NS equation where R ij and ∇ i are Ricci tensor and covariant derivative associated with the distorted 2-sphere. It is checked that all other components of the Ricci tensor R (m = 1, 2, 3, 4). In order to make these vanish g (2,m) must be adjusted.
The expansion scalar θ with perturbation in the Schwarzschild black hole is given by θ =λ This is not positive semi-definite as in the case of Kerr black hole. We checked that this conclusion does not change, even if the metric in [7] is used.