Thermodynamic and classical instability of AdS black holes in fourth-order gravity

# Thermodynamic and classical instability of AdS black holes in fourth-order gravity

Yun Soo Myung a    and Taeyoon Moon Institute of Basic Science and Department of Computer Simulation
Inje University
Gimhae 621-749, Korea
###### Abstract

We study thermodynamic and classical instability of AdS black holes in fourth-order gravity. These include the BTZ black hole in new massive gravity, Schwarzschild-AdS black hole, and higher-dimensional AdS black holes in fourth-order gravity. All thermodynamic quantities which are computed using the Abbot-Deser-Tekin method are used to study thermodynamic instability of AdS black holes. On the other hand, we investigate the -mode Gregory-Laflamme instability of the massive graviton propagating around the AdS black holes. We establish the connection between the thermodynamic instability and the GL instability of AdS black holes in fourth-order gravity. This shows that the Gubser-Mitra conjecture holds for AdS black holes found from fourth-order gravity.

## 1 Introduction

Concerning the thermodynamic analysis of a black hole, the Schwarzschild black hole in Einstein gravity is in an unstable equilibrium with the heat reservoir of the temperature  GPY . Its fate under small fluctuations will be either to decay to hot flat space by emitting Hawking radiation or to grow without limit by absorbing thermal radiations in the infinite heat reservoir York . This means that an isolated black hole is never in thermal equilibrium in asymptotically flat spacetimes because of its negative heat capacity. Thus, one has to find a way of getting a stable black hole which might be in an equilibrium with a finite heat reservoir. A black hole could be rendered thermodynamically stable by placing it in four-dimensional anti-de Sitter (AdS) spacetimes because AdS spacetimes play the role of a confining box. An important point to understand is to know how a stable black hole with positive heat capacity could emerge from thermal radiation through a phase transition. The Hawking-Page (HP) phase transition occurs between thermal AdS spacetimes (TAdS) and Schwarzschild-AdS (SAdS) black hole HP ; BCM , which is known to be one typical example of the first-order phase transition in the gravitational system. Its higher dimensional extension and the AdS/CFT correspondence of confinement-deconfinement phase transition were studied in Witt .

To study the HP phase transition in Einstein gravity explicitly, we are necessary to know the Arnowitt-Deser-Misner (ADM) mass Arnowitt:1962hi , the Hawking temperature, and the Bekenstein-Hawking (BH) entropy. These are combined to give the on-shell Helmholtz free energy in canonical ensemble which determines the global thermodynamic stability. The other important quantity is the heat capacity which determines the local thermodynamic stability. If one uses the Euclidean action approach, one also finds these quantities consistently dbranes .

However, the black hole thermodynamics was not completely known in fourth-order gravity because one has encountered some difficulty to compute their conserved quantities in asymptotically AdS spacetimes exactly. Recently, there was some progress on computation scheme of mass and related thermodynamic quantities by using the Abbot-Deser-Tekin (ADT) method Abbott:1981ff ; Deser:2002jk . The ADM method is suitable for computing conserved quantities of a black hole in asymptotically flat spacetimes, while the ADT method is useful to compute conserved quantities of a black hole in asymptotically AdS spacetimes found from fourth-order gravity Kim:2013zha . After computing all ADT thermodynamic quantities depending on a mass parameter , one is ready to study thermodynamics and phase transition between TAdS and AdS black hole in fourth-order gravity. For with critical mass parameter giving , all thermodynamic properties are dominantly determined by Einstein gravity, while for , all thermodynamic properties are dominantly by Wely-squared term. The former is completely understood, but the latter becomes a new area of black hole thermodynamics appeared when one studies the black hole by using the ADT thermodynamic quantities.

On the other hand, there was a connection between thermodynamic instability and classical [Gregory-Laflamme (GL) Gregory:1993vy ] instability for the black strings/branes. This Gubser-Mitra proposal Gubser:2000ec was referred to as the the correlated stability conjecture (CSC) Harmark:2007md which states that the classical instability of a black string/brane with translational symmetry and infinite extent sets in precisely when the corresponding thermodynamic system becomes (locally) thermodynamically unstable (Hessian matrix111There are two representations when defining the Hessian matrix: and . The matrix () can be expressed in terms of the second-order derivatives of the entropy (mass) with respect to the mass (entropy) and the conserved charges. Here, Hessian matrix denotes a negative eigenvalue of the matrix . or heat capacity ). Here the additional assumption of translational symmetry and infinite extent has been added to ensure that finite size effects do not spoil the thermodynamic nature of the argument and to exclude a well-known case of the Schwarzschild black hole which is classically stable, but thermodynamically unstable because of its negative heat capacity.

Interestingly, it is very important to mention that the stability of the Schwarzschild black hole in four-dimensional massive gravity is determined by using the GL instability of a five-dimensional black string. Although the Schwarzschild black hole stability has been performed in Einstein gravity forty years ago  Regge:1957td ; Zerilli:1970se ; Vishveshwara:1970cc , the stability analysis of the Schwarzschild black hole in massive gravity theory were very recently announced. The massless spin-2 graviton has 2 degrees of freedom (DOF) in Einstein gravity, while the massive graviton has 5 DOF in massive gravity theory. Even a massive spin-2 graviton has 5 DOF, one has a single physical DOF when one considers the -mode of massive graviton. Also, it was proved that the -wave perturbation gives unstable modes only in the higher dimensional black string perturbation Kudoh:2006bp . It turned out that the small Schwarzschild black holes in the dRGT massive gravity Babichev:2013una ; Brito:2013wya and fourth-order gravity Myung:2013doa ; Myung:2013cna are unstable against the metric and Ricci tensor perturbations, respectively. This implies that the massiveness of gives rise to unstable modes propagating around the Schwarzschild black hole. If one may find thermodynamic instability from the ADT thermodynamic quantities of AdS black hole in fourth-order gravity, then it could be compared with the GL-instability found from the linearized Einstein equation Myung:2013bow . If one finds a connection between them, it might imply that the Gubser-Mitra conjecture holds even for a compact object of the SAdS black hole found in fourth-order gravity. This is our main motivation of why we study fourth-order gravity here.

In this work, we investigate thermodynamic and classical instability of AdS black holes in fourth-order gravity. These include the BTZ black hole in new massive gravity, Schwarzschild-AdS black hole and higher-dimensional AdS black holes in fourth-order gravity. All thermodynamic quantities are computed using the ADT method. Here we use the ADT conserved quantities, since they respect the first-law of thermodynamics and the ADT mass and entropy are reliable to use a thermodynamic study of the AdS black holes in fourth-order gravity. Finally, we establish a connection between the thermodynamic instability of AdS black holes and the GL instability of AdS black holes in the linearized fourth-order gravity.

## 2 BTZ black hole in new massive gravity

As a prototype, we consider the BTZ black hole in new massive gravity (NMG) which is known to be a three-dimensional version of fourth-order gravity. The NMG action Bergshoeff:2009hq composed of the Einstein-Hilbert action with a cosmological constant and fourth-order curvature terms is given by

 SNMG = SEH+SFOT, (1) SEH = 116πG3∫d3x√−g (R−2λ), (2) SFOT = −116πG3m2∫d3x√−g (RμνRμν−38R2), (3)

where is a three-dimensional Newton constant and a positive mass parameter with mass dimension 2 []. In the limit of , recovers the Einstein gravity , while it reduces to purely fourth-order term in the limit of . The field equation is given by

 Rμν−12gμνR+λgμν−12m2Kμν=0, (4)

where

 Kμν = 2□Rμν−12∇μ∇νR−12□Rgμν (5) + 4RμρνσRρσ−32RRμν−RρσRρσgμν+38R2gμν.

The non-rotating BTZ black hole solution to Eq.(4) is given by BTZ-1 ; BTZ-2

 ds2BTZ=¯gμνdxμdxν=−f(r)dt2+dr2f(r)+r2dϕ2,  f(r)=−M+r2ℓ2 (6)

under the condition of . Here is an integration constant related to the the ADM mass of black hole. The horizon radius is determined by the condition of and denotes the curvature radius of AdS spacetimes.

Its Hawking temperature is found to be

 TH=f′(r+)4π=r+2πℓ2. (7)

Using the ADT method, one can derive all thermodynamic quantities of its mass Clement:2009gq , heat capacity (), entropy Kim:2013qra , and on-shell (Helmholtz) free energy

 (8)

whose thermodynamic quantities in Einstein gravity have already given by Myung:2005ee ; Myung:2006sq ; Eune:2013qs

 M(r+)=r2+8G3ℓ2, C(r+)=πr+2G3, SBH(r+)=πr+2G3, Fon(r+)=M−THSBH=−r2+8G3ℓ2. (9)

These all are positive regardless of the horizon size except that the free energy is always negative. This means that the BTZ black hole is thermodynamically stable in Einstein gravity. Here we check that the first-law of thermodynamics is satisfied as

as the first-law is satisfied in Einstein gravity

 dM=THdSBH (11)

where ‘’ denotes the differentiation with respect to the horizon size only. In this work, we treat differently from the black hole charge and angular momentum to achieve the first-law (10). Here we observe that in the limit of one recovers thermodynamics of the BTZ black hole in Einstein gravity, while in the limit of we recover the black hole thermodynamics in purely fourth-order gravity which is similar to recovering the conformal Chern-Simons gravity from the topologically massive gravity (TMG) Bagchi:2013lma .

On the other hand, the linearized equation to (4) upon choosing the transverse-traceless (TT) gauge of and leads to the fourth-order equation for the metric perturbation  Myung:2011bn ; Moon:2013jna

 (¯∇2−2Λ)[¯∇2−2Λ−M2(m2)]hμν=0,  Λ=−1ℓ2 (12)

which might imply the two second-order linearized equations

 (¯∇2−2Λ)hμν=0, (13) [¯∇2−2Λ−M2(m2)]hμν=0. (14)

Here the mass squared of a massive spin-2 graviton is given by

 M2(m2)=m2−12ℓ2. (15)

Eq. (14) describes a massive graviton with 2 DOF propagating around the BTZ black hole under the TT gauge.

Expressing all thermodynamic quantities in (8) in terms of the mass squared leads to

which shows clearly that all thermodynamical quantities depend on the sign of . The local thermodynamic stability is determined by the positive heat capacity () and the global stability is determined by the negative free energy (). Hence, it implies that the thermodynamic stability is determined by the sign of the heat capacity, while the phase transition is mainly determined by the sign of the free energy.

For , all thermodynamic quantities have the same property as those for Einstein gravity (2), whereas for , all thermodynamic quantities have the same property as those for fourth-order term (3). We observe from Fig. 1 that for , the BTZ black hole is thermodynamically stable regardless of the horizon size because of .

On the other hand, the classical (in) stability condition of the BTZ black hole was recently determined by the condition of regardless of the horizon size  Moon:2013lea . The case of corresponds to the critical gravity where all thermodynamical quantities are zero and logarithmic modes appear. For , the BTZ black hole is thermodynamically unstable because of as well as it is classically unstable against the metric perturbations. Hence, it shows a clear connection between thermodynamic and classical instability for the BTZ black hole regardless of the horizon size in new massive gravity.

Finally, let us turn to the issue related to a phase transition from the thermal AdS (TAdS) to BTZ black hole. For this purpose, we first consider thermodynamic quantities for the TAdS Myung:2006sq

It turns out that for [fourth-order term (3) contributes dominantly to black hole thermodynamics], the TAdS is always favored than the BTZ black hole because of , where is given by (16). In this case, we might not define a possible phase transition because the ground state is the TAdS. Alternatively, this implies that a gab between and might not allow a continuous phase transition222Concerning this issue, a phase transition between hot flat space and flat space cosmological spacetimes was recently studied in TMG by using on-shell free energies Bagchi:2013lma .. It is noted that a phase transition from the TAdS to the BTZ black hole is possible to occur in new massive gravity for (see arXiv:1311.6985v1 for details), in which the Einstein gravity (2) contributes dominantly to black hole thermodynamics. The corresponding phase transition in this case is similar to that obtained in the literature Myung:2012xc for .

## 3 Thermodynamics of AdS black holes in fourth-order gravity

 SdFO=116πGd∫ddx√−g[R−(d−2)Λ0+αRμνRμν+βR2] (18)

with two parameters and . Here we do not include the Gauss-Bonnet term Deser:2011xc because (18) admits solutions of the higher-dimensional Einstein gravity including the higher dimensional AdS black holes. From (18), the Einstein equation is derived to be

 Gμν+Eμν=0, (19)

where the Einstein tensor is given by

 Gμν=Rμν−12Rgμν+d−22Λ0gμν (20)

and takes the form

 Eμν = 2α(RμρνσRρσ−14RρσRρσgμν)+2βR(Rμν−14Rgμν) (21) + α(∇2Rμν+12∇2Rgμν−∇μ∇νR)+2β(gμν∇2R−∇μ∇νR).

For the Einstein space of and together with , Eq.(19) allows a -dimensional AdS black hole solution

 ds2ST=¯gμνdxμdxν=−V(r)dt2+dr2V(r)+r2dΩ2d−2 (22)

with the metric function

 V(r)=1−(r0r)d−3−Λd−1r2,  Λ=−d−1ℓ2. (23)

The horizon is located at () which means that differs from . Hereafter we denote the background quantities with the “overbar”. In this case, the black hole background spacetimes is given by

 ¯Rμν=Λ¯gμν,  ¯R=4Λ. (24)

The Hawking temperature is derived as

 TdH=V′(r+)4π=14πr+[(d−3)+d−1ℓ2r2+]. (25)

Using the ADT method Abbott:1981ff ; Deser:2002jk , all thermodynamic quantities of its mass Liu:2011kf , heat capacity, entropy Liu:2011kf , and on-shell free energy are given by

where

 m2d=1β,  α=−4(d−1)dβ,  M2d(m2d)=d4[m2dd−1−2(d−2)2dℓ2]. (27)

At this stage, we note that even though all thermodynamic quantities are obtained for arbitrary and , we require a condition of because the classical stability could be achieved only under this condition. This means that we have a single mass parameter by eliminating a massive spin-0 graviton in fourth-order gravity, reducing to the Einstein-Weyl gravity. All thermodynamic quantities in -dimensional Einstein gravity were known to be  Myungtads

 Md(r+) = Ωd−2(d−2)16πGdrd−3+[1+r2+ℓ2], (28) Cd(r+) = dMddTdH=Ωd−2(d−2)rd−2+4Gd[(d−1)r2++(d−3)ℓ2(d−1)r2+−(d−3)ℓ2], (29) SBH(r+) = Ωd−24Gdrd−2+, (30) Fond(r+) = Md−THSBH=Ωd−216πGdrd−3+[1−r2+ℓ2] (31)

with the area of

 Ωd−2=2πd−12Γ(d−12). (32)

We observe from (26) that for , thermodynamic stability of AdS black hole is determined by the higher-dimensional Einstein gravity. On the other hand, for , thermodynamic stability of black hole is determined by Weyl-squared term (conformal gravity).

We check that the first-law of thermodynamics is satisfied as

as the first-law is satisfied in -dimensional Einstein gravity

 dMd=TdHdSBH, (34)

where ‘’ denotes the differentiation with respect to the horizon size only. In this work, we treat differently from the black hole charge and angular momentum to obtain the first-law (33). Here we observe that in the limit of we recovers thermodynamics of the AdS black hole in -dimensional Einstein gravity, while in the limit of we recover that in Weyl-squared term (conformal gravity).

## 4 SAdS black hole in Einstein-Weyl gravity

### 4.1 Thermodynamic instability for small black holes

For a definite description of black hole thermodynamics, we choose which provides a SAdS black hole. Its thermodynamic quantities of mass Lu:2011zk , heat capacity, entropy Lu:2011zk , and on-shell free energy are given by

where all thermodynamic quantities of SAdS black hole are shown in the Eqs. (28)-(31) for . The mass squared takes the form Lu:2011zk

 M2d=m2d3−2ℓ2 (39)

which is negative/postive for .

First of all, we depict the free energy as a function of in Fig. 2. Since the sign of free energy depends on the horizon size critically, we plot the two free energies as function of for large ( and small black hole (), respectively. For , one has and , while for , one has and .

We consider first the case of which is dominantly described by the Einstein gravity. Since the heat capacity of blows up at , we divide the black hole into the small black hole with and the large black hole with As is shown in the solid (familiar) curves in Fig. 3, we have the small black hole with which is thermodynamically unstable because , while the large black hole with is thermodynamically stable because .

Especially for , one has , and . Hence we could describe the Hawking-Page phase transition well as for the SAdS black hole in Einstein gravity HP . We wish to comment that the free energy has the maximum value at . Since the free energy becomes negative (positive) for , we did not choose as a boundary point to divide the black hole into small and large black holes.

On the other hand, for which is dominantly described by conformal gravity Lu:2012xu , the small black hole () is thermodynamically stable because , while the large black hole () is thermodynamically unstable because . See the dotted (unfamiliar) curves in Fig. 3 for observation. It seems that there is no known phase transition from thermal AdS to the SAdS black hole in conformal gravity.

### 4.2 GL instability for small black holes

We briefly review the Gregory-Laflamme -mode instability for a massive spin-2 graviton with mass propagating on the SAdS black hole spacetimes in Einstein-Weyl gravity. Choosing the TT gauge, its linearized equation to (19) takes the form

 ¯∇2hμν+2¯Rαμβνhαβ−M2dhμν=0. (40)

which describes 5 DOF of a massive spin-2 graviton propagating on the SAdS black hole spacetimes. We note that choosing the condition of eliminates a massive spin-0 graviton with 1 DOF.

Before we proceed, we wish to mention that the stability of the Schwarzschild black hole in four-dimensional massive gravity is determined by using the Gregory-Laflamme instability of a five-dimensional black string. It turned out that the small Schwarzschild black holes in the dRGT massive gravity Babichev:2013una ; Brito:2013wya and fourth-order gravity Myung:2013doa are unstable against the metric and Ricci tensor perturbations because the inequality is satisfied as

 Md≤O(1)r0,  r0=2MS. (41)

For the massless case of , Eq. (40) leads to the linearized equation around the Schwarzschild black hole with the TT gauge which is known to be stable in the Einstein gravity.

Choosing the -mode ansatz whose form is given by and as

 hsμν=eΩt⎛⎜ ⎜ ⎜ ⎜⎝Htt(r)Htr(r)00Htr(r)Hrr(r)0000K(r)0000sin2θK(r)⎞⎟ ⎟ ⎟ ⎟⎠, (42)

a relevant equation for takes the same form (see Appendix for explicit forms of )

 A(r;r0,ℓ,Ω2,M2d)d2dr2Htr+BddrHtr+CHtr=0, (43)

which shows the same unstable modes for

 0

with the mass

 Md=√m2d3−2l2. (45)

The condition of (44) could be read off from Fig. 4 when one notes the difference between and : and .

On the other hand, the stable condition of the SAdS black hole in Einstein-Weyl gravity is given by

 Md>O(1)r0. (46)

At this stage, we would like to mention the classical stability of case. In this case, its linearized equation reduces to

 ¯∇2hμν+2¯Rαμβνhαβ=0, (47)

which is exactly the linearized equation around the SAdS black hole in Einstein gravity. From the observation333In the next section 5.2, we introduce the corresponding numerical analysis to observe this GL instability. of Fig. 4, the GL instability disappears at , which may imply that the SAdS black hole is stable against the -mode metric perturbation. The SAdS black hole was known to be stable against the metric perturbation even though a negative potential appeared near the event horizon in odd-parity sector Cardoso:2001bb . Later on, one could achieve the positivity of gravitational potentials by using the -deformed technique Ishibashi:2003ap , proving the stability of SAdS black hole exactly Moon:2011sz . This implies that there is no connection between classical stability and thermodynamic instability () for small SAdS black hole. This situation is similar to the Schwarzschild black hole which shows a violation of the CSC between thermodynamic instability () and the classical stability Regge:1957td ; Zerilli:1970se ; Vishveshwara:1970cc . Therefore, we could not apply the Gubser-Mitra conjecture to the SAdS black hole in Einstein gravity.

Let us see how things are improved in Einstein-Weyl gravity. We note that at the critical point of , all thermodynamic quantities vanish exactly. For and small black hole with , the heat capacity takes the form

which shows thermodynamic instability like that of a small SAdS black hole in Einstein gravity. From the condition of (44), however, we find that a small black hole is unstable against the -mode massive graviton perturbation. This implies that the CSC holds for the SAdS black hole in Einstein-Weyl gravity.

Also the stability condition of (46) is consistent with thermodynamic stability condition for large black hole with in Einstein gravity

As was previously emphasized, there is no connection between thermodynamic instability and classical stability for small SAdS black hole in Einstein gravity. However, the GL instability condition picks up the small SAdS black hole which is thermodynamically unstable in Einstein-Weyl gravity. Hence, we conclude that there is a connection between the GL instability and thermodynamic instability for small black hole in fourth-order (Einstein-Weyl) gravity.

## 5 Higher-dimensional AdS black holes in fourth-order gravity

### 5.1 Thermodynamic instability for small black holes

In this section, we comment briefly on the thermodynamic (in)stability for higher-dimensional AdS black hole. To this end, we first recall the thermodynamic quantities (26), obtained in the -dimensional fourth order gravity. Among them, taking into account the heat capacity together with (29), the small and large black holes can be divided by choosing the blow-up heat capacity at

 r+=r(d)∗=√d−3d−1ℓ. (50)

For [, we have the small black hole for which is thermodynamically unstable because in (26), while we have the large black hole for which is thermodynamically stable because . This is dominantly described by the higher-dimensional Einstein gravity. We would like to mention that for we will establish the connection between the GL instability and thermodynamic instability of the small black hole.

On the other hand, for [ which is dominantly described by Weyl-squared term, the small black hole is thermodynamically stable because , whereas the large black hole is thermodynamically unstable because of . This case requires a newly black hole thermodynamics.

### 5.2 GL instability

In order to investigate the classical instability for higher-dimensional AdS black hole, we first consider two coupled first order differential equations444We note that these first order differential and constraint equations can be obtained from using the perturbation equation(40) and TT gauge condition. Finally we have checked, after some manipulations, that these equations are consistent with the second order equation (43) and for [in the -limit], they reduce to those found in the original literature Gregory:1993vy .

 H′ = [3−d−(d−1)r2/ℓ2rV−1r]H+Ω2V(H++H−) (51) H′− = M2dΩH+d−22rH++[d−3+(d−1)r2/ℓ22rV−2d−32r]H− (52)

with the constraint equation

 r2Ω[4rΩ2−rV′2+(d−2)VV′+2rVM2d+2rVV′′]H−−Ωr2V[2M2dr+(d−2)V′]H+ −2r2V[2(d−2)Ω2−2M2dV+rM2dV′]H = 0, (53)

where

 H≡Htr,   H±≡HttV(r)±V(r)Hrr   with  V(r)=1−(r0r)d−3+r2ℓ2. (54)

At infinity of , asymptotic solutions to Eqs.(51) and (52) are

 H(∞) = C(∞)1r−(d+1)/2+√M2dℓ2+(d−1)2/4+C(∞)2r−(d+1)/2−√M2dℓ2+(d−1)2/4, H(∞)− = ~C(∞)1r−(d−1)/2+√M2dℓ2+(d−1)2/4+~C(∞)2r−(d−1)/2−√M2dℓ2+(d−1)2/4, (55)

where are

 ~C(∞)1 = M2d(1−d)/2+√M2dℓ2+(d−1)2/4C(∞)1, ~C(∞)2 = M2d(1−d)/2−√M2dℓ2+(d−1)2/4C(∞)2. (56)

At the horizon , their asymptotic solutions are given by

 H(r+) = C(r+)1(rd−3−rd−3+)−1+Ω/V′(r+)+C(r+)2(rd−3−rd−3+)−1−Ω/V′(r+), H(r+)− = ~C(r+)1(rd−3−rd−3+)Ω/V′(r+)+~C(r+)2(rd−3−rd−3+)−Ω/V′(r+), (57)

where are

 ~C(r+)1 = (d−3)rd−3+Ω(2Ω−V′(r+))2V′(r+)(M2dr++(d−2)Ω) C(r+)1, ~C(r+)2 = −(d−3)rd−3+Ω(2Ω+V′(r+))2V′(r+)(M2dr+−(d−2)Ω) C(r+)2.

We note that two boundary conditions of the regular solutions correspond to and at infinity and horizon, respectively.

Eliminating in Eqs. (51) and (52) with the help of the constraint (53), one can find the coupled equations with , only. For given dimensions , fixed , and various values of , we solve these coupled equations numerically, which yields possible values of as a function of given by

 Md= ⎷dm2d4(d−1)−(d−2)22ℓ2. (58)

Fig. 5 shows that the curve of possible values of and intersects the -axis at two places: and = where is a critical non-zero mass. The fact that the curve does not intersect the -axis at follows from the stability of the AdS black hole in higher-dimensional Einstein gravity. Explicitly, for , the AdS black hole is unstable (stable) against the metric perturbations. From the observation of Fig. 5, we read off the critical mass depending on the dimension as

 (d45678910Mcd0.861.261.571.832.072.292.49). (59)

For higher-dimensional black strings with , the critical wave number marks the lower bound of possible wavelengths for which there is an unstable mode. Especially for setting, there exists a critical wave number where for , the black string is unstable (stable) against the metric perturbations. There is an unstable (stable) mode for any wavelength larger (smaller) than the critical wavelength :