Dual Fluid for the Kerr Black Hole

Rotating black holes are algebraically special solutions to the vacuum Einstein equation. Using properties of the algebraically special solutions we construct the dual fluid, which flows on black hole horizon. An explicit form of the Kerr solution allows us to write an explicit dual fluid solution and investigate its stability using energy balance equation. We show that the dual fluid is stable because of high algebraic speciality of the Kerr solution.


Introduction
The fluid/gravity correspondence is a framework that connects the knowledge about the gravity equations and fluid dynamics. There are multiple approaches to such correspondence originating from membrane paradigm [1] , AdS/CFT [2,3], shear perturbation [4,5], quasinormal modes [6,7], algebraic speciality [8] and others. Each approach describes an explicit pair for the gravity and fluid equations with an additional mapping procedure for the solutions. In our paper we want to make a step further and use the fluid/gravity correspondence to answer interesting questions about the dual fluid in terms of the geometric data. The two major questions we are attempting to approach are the explicit solutions and solutions stability. Both questions are interesting and natural for the nonlinear equations on both fluid and gravity sides. An ability to write a solution in closed form often teaches us about some new features of the solution such as symmetries or algebraic speciality. Thus an idea to apply the fluid/gravity mapping to know closed form solution on either side to generate more closed form solutions is very attractive. Another advantage of an explicit solutions is an ability to investigate the stability of the solutions. In context of fluid dynamics the stability is closely related to the phenomenon of turbulence.
Among the multiple fluid/gravity mappings the algebraic speciality approach fits best to answer questions that we described. In our paper [8] we described the simplest setup: minimally special geometry and (conformally) flat hypersurface geometry. Unfortunately almost all explicit solutions to Einstein equations are more special [9], i.e. have additional vanishing Weyl tensor components. Furthermore, the horizon geometry is far from being (conformally) flat for most solutions. Thus we need to modify our approach to adopt for the known black hole solutions. While it is possible to describe the most general algebraic constrains and hypersurface geometry we decided to restrict our discussion to the rotating black holes. Such restriction allows us to make discussion much simpler while keeping several interesting phenomenon and being physically relevant on both gravity and fluid sides.
The generic rotating black hole geometry turns out to be very complicated to analyze straight on, so we start our discussion with slowly rotation case, where the angular momentum J is much smaller then the mass squared M 2 . The geometry is an algebraic type D, while the r = const fluid hypersurface preserves the smaller type I i subset of type D constraints. The type I i constraints have type I as a subset so we can construct the dual fluid, while the additional constraints restrict possible fluid solutions to Killing flows. Interestingly, the simple analysis of additional speciality constraints provides a quick way solve the Navier-Stokes (NS) equations and to reproduce the slowly rotating black hole results from [10]. Since slowly rotating Kerr solution can be viewed as a small perturbation over the Schwarzschild geometry our NS system and fluid solution match with the results by Bredberg and Strominger [11].
The fluid dual to the generic rotating black hole solution obeys the modified version of an incompressible NS equation. The additional terms are similar to ones described in fluid/gravity generalizations [12]. Similarly to the toy case higher algebraic speciality provides additional constraints on velocity similar to the Killing equation. A solution to additional constraints provides a quick way of solving the NS equation for a dual fluid. As an example we describe in details both fluid equations and solution for the case of 4d Kerr geometry.
Given an explicit form of the fluid equations and solutions we can investigate the linearized stability of the flow. The most common approach to stability is based on the Reynolds number estimation. The Reynolds number for the slow rotating case is equal to the 3J/M 2 and is parametrically small. We also estimate the critical Reynolds number, where fluid can be unstable, using the energy balance equation. It turns out that the dual fluid to the generic type D black hole solution is always stable.
We begin our discussion with the review of the algebraically special fluid/gravity correspondence in section 2. In section 3 we review the Kerr geometry and describe the dual fluid hypersurface. In section 4 we consider a slow rotating Kerr solution and apply the fluid/gravity map to it. In section 5 we describe the dual fluid to the Kerr black hole and discuss its stability in section 6. In the closing section 7 we discuss the results and possible generalizations.

The fluid/gravity correspondence
In this section we will review the fluid/gravity correspondence and modify it to include the rotating black hole solutions.
Given a solution to the Einstein equations in p+2 dimensions and a time-like codimensionone hypersurface Σ we can construct a symmetric conserved two-tensor T ab ∇ a T ab = 0, T ab = T ba , a, b = 0, ..., p, where ∇ a is a covariant derivative with respect to the induced metric h ab on a hypersurface.
The tensor T ab is often called the Brown-York (BY) tensor and is constructed from the extrinsic curvature with n being the unit normal to Σ and K ≡ K ab h ab . The fluid/gravity correspondence maps the BY tensor into the fluid stress tensor. Fluid equations can be formulated in the form of covariant conservation for the fluid stress tensor. However the (p + 1)(p + 2)/2 components of the stress tensor are expressed in terms of p-components of the fluid velocity and two scalars: density and pressure. In case of gravity-dual fluid the additional constraints for the BY tensor come from metric being algebraically special.
The covariant conservation (2.1) originates from the bulk Einstein equation G µν = 0 µ, ν = 0, ..., p + 1 on the hypersurface Σ with coordinates x a , a = 0, ..., p where we use (p + 1) upperscript for the hypersurface quantities. The minimally special metrics are algebraic type I, what imposes a constraint where C µνλρ is the Weyl tensor and l, k, m i are the null frame vectors. In particular we want the null frame vectors to be with T being unit normal for the time foliation Σ T of Σ. The rest of the null frame vectors m i are orthonormal basis for tangent space to Σ T .
We can express the p + 2-dimensional Riemann tensor on a hypersurface in terms of intrinsic geometry and extrinsic curvature K ab to evaluate The type I constraint (2.5) is written for the Weyl tensor components so generally we need to subtract the trace from the Riemann tensor expression (2.8

Kerr geometry
The Kerr metric in Boyer -Lindquist (BL) coordinates is of the form The Kerr metric is an algebraically special metric of type D. The null frame in BL coordinates is given by the Kinnersley tetrad For the type D metric the only nontrivial components of the Weyl tensor have boost weight zero and can be expressed via the complex Ψ 2 invariant [13, 14] 3.1 r = r 0 = const surface A natural candidate for the fluid hypersurface is the r = r 0 = const hypersurface Σ with the unit normal The normal vector becomes null when ∆(r) = 0, what corresponds to the pair of BH horizons The mean curvature of Σ becomes large as we approach horizons. We can use ∆ as small expansion parameter to describe the geometry of Σ as it approaches the outer horizon r = r + .

Induced metric
The induced metric on Σ is with r, ρ being evaluated at r 0 . Fo the rest of the section we will discuss the hypersurface quantities, so in order to avoid an overusing the 0-subscript we will assume that r = r 0 . The determinant of the induced metric so the hypersurface indeed becomes null as r 0 → r + . We can remove the dtdφ term by choosing the zero angular momentum observable (ZAMO) coordinates The metric (3.7) becomes In particular the leading ∆-expansion

Extrinsic curvature
The extrinsic curvature on the hypersurface is defined via the Lie derivative L n g µν = n σ ∂ σ g µν + ∂ µ n σ g σν + ∂ ν n σ g µσ . (3.12) Since our normal vector has only r-component and the Kerr metric (3.1) has g rt = g rφ = g rθ = 0 the Lie derivative simplifies into For our hypersurface after the coordinate transform (3.9) we have the following extrinsic curvature components (3.14)

Null frame
The fluid/gravity correspondence require algebraic speciality with respect to the null frame constructed from the hypersurface data n, T . In practice we often have a null frame of maximal speciality written for the full bulk metric. Given a null frame we can construct the unit vector but it can fail to be hypersurface orthogonal, i.e. obey Moreover, we need Σ to have large mean curvature K = ∇ µ n µ , so there is no canonical fluid hypersurface for a generic algebraically special geometry. Fortunately for the Kerr geometry we have a family of r = const surfaces that approach the horizon and serve as natural candidates for Σ. For these hypersurface the mean curvature (3.6) becomes large as we approach the horizon, but they do not preserve the full type D speciality. Let us figure out how much speciality can we preserve on Σ.
The null frame (3.2) on Σ changes under the coordinate transformation (3.9) to become (3.17) Moreover we can further rescale vectors k → with unit normal to Σ and the time-foliation normal for the induced metric (3.10) on Σ being The difference between the T ±n √ 2 null frame and the canonical type D null frame (3.18) can be schematically written as √ ∆f i m i . The possible corrections to the algebraic constraints from the different weight components have the following schematic form In the second equality we used the type D constraints and the near-horizon scaling of the boost weight zero components (3.3). Using (2.7) we can estimate the leading order expressions for the Weyl tensor components in terms of the hypersurface data to be Thus the corrections (3.20) does not affect only the leading order for the type I i constraints.
So effectively we can say that the choice of Σ breaks the Kerr geometry speciality from the type D to the subtype I i , which has C lilj = C kikj = 0.

Geometry
The Kerr metric (3.1) at leading order in a becomes The r = r 0 hypersurface has the following geometric data Using the coordinate transformation we can remove the dtdφ term, so the geometry simplifies into The mean curvature becomes large as we approach the black hole horizon at r = 2M since r 0 f 2 0 = (r 0 − 2M). Moreover the leading near-horizon behavior of the mean curvature is related to the Hawking temperature T H for the slowly rotating Kerr solution.
The type D null frame (3.2) after the coordinate transformation (4.3) becomes We can freely rescale our null vectors . imposes additional relations for the dual fluid, which we will describe in fluid variables later.

Dual fluid
Our geometry (4.4) can be written in terms of τ = λ −2 t and λ 2 = f 2 0 = O(r 0 − 2M) to simplify the large mean curvature expansion analysis. The type I constraint (2.8) for metric (4.9) simplifies into ij , D i being Ricci tensor and covariant derivative with respect to the spatial metric γ ij . The large mean curvature K = O(λ −1 ) motivates to search for a perturbative in λ solution to the type I constraint (4.10) using The solution for K ij is The covariant conservation equation (2.3) for our geometry (4.9) becomes (4.14) The equation (4.14) at leading orders O(λ −1 , λ 0 ) order is which is the statement that the BH temperature (4.6) is the same at each point on the horizon. The next order equation is viscosity η and pressure p The solution (4.12) in the new notation is of the form
becomes the Navier-Stokes equation in curved space if we redefine pressure and velocity The equation (4.22) describes the shear perturbation for the Schwarzschild metric [11], so it is not surprising that it describes the dual fluid for the slowly-rotating Kerr black hole.
so we can use the Riemann tensor instead. The leading order expression in fluid variables with k i (x) being a Killing vector for the spatial metric γ ij . The substitution of (4.26) into (4.22) lead to the equation We can integrate over the fluid on compact spatial section of the horizon to conclude that ∂ τ f = 0. So we can summarize that Killing-based solution (4.26) solves the NS equation (4.22) with constant f (τ ), that is imposed by compactness of the Kerr horizon.
The solution is The solution (4.29) is generic for any type I i metric with a hypersurface geometry (4.9) and large mean curvature. Our slowly rotating Kerr geometry (4.4) obeys these properties so we can compare extrinsic curvature from (4.4) and the NS solution (4.29). The dual fluid is τ -independent Killing vector on the round two sphere. Indeed our result for the hypersurface geometry (4.9) match the generic solution for toy example (4.29).
Our toy example illustrates an feature, that are common for all type D

Dual fluid for rotating black hole
The most studied fluid system is the incompressible Navier-Stokes (NS) equation, so we want to use additional physical assumptions to write our fluid system in the form similar to the NS equation. In particular we will use the near-horizon expansion with small parameter λ to control the relative size of various terms and simplify our equations.

Geometry
The induced metric on the hypersurface is of the form (3.11) where we we identify λ 2 r 2 = ∆. The smallness h tt = O(λ 2 ) is a generic feature when Σ approaches the black hole horizon and becomes null. The second property, is related to the fact that all point on the black hole horizon move with the same angular velocity Ω H . Let us do a time redefinition τ = λ 2 t so the metric takes the form We chose τ = const as a time foliation so the unit normal is of the form Using our explicit metric ansatz (5.2) we can rewrite the type I constraint (2.8) (5.4)

Near-horizon expansion
Using the large mean curvature K = O(λ −1 ) of a hypersurface and the λ-expansion The covariant conservation (2.3) for our metric becomes which is the statement that the BH temperature is the same at each point on the horizon.
The next order is Similarly to the slow rotating case let us introduce the fluid velocity v i as viscosity ρ and pressure p The solution (5.6) in the new notation is of the form The time-component of the constraint equations (5.7) becomes the incompressibility equation while the spatial component (5.10) becomes the NS-like equation The equation ( The additional constraint for type I i geometries C kikj = 0 can be written using Riemann The NS equation (5.15) can be rewritten using Σ ij Furthermore, algebraically special solutions of type D conjectured to have some number of Killing vectors [9]. The additional constraint (5.16) can be written as a Killing equation For a given solution V, P we can consider a small perturbation on top of it v → V + v, p → P + p, (6.2) that satisfies the following linearized equation The powerful approach to linearized stability is based on energy balance equation [15].
Let us take the linearized equation (6.3), multiply it by v i and integrate over the time slice of the horizon The lefthandside measures time dependence for the kinetic energy of the fluid perturbation while the rightahndside describes the energy dissipation by viscosity term and energy flow from background solution. The relative size for the two terms in (6.4) can be described in terms of dimensionless combination Re = Lv η , (6.5) so that stability requires The critical value is determined by minimization over divergence free velocities. This bound was first proposed by O.Reynolds [16]. Existence of the minimum is guarantied because of the same (quadratic) velocity dependence for both functionals. Moreover, the contribution to energy balance from the quadratic term, that we dropped in linearized analysis, is a total derivative  for slowly rotating black hole has Re c = ∞ and it is always stable.
For the generic Kerr we have the following linearized system The extra terms are similar to the ones described in [12] and describe various physical effects like Coriolis force.
The energy balance equation is The type I i condition Σ ij = 0 and bulk Killing symmetry V k ∂ k h = 0 simplify energy balance into what makes the fluid, dual to rotating black hole, stable.

Summary of the results
We described the generalization of the algebraically special fluid/gravity correspondence, which includes rotating black holes in four and higher dimensions. On a gravity side rotating black holes are type D algebraically special while the duality construction requires to choose the null frame with smaller speciality. The reduced speciality is of type I i , which is still