Contents

KUNS-2170

Keiju Murata

Department of Physics, Kyoto University, Kyoto 606-8501, Japan

e-mail:murata@tap.scphys.kyoto-u.ac.jp

July 15, 2019

We study gravitational perturbations of the Kerr-AdS spacetime with equal angular momenta. In this spacetime, we found the two kinds of classical instabilities, superradiant and Gregory-Laflamme instabilities. The superradiant instability is caused by the wave amplification via superradiance, and by wave reflection due to the potential barrier of the AdS spacetime. The Gregory-Laflamme instability appears in Kaluza-Klein modes of the internal space and breaks the symmetry . Taken into account these instabilities, the phase structure of Kerr-AdS spacetime is revealed. The implication for the AdS/CFT correspondence is also discussed.

## 1 Introduction and Summary

Recently, AdS black holes in compactified type IIB supergravity have attracted much interest because they describe strongly coupled thermal super Yang-Mills theory via AdS/CFT correspondence [1, 2, 3, 4]. Especially, phase transitions of dual gauge theory are identified with instabilities of AdS black holes and understanding of stability of AdS black holes is important to reveal the strongly coupled gauge theory.

The superradiant instability is caused by property of rotating AdS black holes and information of the internal space is not so important for superradiant instability. However, if the internal space is taken into account, we can find another type of instability, called Gregory-Laflamme instability. Originally, the Gregory-Laflamme instability has been found in the black brane solution [22, 23, 24], but, in the Schwarzschild-AdS spacetime, the situation can be similar to the black brane system. If the horizon radius is much smaller than radius of , the internal space may be considered as . Then, we can consider Sch-AdS spacetime as a black brane and the Gregory-Laflamme instability may appear in Kaluza-Klein modes. The Gregory-Laflamme instability of Schwarzschild-AdS spacetime has been already found in [25]. In this paper, extending their work, we study the Gregory-Laflamme instability of Kerr-AdS spacetime.

We will take into account Gregory-Laflamme and superradiant instabilities and reveal the phase structure of Kerr-AdS spacetime. There are two kinds of instabilities in the Kerr-AdS spacetime. Thus, we can expect that this spacetime has rich phase structure and it will be useful to find the evidence of the AdS/CFT correspondence.

The organization and summary of this paper is as follows. In section 2, we introduce Kerr-AdS spacetimes with equal angular momenta. Especially, the spacetime symmetry is studied. We shall see that the symmetry is in the case of equal angular momenta. In section 3, we study the gravitational perturbation of Kerr-AdS spacetime neglecting the Kaluza-Klein modes of the internal space . We can get the master equations which are relevant for the superradiant instability. These equations are solved numerically and we find the onset of superradiant instabilities given by , where is angular velocity of horizon and is curvature scale of AdS spacetime. In section 4, we study Gregory-Laflamme instability of Kerr-AdS spacetime. We consider the gravitational perturbation including Kaluza-Klein modes in order to see the Gregory-Laflamme instability and get the ordinary differential equations in which three variables are coupled. These equations are solved numerically and we find the Gregory-Laflamme instability. In section 5, we reveal the phase structure of Kerr-AdS spacetime taking into account superradiant and Gregory-Laflamme instabilities. The result is in Figure.1, and are angular velocity and temperature of Kerr-AdS black holes. In Figure.1, and are normalized by the curvature scale of the AdS spacetime, . The solid and dashed lines are onset of the Gregory-Laflamme and superradiant instabilities, respectively. These lines cross each other and we can see five phases in this diagram. In the “Stable” region, Kerr-AdS black holes are stable. In the “SR” and “GL” region, black holes are unstable against superradiant and Gregory-Laflamme instabilities, respectively. In “SR&GL” region, black holes are unstable against both of them. In “No Black Holes” region, there is no black hole solution. The final section is devoted to the conclusion.

## 2 Kerr-AdS5 black hole in Type IIB Supergravity

### 2.1 Kerr-AdS5×S5 spacetime with equal angular momenta

In this section, we introduce Kerr-AdS spacetime as a solution of type IIB supergravity. The equations of motion of type IIB supergravity are given by

 RMN=148FMP2P3P4P5FNP2P3P4P5−1480gMNFP1P2P3P4P5FP1P2P3P4P5 , (2.1) ∇P1FP1P2P3P4P5=0 , (2.2)

where . The form is RR 5-form satisfying and . We concentrate on the metric and RR 5-form field in type IIB supergravity, while other components, such as dilaton, NSNS 3-form, RR 1-form and 3-form, have been set to be zero. We will consider Kerr-AdS spacetime which is a solution of (2.1) and (2.2). The Kerr-AdS spacetime can have two independent angular momenta generally, but, for simplicity, we will consider the case of equal two angular momenta. Then, the spacetime symmetry is enhanced and stability analysis will be possible. The metric of Kerr-AdS spacetime with equal angular momenta is given by111To obtain this metric from Kerr-AdS spacetime given in [26, 27, 28, 29], we need some coordinate transformation and redefinition of a parameter. These are summarized in appendix A.

 ds2=−(1+r2L2)dt2+dr2G(r)+r24{(σ1)2+(σ2)2+(σ3)2}+2μr2(dt+a2σ3)2+L2dΩ25 , (2.3)

where is defined by

 G(r)=1+r2L2−2μ(1−a2/L2)r2+2μa2r4 . (2.4)

Then, RR 5-form is

where is volume form of and is volume form of AdS part of (2.3). Because of the relation, , the form satisfies the self dual condition. In (2.3), we have defined the invariant forms of as

 σ1=−sinψdθ+cosψsinθdϕ ,σ2=cosψdθ+sinψsinθdϕ ,σ3=dψ+cosθdϕ , (2.6)

where , , . It is easy to check the relation . The dual vectors of are given by

 e1=−sinψ∂θ+cosψsinθ∂ϕ−cotθcosψ∂ψ ,e2=cosψ∂θ+sinψsinθ∂ϕ−cotθsinψ∂ψ ,e3=∂ψ , (2.7)

and, by the definition, they satisfy .

The horizon radius can be determined by . The angular velocity of Kerr-AdS back hole is given by

 ΩH=2μar4++2μa2 . (2.8)

For existence of horizon, the angular velocity has the upper bound,

 ΩH≤(12r2++1L2)1/2≡ΩmaxH . (2.9)

In term of and , the two parameters in the metric (2.3) can be rewritten as

 a=r2+ΩH1+r2+/L2 ,μ=12r2+(1+r2+/L2)21−(Ω2HL2−1)r2+/L2 . (2.10)

We will use parameters mainly.

### 2.2 Spacetime Symmetry

Now, we study the symmetry of (2.3). It will be important for separation of variables of the gravitational perturbation equations in section 3 and 4. Apparently, the metric (2.3) has the time translation symmetry and symmetry comes from part of (2.3). Additionally, the spacetime has the symmetry characterized by the Killing vectors :

 ξx=cosϕ∂θ+sinϕsinθ∂ψ−cotθsinϕ∂ϕ ,ξy=−sinϕ∂θ+cosϕsinθ∂ψ−cotθcosϕ∂ϕ ,ξz=∂ϕ . (2.11)

The symmetry can be explicitly shown by using the relation , where is a Lie derivative along the curve generated by the vector field .

From the metric (2.3), we can also read off the additional symmetry, which keeps the part of the metric, and is generated by . The generator satisfies and and, thus, . Therefore, the symmetry of Kerr-AdS spacetime with equal angular momenta becomes .

For later calculations, it is convenient to define the new invariant forms

 σ±=12(σ1∓iσ2) . (2.12)

Then, the dual vectors for are

 e±=e1±ie2 . (2.13)

By making use of these forms, the metric (2.3) can be rewritten as

 ds2=−(1+r2L2)dt2+dr2G(r)+r24{4σ+σ−+(σ3)2}+2μr2(dt+a2σ3)2+L2dΩ25 . (2.14)

We will use this expression in the following sections.

In the following sections, we will study the stability of Kerr-AdS spacetime with equal angular momenta (2.3). In this spacetime, we can expect two kinds of instabilities. One of them is the superradiant instability which is caused by the wave amplification via superradiance, and by wave reflection due to the potential barrier of the AdS spacetime. This instability should be seen, even if Kaluza-Klein modes are neglected. The other instability is the Gregory-Laflamme instability which is instability of the internal space , that is to say, the Gregory-Laflamme instability is instability of Kaluza-Klein modes. First, we shall see the superradiant instability of Kerr-AdS spacetime in this section.

### 3.1 Perturbation equations and separability

To see the superradiant instability, we can neglect Kaluza-Klein modes of . In addition, we will consider only metric fluctuation on the AdS part of the spacetime, that is,

where are indexes on AdS and is volume form of . For the perturbations, the equation (2.2) is trivially satisfied and equation (2.1) becomes

 δGμν−6L2hμν=0 . (3.2)

where is perturbation of the Einstein tensor of five-dimensional metric , which is defined by

 δGμν=12[∇ρ∇μhνρ+∇ρ∇νhμρ−∇2hμν−∇μ∇νh−gμν(∇ρ∇σhρσ−∇2h−Rρσhρσ)−Rhμν] , (3.3)

where denotes the covariant derivative with respect to and . Tensors and are Ricci tensor and Ricci scalar of . We take AdS part of (2.14) as a background metric . The equation (3.2) is nothing but the perturbation of five-dimensional Einstein equations with the negative cosmological constant.

The perturbation equation (3.2) is a partial differential equation of . However, in previous works [30, 18, 19, 21], it was shown that the perturbation equations can be reduced to ordinary differential equations by focusing on the symmetry of the background spacetime, . Here, we will briefly review these works.

Let us define the two kinds of angular momentum operators

 Lα=iξα ,Wa=iea . (3.4)

where and . They satisfy commutation relations

 [Lα,Lβ]=iϵαβγLγ ,[Wa,Wb]=−iϵabcWc ,[Lα,Wa]=0 , (3.5)

where and are antisymmetric tensors which satisfy . The Casimir operators constructed by and are identical and we define . The symmetry group, is generated by and . Here, we should notice the fact

 LW3σ±=±σ± ,LW3σ3=0 . (3.6)

It means that and have charges and . Since operators , and commute each other, these are simultaneously diagonalizable. The eigenfunctions are called Wigner functions defined by

 L2DJKM=J(J+1)DJKM ,LzDJKM=MDJKM ,W3DJKM=KDJKM , (3.7)

where indexes and are defined for and . The following relations are useful for later calculations

 W+DJKM=iϵKDJK−1,M ,W−DJKM=−iϵK+1DJK+1,M ,W3DJKM=KDJKM , (3.8)

where we have defined and . From this relation, we get the differential rule of the Wigner function as

 ∂+DJKM=ϵKDJK−1,M ,∂−DJKM=−ϵK+1DJK+1,M ,∂3DJKM=−iKDJKM , (3.9)

where we have defined and .

Now, we consider the mode expansion of . The metric perturbations can be classified into three parts, where and . They behave as scalar, vector and tensor for coordinate transformation of , respectively. The scalar can be expanded by Winger functions immediately as

 hAB=∑KhKAB(xA)DK(xi) . (3.10)

Here, we have omitted the indexes because the differential rule of Wigner function (3.9) cannot shift and therefore the modes with different eigenvalues are trivially decoupled in the perturbation equations.

To expand the vector part , we need a device. First, we change the basis to , that is where . Then, because is scalar, we can expand it by the Wigner function as

 hAi(xμ)=hA+(xμ)σ+i+hA−(xμ)σ−i+hA3(xμ)σ3i=∑K[hKA+(xA)σ+iDK−1+hKA−(xA)σ−iDK+1+hKA3(xA)σ3iDK] . (3.11)

In the expansion of , and , we have shifted the index of Wigner functions, for example, has been expanded as . The reason is as follows. The invariant forms and have the charge and , respectively (see Eq. (3.6)), while the Wigner function has the charge (see Eq. (3.7)). Therefore, by shifting the index , we can assign the same charge to , and in Eq. (3.11).

The expansion of tensor part can be carried out in a similar way as

 hij(xμ) = ∑K[hK++σ+iσ+jDK−2+2hK+−σ+iσ−jDK+2hK+3σ+iσ3jDK−1 (3.12) +hK−−σ−iσ−jDK+2+2hK−3σ−iσ3jDK+1+hK33σ3iσ3jDK] .

To assign the same charge to each term, we have shifted the index of Wigner functions.

Substituting Eqs. (3.10), (3.11), (3.12) into the perturbation equations (3.2), we get the equations for each mode labelled by , , . Because of symmetry, different eigenmodes cannot appear in the same equation.

It is interesting that we can find master variables from above information. First, we should notice that coefficients of the expansion have different indexes and, therefore, coefficients of components , and are restricted as follows:

 h++hA+,h+3hAB,hA3,h+−,h33hA−,h−3h−−|K−2|≤J|K−1|≤J|K|≤J|K+1|≤J|K+2|≤J

From this table, we can see that, in mode, there is only one variable , respectively. Therefore, the modes always reduce to a single master equation. We will study the stability of these modes. In fact, modes also reduce to a single master equation. The stability of modes are studied in appendix B and we will see that these modes are irrelevant to see the onset of the superradiant instability.

### 3.2 Master Equations

We will derive the master equation for modes. Because of the relation , we will consider modes only. Then, we can set as

 hμν(xμ)dxμdxν=h++(r)e−iωtDJ(xi)σ+σ+ , (3.13)

where . This field is gauge invariant. We substitute Eq. (3.13) into Eq. (3.2) and use the differential rule of Wigner functions (3.9). Then, component of (3.2) is given by

 12r10G(r)[−r10G(r)2h′′++−r5G(r)(6μr2λa2−10μa2+6μr2−λr6−r4)h′++−{−4λ2r12+(4λ(3+3J+J2)+ω2)r10−4(J+1)(J+2)r8−2μ(−4+16λa2+4Jλa2−12J−4J2+4a(J+2)ω−a2ω2)r6+8μ(2μ+2μλ2a4+4μa2λ+Ja2+4a2)r4−48a2μ2(1+λa2)r2+32μ2a4}h++]e−iωtDJ(θ,ϕ,ψ)=0 . (3.14)

This equation can be rewritten as

 −d2Φdr2∗+V(r)Φ=[ω−2(J+2)Ω(r)]2Φ , (3.15)

where we have introduced the new variable,

 Φ=(r4+2μa2)1/4r3/2h++ . (3.16)

and the tortoise coordinate,

 dr∗=(r4+2μa2)1/2r2G(r)dr . (3.17)

The function and potential are given by

 Ω(r)=2μar4+2μa2, (3.18)

and

 V(r)=G(r)4r2(r4+2μa2)3[15r14/L2+(4J+7)(4J+5)r12+6μ(3+11a2/L2)r10+2μa2(16J2+32J+5)r8−4μ2a2(10−17a2/L2)r6−4μ2a4(16J+35)r4+8μ3a4(1−a2/L2)r2−40μ3a6] . (3.19)

We can obtain the asymptotic form of and as

 Ω(r)→ΩH(r→r+) ,Ω(r)→0(r→∞) , (3.20)

and

 V(r)→0(r→r+) ,V(r)→15r24L4(r→∞) , (3.21)

where is the angular velocity of the horizon which is defined in Eq. (2.8). Therefore, the asymptotic form of the solution of master equation (3.15) becomes

 Φ→e±i{ω−2(J+2)ΩH}r∗(r→r+) ,Φ→r−1/2±2(r→∞) . (3.22)

We will solve (3.15) numerically and show the superradiant instability.

### 3.3 Stability analysis

#### 3.3.1 A Method to Study the Stability

We will find the instability by shooting method. Then, since the master equation (3.15) is not self adjoint form, we should put () and there are two shooting parameter, and . However, if the purpose is to find the onset of instability, the number of shooting parameter can be reduced to one [31, 13, 21].

We have separated the time dependence as in (3.13). Therefore, unstable mode satisfies . Thus, the boundary condition for regularity at the horizon becomes

 Φ→e−i{ω−2(J+2)ΩH}r∗(r→r+) . (3.23)

Then, the general form of wave function at infinity becomes

 Φ→Z1r−5/2+Z2r3/2(r→∞) , (3.24)

where are constants. For regularity at infinity, the condition must be satisfied. Therefore, the boundary conditions which unstable mode satisfies are

 Φ→e−i{ω−2(J+2)ΩH}r∗(r→r+) ,Φ→Z1r−5/2(r→∞) . (3.25)

Now, we consider the marginally stable mode. then, we can set . In this case of , Wronskian of is conserved, that is,

 Im[Φ∗ddr∗Φ]r=r2r=r1=0 , (3.26)

for any and . We take and . Then, from Eq. (3.26), we can get the relation,

 2(J+2)ΩH−ω=−4L−2Im(Z1Z∗2) . (3.27)

where we have used the asymptotic form of (3.23) and (3.24). To avoid divergence at infinity, must be satisfied. Then, we can get

 ω=2(J+2)ΩH . (3.28)

Therefore, equation which we should solve is

 −d2Φdr2∗+^V(r)Φ=0 , (3.29)

where

 ^V(r)≡V(r)−4(J+2)2(ΩH−Ω(r))2 . (3.30)

The boundary condition can be obtained by substituting into (3.30) and it is given by

 Φ′(r+)Φ(r+)=r4++2μa2r4+V′(r+)G′(r+)2 . (3.31)

There is only one shooting parameter, , in (3.29).

#### 3.3.2 Limit of Small Kerr-AdS Black Holes

Before the numerical calculation, it is important to solve master equation (3.29) analytically in some limit [32]. It may be useful to check the numerical calculation. We consider Kerr-AdS black holes in the limit of . Then, master equation (3.15) can be solved exactly. The solution which approach to zero at infinity is

 Φ=(r/L)7/2+2J(1+r2/L2)J+3F((J+2)ΩHL+J+3,−(J+2)ΩHL+J+3;3;11+r2/L2) , (3.32)

where is Gauss hypergeometric function. Then, the asymptotic form of becomes

 Φ=2(2J+2)!Γ[(J+2)ΩHL+J+3]Γ[−(J+2)ΩHL+J+3](rL)−2J−5/2−4(−1)2J+3(2J+3)!Γ[(J+2)ΩHL−J]Γ[−(J+2)ΩHL−J](rL)2J+7/2ln(rL). (3.33)

For the regularity at horizon, the first term of (3.33) must vanish. Thus, we can get where . This calculation is to see the onset of the instability and the lowest value of is important. The lowest value of is given by

 ΩHL=J+3J+2 . (3.34)

Numerical result must approach this value in the limit of .

#### 3.3.3 Onset of superradiant instability

Now, we shall solve (3.29) numerically. Using the Lunge-Kutta algorithm, we integrate the Eq. (3.29) from the horizon to infinity with various . The boundary conditions at the horizon are given by (3.31). Then, the general form of the wave function at infinity is given by (3.24). We can see that, at some value of , the flip the sign. It means that mode exists. We will search such numerically and plot the result in - diagram. The result is in Figure.2. The curves represent borderline of stability and instability of each mode, that is, each mode is stable below the curve, while they are unstable above the curve. From this figure, we can read off that, in the limit of , these curves for each mode approach . This result is consistent for analytical calculation in section 3.3.2. We can also see that, for higher mode, the instability occurs at a lower angular velocity. These curves seem to approach for large . These properties are the same for cases [13].

It is surprising that these results have been already seen in dual gauge theory [34, 33]. In [34], effective mass term for scalar fields of dual gauge theory have been obtained as

 m2eff=(2J+1)2L−2−4Ω2HK2 . (3.35)

Because of , if satisfied, is positive for any and . However, if , can be negative for large and modes. Thus, we see that, for , dual gauge theory is unstable and higher mode becomes tachyonic first as increases. These results are the same for superradiant instability of Kerr-AdS black holes.

## 4 Gregory-Laflamme instability of Kerr-AdS5×S5 spacetimes

### 4.1 Perturbation Equation

In the previous section, we have seen the superradiant instability of Kerr-AdS spacetime. The superradiant instability breaks the symmetry of Kerr-AdS. In this section, we will consider the Gregory-Laflamme instability of Kerr-AdS spacetimes. This instability breaks the symmetry of . Thus, we must see the Kaluza-Klein modes of the perturbations which have been neglected in the previous section.

We will consider only the metric fluctuations on the AdS part of the spacetime, that is,

where is spherical harmonics on which satisfy

 ∇2S5Yℓ=−ℓ(ℓ+4)Yℓ , (4.2)

here is the Laplacian of and . The in (4.1) is the volume form of . Since, in the case of Schwarzschild-AdS, the Gregory-Laflamme instability has been found in these fluctuations [25], the instability of Kerr-AdS must also appear in these fluctuations. Here, we should notice that, in (4.1), depends on the coordinates on . It is essential to see the Gregory-Laflamme instability. Then, from (2.1), we can obtain the perturbation equations as

 δGμν=6L2hμν−εL2(hμν−12h) , (4.3)

where and is defined in (3.3). From (2.2), we can get

 h(xμ)∂aYℓ(Ω5)=0 , (4.4)

where is index on . In the case , (4.4) is trivially satisfied and (4.3) reduces to (3.2). For , (4.4) implies . Then, as a constraint equation of (4.3), we can get transverse condition of as shown in the appendix C. Thus, even for Kaluza-Klein modes, we can use transverse traceless conditions,

 ∇νhμν=gμνhμν=0 . (4.5)

To separate variables of equations (4.3) and (4.5), we will use the formalism in the section 3.1 again. We can expand by the Wigner function and obtain ordinary differential equations labeled by . We will not study the stability of all modes, but we will consider only mode. The Gregory-Laflamme instability of Schwarzschild-AdS spacetime was found in the s-wave of AdS [25] and, thus, we can expect that, in the Kerr-AdS spacetime, the instability appears in most symmetric mode, . The metric perturbation for this mode is given by

 hμν(xμ)dxμdxν=htt(r)dt2+2htr(r)dtdr+hrr(r)dr2+2ht3(r)dtσ3+2hr3(r)drσ3+2h+−(r)σ+σ−+h33(r)σ3σ3 , (4.6)

where we assume that the metric perturbation does not depend on , in order to see the onset of Gregory-Laflamme instability. The stability of mode of this perturbation is shown in Appendix B.1 and, thus, we will consider . We substitute (4.6) into (4.3). Then, from and components of the perturbation equation, we can obtain

 htr=hr3=0 . (4.7)

Now, we introduce dimensionless variables, and , as

 htt=−(1+r2L2−2μr