Instability of Charged Lovelock Black Holes: Vector Perturbations and Scalar Perturbations

We examine the stability of charged Lovelock black hole solutions under vector type and scalar type perturbations. We find the suitable master variables for the stability analysis; the equations for these variables are the Schrodinger type equations with two components and these Schrodinger operators are symmetric. By these master equations, we show that charged Lovelock Black holes are stable under vector type perturbations. For scalar type perturbations, we show the criteria for the instability and check these numerically. In our previous paper, we have shown that nearly extremal black holes have the instability under tensor type perturbations. In this paper, we find that black holes with small charge have the instability under scalar type perturbations even if they have relatively large mass.


I. INTRODUCTION
The braneworld scenario with large extra dimensions predicts that higher dimensional black holes might be produced at colliders [1]. Therefore, higher dimensional black holes become attracting subjects and some aspects of these have been inspected so far. For example, exact solutions are investigated in higher dimensions. In higher dimensions, besides Schwarzschild black hole, Reissner-Nordström black holes [2] and rotating black holes [3], various solutions are found: black ring solution [4], black di-ring [5], black saturn [6] and so on. For these solutions, from the standpoint of black hole creations, it is important to examine the stability of such solutions because stationary solutions with the instability are not attractors of time evolution. This suggests that such black holes should not be realized.
So far, various stability analyses for black hole solutions have been performed. One of the most notable analysis is that of Tangherlini-Schwarzschild solutions by Kodama and Ishibashi [7]. They have derived master equations for all type perturbations. These are Shrödinger type equations and they have shown that these Shrödinger operators are all positive definite using the S-deformation approach which they have developed by Friedrichs extension. These results show that Schwarzschild black holes are also stable in higher dimensions. They have also examine the stability of higher dimensional Reissner-Nordström black holes [8]. By the S-deformation, they have also shown that this charged solution is stable under tensor and vector type perturbations. For scalar type perturbations, it has been shown that this black hole is stable in 4 and 5 dimensions. For this solution, the stability has also studied numerically and it has been found that black holes with large negative cosmological constants and large charge are unstable in more than 7-dimensions [9]. On Myers-Perry black hole solutions, in D ≥ 6, there found the instability for singly rotating solutions when the spin parameter is large enough [10]. In even dimensions, stability of near horizon geometry of rotating black hole with equal angular momenta are investigated [11].
It has been suggested that scalar mode for base space has the instability. Recently, the stability of black ring solution is examined by using local Penrose inequality and it is shown that the fat branch is unstable [12].
The stability analyses we have introduced above are all premised on Einstein theory. In fact, the stability of black hole solutions have been examined mainly in Einstein theory.
It is as important as such analyses to investigate the stability in more general theories.
In 4-dimensions, Einstein theory is characterized by two properties; the action has the general coordinate covariance and equation of motion consists of metric, the first derivative of metric and the second derivative of metric [13]. Then it is natural to extend the four dimensional gravitational theory to higher dimensional one keeping these two properties. In higher dimensions, the most general theory which satisfies above two features is not Einstein theory; it is Lovelock theory [14]. Then it is important to generalize the stability analysis of black hole solutions in Einstein theory to these in Lovelock theory.
Same as Einstein theory, a spherical symmetric solution is known in Lovelock theory [15,16]. This solution is called as Lovelock black hole solution. For this Lovelock black hole solution, the stability has been analyzed in [17][18][19]. In these papers, it has been shown that black holes with sufficiently small mass are unstable under scalar type perturbations in odd dimensions and unstable under tensor type perturbations in even dimensions. This critical mass differs with dimensions and Lovelock couplings. These instabilities become stronger as wavelength becomes smaller, and the time scale of the instability converge to 0 in small scale limit. Under vector perturbations, this solution is stable in all dimensions, which is independent of mass.
Since black hole creations originate from protons at colliders, it is also important to take account of Maxwell-charge. In Lovelock theory with U(1) field, a charged black hole solution is known [15]; this has spherical symmetry and a time-like Killing vector, and this Reissner-Nordström like solution is called as charged Lovelock black hole solution. Then, in this paper, we'd like to extend the stability analysis for Lovelock black hole solutions to charged Lovelock black hole solutions. For this charged solution, stability analysis under tensor type perturbations has been examined by us and we have shown that black holes are unstable if they have nearly extremal mass [20]. In this paper, we extend our previous discussion to vector type perturbations and scalar type perturbations; that is, we derive master equations for these type perturbations and examine the stability using master equations.
The organization of this paper is as follows. In section II, we review Lovelock theory, present the charged Lovelock black hole solutions and check the behavior of these solutions.
We mainly concentrate on asymptotic flat branch. In section III, we review the analysis for tensor type perturbations [20]. We examine tensor perturbations and show the criteria for stability under this type perturbations. In section IV, we derive master equation for vector type perturbations and show that there is no instability under this type perturbation. In section V, we concentrate on scalar type perturbations. We show that master equations can be summarized as a Schrödinger type equation with two components, and using this equation we present criteria for stability. In section VI, we numerically examine the conditions for the instability presented in section III and V. In this paper, we only check in 5 − 8 dimensions.
In the final section VII, we summarize this paper.

II. CHARGED LOVELOCK BLACK HOLES
In this section, we introduce Lovelock theory and present charged black hole solutions in Lovelock-Maxwell theory. These solutions are expressed as the roots of the polynomial equation and we confirm that one of the roots is asymptotic flat. For this asymptotically flat root, we briefly check the behavior, singularity and horizons.

A. Lovelock-Maxwell System
In Ref. [14], D.Lovelock have constructed the gravitational theory whose equation of motion consists of the metric, the first derivative of the metric and the second derivative of the metric. The Lagrangian for this theory is where Λ corresponds to a cosmological constant and β m s are arbitrary constants which we call Lovelock couplings. We add the coefficients (2m)!/2 m m 2m−2 p=1 (n − p) for convenience. In the above Lagrangian, n is related to dimension D as n = D − 2 and k corresponds to the maximum order defined as k ≡ [(D − 1)/2] where [x] is the Gauss symbol. There exists maximum order k due to the antisymmetric property of δ ρm . When we fix the maximum order k, the dimension is restricted as n = D − 2 = 2k − 1, 2k; for example, the second order Lovelock theory is the most general in n = 3 or n = 4, and the third order one is in n = 5 or n = 6. By the ambiguity of the overall factor of the action, we take the unit β 1 = 1 in this paper.
In this paper, we want to concentrate on Lovelock-Maxwell system. This system is described by the action where F µν is the field strength of Maxwell field A µ . In the action (1), the dynamical variables are g µν and A µ . The variations by these variables lead where G µ ν , which we call Lovelock tensor, and T µ ν , which means energy momentum tensor for U(1) field, are defined as The field strength is defined as F = dA, then F µν must satisfy the identity The above equations (2), (3) and (6) are our basic equations.

B. Charged Lovelock Black Holes
For the basic equations, black hole solutions with two parameters are known [15]. We assume the static spherical symmetric metric with spherical symmetric electric field F tr = E(r), other components = 0 .
In these, γ ij corresponds to the metric for S n .
We can easily check that these ansatz satisfies (6). Then we concentrate on the others and these lead following equations; and the other components are identical. In (9), ψ is related to f (r) as f = 1 − r 2 ψ and P[ψ] is defined as .
The second equation of (9) is derived by a derivative of the third equation with the first equation, so we only consider the first and third equations. The first equation can be easily integrated and the result is In this equation, n(n − 1)Q is an integral constant and this constant corresponds to the charge, which can be seen from the behavior of E(r). Substituting (11) into the third equation of (9), we can gain (r n+1 P[ψ]) ′ = (n − 1)Q 2 /r n , or integrating both sides reads where M is an integral constant. We will see that M corresponds to mass when checking the asymptotic behavior of the solution [21,22].
We must solve the polynomial equation (12) for solution of Lovelock-Maxwell system. In order to solve the polynomial equation (12), in this paper, we assume some conditions for Lovelock couplings β m and M for simplicity. First we consider mass of black hole is positive, that is, M > 0. Second we set cosmological constant Λ = 0. In Λ = 0, as we will see later, there must exist an asymptotic flat branch. Third, for simplicity, we assume the positivity of Lovelock couplings, that is,

C. Asymptotic Flat Branch
Because (12) is k-th order polynomial, the polynomial equation (12) should have at most k solutions. However, assuming all Lovelock couplings are positive and Λ = 0, one of the roots corresponds to an asymptotic flat solution.
For instance, we see the above statement when k = 2. In this case, (12) reduces into the 2nd order polynomial equation, so this can be easily solved as Let's consider the limit r → ∞. The first root behaves as ψ → M/r n+1 and the second converges as ψ → −2/β 2 . Therefore, the function f (r) = 1 − r 2 ψ(r) behaves as f = 1 − M/r n−1 for the first root and f = 1 + 2r 2 /β 2 for the second one. This shows that the first branch expresses an asymptotic flat solution and the other is an asymptotic AdS solution.
Regrettably, we can not write the roots of (12) with general k explicitly. Nevertheless, we can understand the existence of an asymptotic flat solution as the solution of (12). In order to check this statement, we want to introduce the graphical method. In  (7) with this asymptotic behavior shows that this positive ψ corresponds to an asymptotic flat solution. This asymptotic behavior of f (r) also explains that M corresponds to ADM mass.
In the last of this subsection, we briefly check the behavior of our asymptotic flat root not in the asymptotic region. Let us consider by the graphical method with Fig.1 again.
Because M(r) behaves as 0 → M max → 0 when r moves ∞ → r max → r 0 , our ψ(r) varies as 0 → ψ max → 0. When r becomes smaller than r 0 , M(r) becomes negative and so ψ(r) also takes negative values.

D. Singularities
In ψ < 0 or in r < r 0 , our asymptotic flat solution has the curvature singularity. To confirm this, we examine the Kretschmann invariant This value diverges at r = 0; and the singularity also exists where f ′ or f ′′ diverge. For example, f ′ has the term like r 2 ψ ′ . Form the the derivative of (12), this can be estimated as r 2 ψ ′ = r 2 M ′ /∂ ψ P. Then, besides r = 0, R µνλρ R µνλρ also diverges where the derivative of P[ψ] with respect to ψ becomes 0. If P[ψ] takes a extreme value at ψ 0 , because P[ψ] is monotonically increasing in ψ ≥ 0, such ψ 0 must be negative; that is, there is the singularity at r s (< r 0 ). If P[ψ] is monotonically incresing for all ψ, there is the curvature singularity at r = 0. Therefore, whether P[ψ] has extreme values or not, our asymptotic flat branch has the curvature singularity somewhere in 0 ≤ r < r 0 .

E. Horizons
Singularities must be wrapped by the event horizon from the standpoint of cosmic censorship. In this subsection, we consider horizons and present the condition for existence of horizons.
Our asymptotic flat solution has the event horizon at f (r) = 0. This branch also satisfies where r H are horizon radii and ψ H is defined as ψ H ≡ ψ(r H ). The first equation shows, if horizons exist, the corresponding ψ H must be positive. Our ψ(r) is positive in r > r 0 , then r H , if exists, must satisfy r H > r 0 . As we have emphasized, the singularity exists somewhere in r < r 0 . So we do not worry about the naked singularity if (15) has roots.
Here, we consider the criteria for the existence of roots of eq. (15). Eliminating ψ H by the takes an extremal minimum at r = r ex . Because of (16), the cross points mean horizon radii r H .
first equation of (15), the second equation becomes equation for r H as follows; The first term of α(r) is negative power of r and its coefficient is positive. Under our assumption (13), because n = 2k or 2k − 1, the other terms are positive power of r and their coefficients are positive. Therefore, y = α(r) behaves as Fig.3; α(r) diverge near r = 0, takes extremal minimum at r = r ex and monotonically increase in r > r ex . Then, we can denote that (16) has two roots when M is larger than M ex where In the two roots, the larger one corresponds to the outer horizon and we call this r out hereafter. Note that α(r) only depends on Q except for Lovelock couplings. Therefore, r ex is determined when we fix Q; this shows that M ex depends only on charge.
Finally, we check the behaviors of some functions for the later discussions. First, we examine the behavior of M(r) outside of r out . The l.h.s of the first equation of (16) is monotonically decreasing function and the r.h.s is monotonically increasing in r < r max and monotonically decreasing in r > r max . Then, when (16) has two roots, it is forbidden that both of them are smaller than r max ; at least, the larger root r out must satisfy r out > r max . Hence, Then, in Fig.1, while r > r out , the dotted line y = M(r) falls monotonically as r becomes larger and so ψ(r) decreases monotonically in this region. Therefore, from the relation when we consider the outside of r out .

III. TENSOR TYPE PERTURBATIONS
Thanks to the spherical symmetry of background (7), tensor-type, vector-type and scalartype perturbations are decomposed and we can examine them separately. We have already examined the tensor perturbations in [20]. In this section, we review our previous analysis briefly. Note that we only consider the case when there exist horizons; that is, M > M ex .

A. Master Equation
Under tensor type perturbation, there is no perturbation for Maxwell field. Then we only consider the gravitational perturbations In this expressions, φ corresponds to the master variable. T ij is the tensor harmonics which is characterized by the traceless condition T i i = 0, the divergence-free condition T ij |j = 0 and the eigenequation T ij Note that | is the covariant derivative for γ ij and ℓ is the integer which satisfies ℓ ≥ 2 .
Using above metric perturbations, the first order equation δG i j = 0 leads [19] T where T (r) is which is always positive in r > r out due to (19).
For this equation, as we have shown in [19,20], there exist ghost like instabilities if T ′ has negative regions. For example, the coefficient of kinetic term in (21) is proportional to T ′ , so this term has the wrong sing while T ′ is negative. Then, here we also assume T ′ (r) > 0 in r > r out for avoiding the ghost instability.
Under the ghost-free condition, we can change the normalization of φ as Ψ(r) = φ(r)r T ′ (r). Using this variable, converting r to r * which defined as dr * /dr = 1/f and Fourier transforming like Ψ → Ψe iωt , (21) is recast as where Eq. (23)

B. Stability Analysis
In this subsection, we show "there exist negative spectra if T ′′ has negative region in r > r out " when T ′ is always positive. To show this, we define the inner product as and use the inequality where ω 2 0 is the lower bound of spectra and χ is an arbitrary smooth function with compact supports. From this inequality, we can show that there exist negative spectra if we find a function χ such that (χ, Hχ) becomes negative under our assumptions.
We assume T ′′ has negative regions and define I as a closed set on such regions. Under these, we chose χ 0 as a smooth function which has a compact support on I. For this χ 0 , In this calculation, we use Gauss divergence theorem and neglect boundary terms because χ 0 is smoothly connecting to 0 at ∂I. In (27), the first term must be positive and the second integral is negative because we assume T ′ > 0 in r > r out and T ′′ < 0 on I. Therefore, taking ℓ → ∞, (χ 0 , Hχ 0 ) must become negative. Because of (26), this means negative spectra exist in sufficiently large ℓ modes. Then we can declare that black holes are unstable if T ′′ takes negative values somewhere in r > r out .
Inversely, it can be also shown that charged Lovelock black holes are stable if T ′′ is always positive in r ≥ r out . As shown in Ref. [7], it is sufficient for the stability to show that (Φ, HΦ) . We can check this criterion by the same calculation of (27) and the positivity of T ′′ , so we can say black holes are stable if T ′′ is always positive.
We want to summarize this section. For avoiding the ghost instability, we must assume T ′ is always positive. Under this assumption, charged Lovelock black holes are stable if and only if T ′′ always takes positive values in r > r out . Hence, what we have to do is a probe of the behaviors of T ′ and T ′′ , and we will check these in the section VI.
In the end of this section, we want to comment about T ′ in the 2nd order Lovelock theory. We have already shown in our previous paper [20], there is no ghost instability for this case. This can be checked by the direct calculations; eq.(14) leads T (r) = r n−1 ∂ ψ P[ψ] = r n−1 1 + 2β 2 M(r), then T ′ is calculated as

IV. VECTOR TYPE PERTURBATIONS
In this section, we examine the vector type perturbations. We here also assume M > M ex for horizons and T ′ > 0 for no ghosts under tensor type perturbations. We only consider the perturbations in r > r out . Under these assumptions, we derive master equations for vector type perturbations. Using this equations, we show that charged Lovelock black hole solutions are stable for vector type perturbations when T ′ is always positive.

A. Gravitational Perturbations
Firstly, we'd like to consider the metric perturbations. In this paper, we use the Regge-Wheeler gauge in which metric perturbations are expressed as In these, V i is the vector harmonics which is characterized by the transverse condition V i |i = 0 . For vector type perturbations, other than δG t i , δG r i and δG i j are trivial. From the above metric, we can calculate the non-trivial components as This is almost same as the results of [19] except for the background electric field. This gap mainly arises from the difference of the identity

B. Perturbation of Maxwell field
Next we examine the vector perturbations of Maxwell field. We start from the perturbation of vector potential where V i is the vector harmonics. Note that C is gauge invariant under U(1) gauge because there is no gauge freedom for vector perturbations. Using above δA µ , we can easily calculate the first order of the field strength as Here, we derive the evolution equation for first order variable C from Maxwell equations.
It is easy to check the above field strength satisfies the identity δF [µν;λ] = 0. Therefore, δ(F µν ;ν ) = 0 is important for the evolution equation. In these equations, µ = t, r components are trivial and µ = i components read In order to gain evolution equations for gravitational field, we must calculate the first order of the energy momentum tensor (5). This tensor consists of δF µν , δg µν and background variables, so this tensor can be calculated from (32), (29), (7) and (8), and the results are as follows; Then, from (30) and (34), the first order Lovelock equation δG µ ν = δT µ ν reads Substituting (36) into the first equation of (35) and integrating this with respect to r reads Here we use (r n E) = const. and C 1 (t) is a constant of integral. Same as this, substituting (36) into the second equation of (35) and integrating this with respect to t reads Comparison (37) with (38) shows that C 1 (t) = C 2 (r) = const. and this constant can be absorbed into φ. Therefore the three equations (35) are reduced into one equation Same as this substitution, (36) makes Maxwell equation (33) 1 As we have seen, φ determine the perturbation of gravitational field and C does Maxwell field. Therefore, these are the master variables and (39) and (40) are the master equations for vector type perturbations.
Finally, we'd like to alter these two equations into a Scrödinger equation with two components. To do so, we must change three points: Firstly, we change the normalization In this, we use the assumption that T ′ is always positive. Secondly, we switch radial coordinate r to r * . Finally, we Fourier transforme like Ψ → Ψe iωt and ζ → ζe iωt . Then, (39) and (40) become a Scrödinger equation with two components as where and Note that these equations are not decomposed due to higher curvature collections. In Einstein limit, owing to T (r) = r n−1 , the above potential matrix can be diagonalized by constant eigenvectors. This indicates that our Schrödinger equation can be decomposed into two equations by taking suitable linear combinations of ψ and ζ. In fact, we can do so by using the combination α ± ψ − ζ with which is consistent with [8]. Against this, because T (r) is more complicated in general Lovelock theory, we must consider the above coupling system.

D. Stability Analysis
In this subsection, we show that the Schrödinger equation (42) has no negative eigenvalue states.
In order to show this, for Ψ = (Ψ, ζ) T , we define the inner product as Here, we prove that charged Lovelock black holes are stable for vector type perturbations.
In order to show this, we prove H is an essentially positive-definite self-adjoint operator. For this, because H with C ∞ 0 (r * ) × C ∞ 0 (r * ) is a symmetric operator, it is sufficient to check this operator is positive-definite [7].
We assume Ψ 0 ∈ C ∞ 0 (r * ) × C ∞ 0 (r * ), then ( Ψ 0 , H Ψ 0 ) can be estimated as where we use Gauss theorem in the second equality and neglect boundary terms because Ψ 0 and ζ 0 are in C ∞ 0 (r * ). This calculation shows that H with C ∞ 0 (r * ) × C ∞ 0 (r * ) is positivedefinite. Then, since H with C ∞ 0 ×C ∞ 0 is essentially self-adjoint, H can be uniquely extended to a positive-definite self-adjoint operator, so there is no instability under vector type perturbations.

V. SCALAR TYPE PERTURBATIONS
In this section, we derive the master equations and present conditions for the instability under scalar type perturbations. In this section, we assume M > M ex and also assume that T ′ is always positive outside r out .

A. Gravitational Perturbations
Firstly, we consider the metric perturbations. In this paper, we take the Zerilli gauge in which metric perturbations are described as where Y is the scalar harmonics which is characterized by the eigenequation Y |l |l = −κ s Y with κ s = ℓ(ℓ + n − 1) and ℓ = 0, 1, 2, · · · .
For deriving the master equation, it is sufficient that we calculate δG j i (i = j), δG r t , δG t t , δG i r and δG r r [18]. By the above metric perturbations, we can derive the following results [19]; These are same as our previous calculation for neutral black holes except for the detail expression of f (r).

B. Scalar Perturbations for Maxwell Field
Next, we examine the scalar perturbations for Maxwell field. We start form perturbations of the field strength which has the U(1) gauge invariance. We describe this as where Y is the scalar harmonics. Note that δF ij = 0 because we can not construct antisymmetric tensors from scalar functions.
From (54), we can define a new variable B as Substituting (55) into (52) and integrating this with respect to r, (52) becomes where C 1 (t) is a constant of integral. We use r n E = const. in this integral. Same as this, (53) becomes Here C 2 (r) is also a constant of integral. Then, a comparison with above two equations reads C 1 (t) = C 2 (r) = const. and so this term can be absorbed into B. Therefore,  (51) and (55). The result is Finally, for the first order perturbations of Lovelock equations, we calculate the first order of the energy momentum tensor (5). Eq. (5), the background electric filed E(r), metric perturbations (48) and perturbations of Maxwell field (50) yield Note that we use the relation (56) in (t, t), (r, r) components and also use the relation (55) in (r, i) components.
Then, from (59), (60) and (61), we can express H 0 , H and K as follows [19]; where Above equations show that two variable φ and B express the perturbative variables and so these are master variables. Therefore, we must construct evolution equations for these variables in order to examine the stability of background solution. That for B has already been derived as (57), but this include metric perturbations. Then, substituting (65) into The evolution equation for φ is derived from nrf ×(62)+(63) with (65) (see [19]) and the result isφ These coupled two equations (67) and (68) determine the behavior of φ and B. These two functions determine all perturbative variables, so these two equations are the master equations.
Here, we derive a Schrödinger equation with two components like (42) from these two equations. Against the case for vector type perturbations, it is more complicated because there is φ ′ in evolution equation for B. Therefore, in order to transform these equations into a Schrödinger type equation, we must eliminate this φ ′ . For this, we must consider linear combinations of the master variables φ and B. For example, the following combinations are fit for our purpose; By using these variables and tortoise coordinate r * , we can obtain where and

D. Condition for Instability
In this subsection, we show the criterion for the instability under scalar type perturbations.
Here, we show that "if 2T ′2 − T T ′′ takes negative values somewhere in r > r out , charged Lovelock black holes have the instability" when T ′ is always positive. In order to show this, we here also define the inner product as where Ψ = (Ψ, ζ) T . For this proof, it is convenient to use the following inequality; for any This inequality suggests that there exist instabilities if we can find trial function Ψ test which We assume 2T ′2 − T T ′′ has negative regions and define I as a closed set in the region 2T ′2 −T T ′′ < 0. Then, we choose a trial function as Ψ test = (Ψ 0 , 0) T where Ψ 0 is a sufficiently smooth function with compact support on I. Using this test function, ( Ψ test , H Ψ test ) is evaluated as Note that we neglect the boundary terms in second equality because Ψ 0 is zero at the boundary of I. Furthermore, using the relation

the second line of the last equation can be evaluated as
We'd like to summarize this section. We assume T ′ > 0 in r > r out . Under this assumption, we derived master equations. We can unify these equations as a Schrödinger equation with two components. We show that this Schrödinger operator has negative spectra when 2T ′2 − T T ′′ has negative region. Therefore, we can denote that 2T ′2 − T T ′′ is crucial for the stability of charged Lovelock black holes. This criteria is same as neutral case [19].
We have not shown the inverse statement so far. Then, even if 2T ′2 − T T ′′ is always positive, we can not declare this black hole is stable. For example, in Einstein case, T (r) is r n−1 , so 2T ′2 − T T ′′ = n(n − 1)r 2(n−2) > 0. Then we can not say anything for Einstein case.
Same as this, in 6-dimensions, this function can be evaluated as so we cannot say anything for scalar perturbations in 6-dimensions .

VI. NUMERICAL RESULTS
In section III and section V, we have shown that the behavior of T (r) is crucial for instability. In detail, T ′ is critical for ghost, T ′′ is for tensor perturbations and 2T ′2 − T T ′′ is for scalar perturbations. In this section, we check the behavior of these functions. For neutral cases, we can examine analytically because we can reduce these functions into the polynomial functions of ψ [19]. However, such reduction can not be performed in charged case. Therefore, we numerically check the behaviors of T ′ , T ′′ and 2T ′2 − T T ′ for various (|Q|, M) with some Lovelock couplings.
In numerical calculation, we must use dimensionless parameters. So far we have discussed in the unit β 1 = 1; in this unit, we can not fix the scale of length. Same as our previous paper [20], we here also use β 2 for fixing this scale. This constant has a dimension of length squared. Therefore, we can relate r, ψ, M, Q and Lovelock couplings β m s to the dimensionless parameters as follows; Hereafter, we show some results for 5 ∼ 8 dimensions. The strategy of our numerical calculation is basically same as our previous paper except for checking 2T ′2 −T T ′′ [20]. Note that µ ex is the dimensionless extremal mass parameter which can be calculated from (17).
µ tensor is a border between stable and unstable for tensor type perturbations and µ scalar corresponds to that for scalar type perturbations.

A. 5-dimensions
As we have mentioned above, we use β 2 for fixing the scale of length. Then we need not regard the Lovelock couplings in 5-dimensions.
We present the numerical results for 5-dimensions in Fig.4. For this figure, we check the region where µ ex (|Q|) ∼ µ ex (|Q|) + 3 for each |Q| and the mesh size is dµ = dQ = 10 −3 .
As shown in our previous paper [19], the tensor-unstable region lies thinly on the extremal mass µ ex (|Q|) and µ tensor (|Q|) converges to 0.5 when |Q| → 0. In this limit, µ ex also converges to this value. Note that this thin region is over at |Q| ∼ 3 which has been checked in our previous analysis.

B. 6-dimensions
In 6-dimensions, same as 5-dimensional case, we need not alter Lovelock coefficients.
As we have checked in section III, T ′ is always positive so there is no ghost. It has also mentioned in section V that 2T ′2 − T T ′′ is positive definite in 6-dimensions, which means we can not find the instability for scalar type perturbations. Then we can only detect the instability under tensor perturbations and the results are same as our previous analysis [20]; same as the tensor-unstable region in 5-dimensions, this region exists just on µ ex (|Q|) and disappears at |Q| ∼ 3.28. However, there also exists a difference; µ tensor converges to 0.27 in |Q| → 0 while µ ex → 0. Therefore, we can say that black hole with µ ∼ µ ex has the instability under tensor type perturbations when 0 ≤ |Q| < 3.28. Especially, black hole has the instability also in neutral case against the 5-dimensional case. This diagram is almost same as 5-dimensional cases; the tensor-unstable region clings to µ ex (|Q|) and the scalar one does to µ-axis. Our previous analysis shows that the tensorunstable region exists in 0 < |Q| < 4.695 [20]. The upper bound of the scalar-unstable region is µ ∼ 3.99 and this region vanishes at |Q| ∼ 0.516. When c 3 changes, the upper bound etc. change but the appearances of these unstable regions do not change.
When c 3 = 0.2, there are no ghost regions. However, when c 3 is larger than 0.25, we can find ghost region near the origin of the diagram (see Fig.14 of our previous paper [20]).
Therefore, same as 5-dimensions, we can roughly say that black hole suffers from the instability under scalar perturbations if Q ∼ 0 and µ is smaller than a certain value and has instability under tensor modes when µ is as small as µ ex . Against 5-dimensional case, there exists c 3 dependance for ghost regions.
This figure is almost same as that for 6-dimensional case. In this figure, there are no parameters which make 2T ′2 − T T ′′ negative, so we can only find the instability under tensor type perturbations. For tensor type perturbations, the unstable region slightly lies on µ ex (|Q|) and there also exists a gap between µ tensor and µ ex in |Q| → 0. These properties are similar to 6-dimensional case.
Against the above results, there exist a scalar-unstable region in Fig.8. This figure is calculated with c 3 = 1 and check the region in µ ex (|Q|) < µ < µ ex (|Q|) + 30 with the mesh size dµ = dQ = 10 −3 . This figure is very similar to 5-dimensional diagram; a tensor-unstable region exists just on extreme line µ ex (|Q|) and a scalar-unstable region localizes near µ-axis.
Furthermore, there is no ghost region in c 3 = 1.
In Fig.7 and Fig.8, there are no ghost regions. However, when c 3 becomes larger than 5.92, a ghost region appears near the origin of the diagram (see Fig.18 of [20]). Then, the appearance is similar to 7-dimensional case with sufficiently large c 3 .
In 8-dimensions, as we have presented, the appearance of diagrams are very responsive to the Lovelock coupling c 3 . When c 3 is very small, this is same as 6-dimensional diagram.
As c 3 becomes larger, the looks of the diagrams change from 5-dimensional results to 7dimensional diagrams with large c 3 . It is still an open issue why such dramatical changes occur in 8-dimensions.

E. Summary of Numerical Results
In this section, we have numerically checked the condition for the instability and ghost.
Here, we summarize the results.
We numerically examine the behavior of T (r) for various Lovelock  is manner of tensor-unstable region near |Q| = 0 when there is no scalar-unstable region.
For this case, in our numerical calculation, there must exist slight mass range in which black holes have the instability under tensor perturbations. Therefore, whether scalar-unstable region exists or not, when black holes with nearly extreme mass have slight charge, they must be unstable; which type instability they have depends on parameters and dimensions, but they have at least one type instability.
These results also lead some open questions. The first is dimensionality. One of future works is the exploration of more general conditions for stability under scalar type perturbations. In this paper, we have shown that black holes are unstable if 2T ′2 − T T ′′ has negative region. However, the inverse statement has not been proved. Hence, so far, we can not say anything when 2T ′2 − T T ′′ is always positive. Furthermore, by this criterion, we can not detect the instability under scalar type perturbation in Einstein theory [9]. In this sense, it is interesting to find out more general conditions.
It is interesting to investigate the relation between dynamical instability we have shown in this paper and thermodynamics. On the thermodynamics for Lovelock black holes, variation of Lovelock coefficients is also examined in Ref. [23]. In our paper, we have found that the appearances of Q − µ diagrams change dramatically as Lovelock coupling c 3 varies in 8 dimensions. Then it is interesting if such dramatical changes are found in the thermodynamics.
The relation between instability and gravitational collapses might be important. In Lovelock theory, collapses of dust clouds have been examined [24]. Furthermore, these are extended to charged dust clouds [25]. In these, dependence of dimensions are found for, for example, naked singularity formations. Our results also depends dimensions, so there may exit relations between the instability and the gravitational collapse in Lovelock gravity. In dust collapses, the authors of above papers have also pointed out the tendency that higher curvature collections suppress formations of apparent horizons. These results should express that higher curvature collections make attractive force weaker, and this property might be related to the instability of Lovelock black holes we discussed in this paper.