Fatal Effects of Charges on Stability of Black Holesin Lovelock Theory

Fatal Effects of Charges on Stability of Black Holes
in Lovelock Theory

Tomohiro Takahashi

1 Introduction

One of the most exciting predictions of the braneworld with large extra-dimensions is the possibility of higher dimensional black hole creation at the LHC. [1]  Then, it is important to study higher dimensional black holes. Especially, stability analysis of higher dimensional black holes is important because unstable black hole solutions are not attractor solutions; that is, black holes with instability must not be created at the LHC.

The stability of higher dimensional black holes in Einstein theory has been intensively studied. The most famous one is stability analysis of higher dimensional Schwarzschild black hole by Kodama and Ishibashi.[2] They have shown that higher dimensional Schwarzschild black holes are stable for all type perturbations. They have also extended their works to charged black holes in higher dimensions and derived master equations.[3] Furthermore, for charged black holes, numerical studies have also been done.[4] For the stability analysis of rotating black holes in higher dimensions, a group theoretical method is developed,[5]  which is used for 5-dimensional rotating black holes with equal angular momenta[6]  and also used to study the stability of squashed black holes.[7, 8, 9] The stability of a special class of rotating black holes in more than 5-dimensions is also studied. [10, 11, 12]

As mentioned above, many studies have been done based on Einstein theory. It is no less important than these to examine the stability of higher dimensional black holes in more general gravitational theories. This is because Einstein theory is not the most general gravitational theory which contains terms only up to the second order derivatives of metric in the equations of motion. The most general theory in this sense is Lovelock theory.[13]  Furthermore, because black hole productions occur at the fundamental scale, Einstein theory is not reliable any more. In fact, it is known Einstein theory is only a low energy limit of string theory.[14]  In string theory, there are higher curvature corrections in addition to the Einstein-Hilbert term.[14]  Thus, it is natural to extend gravitational theory into those with higher curvature corrections in higher dimensions. Lovelock theory belongs to such class of theories.[13, 15]  Then, it is worthy to extend the stability analysis to this general Lovelock theory.

In Lovelock theory, it is known that there exist static spherical symmetric black hole solutions [15, 16] (and also topological black hole solutions are found [17]). For this static spherical symmetric black holes, stability analyses under all type perturbations have been performed. [18, 19, 20, 21] It is shown that there exists the scalar mode instability in odd dimensions, the tensor mode instability in even dimensions and no instability under vector type perturbation in all dimensions. In 2nd order Lovelock theory, stability analysis is extended to black string. [22] They have shown that scalar perturbations have an exponentially decaying behavior under s-mode approximations.

However, because black holes may be produced by proton-proton collision at the LHC, it is natural to suppose these black holes have charges. In Lovelock-Maxwell theory, there exist static spherical symmetric charged black hole solution, namely charged Lovelock black hole solution. [15] Then, it is important to extend the above discussion to this charged solution.

In this paper, we study the linear stability of charged Lovelock black hole solutions. Our purpose is examining the response of instability to the charge and considering the black holes creation in Lovelock theory at LHC. In this paper, for the first step, we only concentrate on tensor type perturbation . Furthermore, we mainly consider the second order and third order Lovelock theory for simplicity, although it is important to consider higher Lovelock terms. [23]

The organization of this paper is as follows. In section 2, we review Lovelock theory and confirm the existence of asymptotically flat black hole solutions. In section 3, we consider tensor perturbations and denote the condition for stability of charged Lovelock black holes. The method using in this section is the same as our previous paper.[21]  In section 4, we check the criteria presented in section 3 numerically. We mainly examine the second order and third order Lovelock theory in this section. The final section 5 is devoted to the conclusion.

2 Charged Lovelock Black Hole Solutions

In this section, we review Lovelock theory and introduce charged black hole solutions.

In Ref. 13, the most general symmetric, divergence free rank (1,1) tensor is constructed out of a metric and its first and second derivatives. The corresponding Lagrangian can be constructed from -th order Lovelock terms

(2.1)

where is the Riemann tensor in -dimensions and is the generalized totally antisymmetric Kronecker delta defined by

Then, Lovelock Lagrangian in -dimensions is defined by

where we defined the maximum order and are arbitrary constants. Here, represents the maximum integer satisfying . Then, fixing , satisfies or . Hereafter, we set , and , for convenience.

Then, the action for Lovelock-Maxwell system is

(2.2)

where means field strength of electromagnetic field and is vector potential. Taking variation of this action with respect to , we can derive

(2.3)

where is Lovelock tensor defined as

(2.4)

and

(2.5)

is energy momentum tensor of Maxwell field. Furthermore, by varying (2.2) by , we can get parts of Maxwell equations

(2.6)

and the rests of Maxwell equations can be derived from the identity , which means

(2.7)

As shown in Ref 15, there exist static spherical symmetric solution of these equations. Let us consider the following metric

(2.8)

where means metric of and . Using this, Lovelock tensor is calculated as follows:

(2.9)

where is defined as

and is

(2.10)

For Maxwell field, we assume spherical symmetric electric field, that is,

(2.11)

Substituting this ansatz for (2.6) shows that satisfies

(2.12)

where is constant of integral which means charge of black hole. The normalization factor is only for convenience. Note that (2.11) and (2.12) satisfies the identity (2.7). Hence, this is exact solution for Maxwell equations. The energy-momentum tensor for this solution is

(2.13)

The results (2.9) and (2.13) leads Lovelock equation (2.4) as

(2.14)

Note that these equations are not independent. In fact, a derivative of the first equation with respect to leads the second equation. Therefore, it is sufficient to consider only the first equation. Integrating this equation leads

(2.15)

In (2.15), is constant of integral which is related to ADM mass as

(2.16)

where we used a unit . We can get this result from the asymptotic behavior [24] and also gain by a bzackground-independent formalism. [25]

In this paper, we want to concentrate on asymptotically flat, i.e. , solutions with a positive ADM mass because such black holes could be created at the LHC. We also assume that Lovelock coefficients satisfy

(2.17)

for simplicity. Furthermore, for numerical calculation, we nondimensionalize all variables. Our choice means only scale of length is not fixed. There exist many candidates, but we use for nondimensionalization in this paper. For example, radial can be nondimensionalized as . From , should be nondimensionalized as and this nondimensionalization leads that eq. (2.15) is expressed as

(2.18)

where

(2.19)

are nondimensionalized Lovelock coefficients, nondimensionlized mass parameter and nondimensionalized charge parameter respectivrely. After this, we miss tilde, that is, means nondimensionlized radius for example.

Fig. 1: In this figure, the curve means and the dotted line is . The cross points of these mean the roots of (2.18). Note that radial is fixed in this figure so is constant.

It is easily seen that (2.18) has many branches. However, under our assumptions (2.17), an asymptotic flat solution must exist. For example, in case, (2.18) leads two branches

(2.20)

In these two branches, the upper branch is asymptotic flat branch because this behaves as in asymptotic region.

Against case, it is difficult to solve (2.18) in general cases. However, we can recognize that there must exist an asymptotic flat branch by using graphically method which is expressed in Fig.1. In this figure, we fix and seek the corresponding roots , that is, . In detail, we set the both sides of (2.18) as and draw two curves and in plane. The cross points of these curves correspond . In order to check the existence of asymptotic flat branch, we must take care two points. The first point is behavior of in . Using our assumptions (2.17), it is easily seen that and is monotonically increasing in . The second point is behavior of , especially in asymptotic region.

Fig. 2: This figure shows the behavior of . is always negative in . is monotonically increasing in , at and is monotonically decreasing in . In asymptotic region , this function approaches

This function behaves as Fig.2. From this figure, it is easily seen that is always positive in asymptotic region and as . Combining these two points and Fig.1, we can know that there must exist a cross point of and in as long as we consider asymptotic region; that is, a root of (2.18) which satisfy must exist in asymptotic region. And this root also satisfies as because in this limit. Considering eq.(2.18), means and which means this root expresses asymptotic flat. Note that this result also reminds us that the constant of integration is proportional to ADM mass. After this, we only consider this asymptotic flat branch.

From Fig.1, this asymptotic flat branch behaves as r moves , and becomes negative in because is negative in this region. When becomes still smaller, this branch runs into singularity where Kretschmann invariant diverges. This variable can be calculated as

from metric ansatz (2.8). Hence, there are singularities at and also exist where derivatives of diverge. Especially, for the latter, the first derivative of is

(2.21)

so Kretschmann invariant diverges at , that is, singularities also exist where becomes extreme value.

Fig. 3: (a): has no extreme value in . (b): has extreme values in . Whether has extreme value or not depends on Lovelock coefficients and .

This means that position of singularity is different between Fig.3-(a) and Fig.3-(b). If has no extreme values in (Fig.3-(a)), there is a singularity at : if has extreme values at (Fig.3-(b)), there is a singularity at . In either case, there is a singularity in .

From the viewpoint of cosmic censorship conjecture, there must exist horizons outside of singularities. The branch we concentrate on is static and asymptotic flat, so event horizon coincides with killing horizon. Then horizon radius is characterized as

(2.22)

which means must be positive. This result and behavior of show that , if exists, is larger than ; that is, there exists no naked singularity if has solutions. Hence, what we have to examine is conditions for existence of horizons. Horizons are characterized by (2.22) and also satisfies

Therefore horizons are determined by two equations

(2.23)
Fig. 4: In this figure, we draw two lines and in plane. The cross point means . Under our assumption (2.17), is monotonic function which diverges near and approaches as .

Substituting the first equation for the second equation leads

(2.24)

The solutions of this equation correspond horizon, so the behavior of determines the condition for existence of horizons. The derivative of with respect to is

(2.25)

and solving with respect to leads

(2.26)

Note that or . Then powers of all terms in are negative and their coefficients are positive. Therefore, is monotonically decreasing function like Fig.4. From this figure, there must exist a root of (2.26). We call this because becomes extreme minimum at which can be seen from (2.25).

Fig. 5: In this figure, and are drawn. The cross points of these lines correspond . From this figure, it can be easily seen that there exist horizons if .

Then, behaves as Fig.5. In this figure, we solve eq.(2.24) graphically. The cross points of and mean horizons. From this figure, it can be easily seen that there exist horizons if where is defined as

(2.27)

Note that and in Fig.5 means outer horizon and inner horizon respectively because corresponds horizon radius as .

Finally, for example, we examine in 5-dimensions ( and case). From eq.(2.26), in 5-dimensions is

so substituting this for (2.27) leads

(2.28)

3 Condition for Stability under Tensor Type Perturbations

In this section, we consider conditions for stability under tensor type perturbations. Note that we only consider which means there exists horizon and we only consider the perturbation outside of the outer horizon.

The background metric (2.8) has dimensional spherical symmetry, so perturbations are decomposed into scalar type, vector type and tensor type. In this paper, we concentrate on tensor type perturbations which are characterized as

(3.1)

where means master variable and is tensor harmonics. Tensor harmonics satisfies , and where . Here means covariant derivative with respect to metric . Note that there is no tensorial perturbation for Maxwell field , which means that the first order perturbation of energy momentum tensor . Therefore, the first order perturbation of EOM is and this can be calculated as follows; [21]

(3.2)

Here is defined as

(3.3)

Now we will present the condition for the stability of the solutions we are considering in this paper.

As we will soon see, the master equation (3.2) can be transformed into the Schrdinger form. To do this, we have to impose the condition

(3.4)

In fact, this is necessary for the linear analysis to be applicable. In the case that there exists such that and , we encounter a singularity. Using approximations , and , (3.2) approximately becomes

(3.5)

This shows that near , behaves as , where and are constants of integration. Hence, the solution is singular at for generic perturbations. The similar situation occurs even in cosmology with higher derivative terms.[26, 27]  In those cases, this kind of singularity alludes to ghosts. Indeed, if there is a region outside the horizon, the kinetic term of perturbations has a wrong sign. Hereafter, we call this the ghost instability.

When the condition (3.4) is fulfilled, introducing a new variable and switching to the coordinate , defined by , we can rewrite Eq.(3.2) as

(3.6)

where

(3.7)

is an effective potential.

For discussing the stability, the “S-deformation” approach is useful.[2, 18]  Let us define the operator

(3.8)

acting on smooth functions defined on . Then, (3.6) is the eigenequation and is eigenvalue of . We also define the inner products as

(3.9)

In this case, for any smooth function with compact support in , we can find a smooth function such that

(3.10)

where we have defined

(3.11)

Following Ref. 18, we choose to be

(3.12)

Then, we obtain the formula

(3.13)

Here, the point is that the second term in (3.13) includes a factor , but does not include . Hence, by taking a sufficiently large , we can always make the second term dominant.

Now, let us show that the sign of determines the stability. If on , the solution (2.8) is stable. This can be understood as follows. Note that , then we have for this case. That means for arbitrary if on . We choose, for example, as the lowest eigenstate, then we can conclude that the lowest eigenvalue is positive. Thus, we proved the stability. The other way around, if at some point in , the solution is unstable. To prove this, the inequality

(3.14)

is useful. This inequality is correct for arbitrary , which is not necessarily an eigenfunction of , as far as the left hand side of Eq. (3.14) make sense. If at some point in , we can find such that

(3.15)

In this case, (3.13) is negative for sufficiently large . Then, the inequality (3.14) implies and the solution has unstable modes. Thus, we can conclude that the solution is stable if and only if on .

From the above logic, if has a negative region, negative states exist. Therefore, this instability is dynamical. Then, we call this as dynamical instability in order to distinguish this from the ghost instability which is caused by negativity of .

We want to summarize this section. If has negative region outside the outer horizon , this solution has the ghost instability. Even if is always positive, this solution has the dynamical instability if has a negative region outside the outer horizon. Therefore, charged Lovelock black holes are stable under tensor perturbations if and only if and are always positive outside the outer horizon.

Note that is calculated as in case from (2.20). This leads

(3.16)

which means there exist no ghost instabilities in 5 and 6 dimensions.

4 Numerical Results

In this section, we examine “whether has negative region or not” and “whether has negative region or not” numerically. The detail is as follows. Note that and are parameters for numerical calculations.

Firstly, we fix . Using this , extreme mass can be determined from (2.27). Then, we change from to by . For each , we can gain with numerically from (2.18) and determine and . Then, checking whether these functions have negative region or not, we determine, for example, the border between stable and unstable which we call . Then, we change by and do the same calculation. We repeat this calculation until .

Note that we examine the region and the mesh size is .

4.1 5-dimensional Case

Numerical results of 5-dimensional case are Fig.6 and Fig.7. The former is diagram near and the latter is that of . It can be seen that black holes with are stable, which agrees with our previous results. [21] When black holes are a little charged up, however, there exists an unstable region near extreme mass (Fig.6) and this region vanishes in (Fig.7). As we have already discussed, there is no ghost instability in 5-dimensions. Hence, these figures show that nearly extreme black holes are unstable if .

Fig. 6: diagram near in 5-dimensions. This is calculated with and . And is when and in . The linearity of extreme mass line is guaranteed by (2.28)
Fig. 7: diagram near in 5-dimensions. This is calculated with and . In this figure, the difference between and extreme mass is where and there is no instability where .

4.2 6-dimensional Case

Fig.8 and Fig.9 are the numerical results in 6-dimensions. The former is diagram near and the latter is that of . When black holes are neutral, there is unstable region in ; this agrees with our previous results. [21] When black holes are a little charged up, there also exist an unstable region (Fig.8). However, this region vanishes in (Fig.9). As we have already discussed, there is no ghost instability in 6-dimensions. Therefore, it is shown numerically that nearly extreme black hole is unstable when its charge satisfies .

Fig. 8: diagram near in 6-dimensions. This is calculated with and . And is when and in .
Fig. 9: diagram near in 6-dimensions. This is calculated with and . In this figure, the difference between and extreme mass is in and there is no instability in .

4.3 7-dimensional case

The previous works tell that neutral black holes do not have ghost instability if and have this instability if . Then, in this paper, we concentrate on case and case.

4.3.1 case

Fig.10 and Fig.11 are the numerical results when in 7-dimension. The former is diagram near and the latter is that of . In Fig.10, there are no unstable region and no ghost region, which are consistent with our previous results. [21] However, when a little charged up, black hole has dynimical instability if its mass is as small as extreme mass. And this instability vanishes when (Fig.11). Note that ghost region cannot be detected by our numerical calculation with , =1 and . Therefore, same as 5-dimensional case, these result means that there exists an unstable region near extreme mass in .

Fig. 10: diagram near when in 7-dimensions. This is calculated with and . And is when and in .
Fig. 11: diagram near when in 7-dimensions. This is calculated with , and . In this figure, the difference between and extreme mass is in and there is no instability in .

4.3.2 case

Fig.12 and Fig.13 are numerical results when in 7-dimensions. The former is diagram near and the latter is that of . In Fig.12, there exists a ghost region if and . This result agrees with the previous work [21]. As black hoes are charged up, this ghost region diminishes and unstable region appears. And this unstable region vanishes if (Fig.13).

Fig. 12: diagram near when in 7-dimensions. is calculated with and . Ghost and extreme mass are done with and . This figure shows that there exists a ghost region if and there exists an unstable region if .
Fig. 13: diagram near when in 7-dimensions. This is calculated with , and . In this figure, the difference between and extreme mass is in and there is no instability in .

4.4 8-dimensional Case

If black holes are neutral, in 8-dimensions, the previous works show that there is no ghost instability when and there exists if . Therefore, we concentrate on case and case in this paper.

4.4.1 case

Fig.14 and Fig.15 are the numerical results when in 8-dimension. The former is diagram near and the latter is that of . In Fig.14, black holes with and are unstable, which agrees with our previous works. [21] When a little charged up, black hole has also instability if its mass is as small as extreme mass. However this instability vanishes when (Fig.15). Note that ghost cannot be found in our numerical calculation with , =5 and . Therefore, same as 6-dimensional case, these result means that there exists an unstable region near extreme mass in .

Fig. 14: diagram near when in 8-dimensions. This is calculated with and .
Fig. 15: diagram near when in 8-dimensions. This is calculated with and . In this figure, the difference between and extreme mass is in and there is no instability in .

4.4.2 case

Fig.16 and Fig.17 are numerical results when in 8-dimensions. The former is diagram near and the latter is that of . In Fig.16, there exists a ghost region if