aa r X i v : . [ h e p - t h ] O c t arXiv:0709.1028 [hep-th]WU-AP/270/07 Analytic evidence for the Gubser-Mitra conjecture
Umpei Miyamoto
Department of Physics, Waseda University, Okubo 3-4-1, Tokyo 169-8555, Japan [email protected]
Abstract
A simple master equation for the static perturbation of charged black stringsis derived while employing the gauge proposed by Kol. As the charge is varied itis found that the potential in the master equation for the perturbations becomespositive exactly when the specific heat turns positive thus forbidding a boundstate and the onset of the Gregory-Laflamme instability. It can safely be said thatthis is the first analytic and explicit evidence for the Gubser-Mitra conjecture,correlating the classical and thermodynamic instabilities of black branes. Possiblegeneralizations of the analysis are also discussed.
Introduction
It has been known that black objects with an extended event horizon, such as blackbranes and strings, suffer from a classical instability, called the Gregory-Laflamme (GL)instability [1, 2]. The ultimate fate of this instability remains a matter of investigationin spite of extensive studies (see Refs. [3, 4] for reviews and related works). One of theinteresting aspects of this instability is the correlation with the local thermodynamicinstability of background spacetimes. This correlation of instabilities is known as theGubser-Mitra (GM) conjecture [5, 6] (or the correlated stability conjecture [7]): the GLinstability for black objects with a non-compact translational symmetry occurs if andonly if the background black objects are locally thermodynamically unstable. Supportsbased on a general semiclassical argument [8, 9] and a lot of evidences have been knownfor this conjecture [5–7, 10, 11]. However, mainly due to the difficulty to solve thecomplicated perturbation equations of black branes analytically, any conclusive proofof the conjecture has not been reported, as far as the author knows. Since the GMconjecture plays crucial roles in the stability argument of black objects in string theoryas well as in the holographically dual gauge theories [14, 15], it is important to prove theconjecture or have analytic evidence for it.Now we turn to the complexity/simplicity of the perturbation equations of blackbranes. Much effort has been devoted to simplifying the perturbation equations andto identifying analytically the GL modes. Among them, the expression of perturbationequation obtained by Kol [16] would be simplest. See [16] and references therein forthe comparison between various gauges adopted in the literature. His gauge choice wasoptimized by the action formalism in which gauge fixing is postponed as much as possibleand the action is transformed into a canonical form making use of its own invariance [16].The resultant expression of perturbation equation is so simple that it allows for analyticinvestigations or accurate numerical ones. For example, an approximation formula ofthe dimensional dependence of GL marginal mode, which is sufficiently accurate in alldimensions, was obtained in [17], while solving the perturbation equation for generaldimensions is difficult even in such a gauge.In this letter, we adopt the gauge mentioned above for the static perturbation of(non-dilatonic) magnetically charged black strings to obtain a single master equation.We show that if such a master equation is written in Schr¨odinger form, the potentialexhibits a significant feature: the potential becomes positive definite to forbid the criticalmode of GL instability for locally thermodynamically stable black strings. That is, onesees an explicit realization of the GM conjecture in this system. It should be mentioned that possible counterexamples and refinement of the conjecture have beenproposed [12, 13]. In this article, however, we only consider the original version, presented above. Magnetic black strings
We consider the following ( d + 1)-dimensional action ( d ≥ I = 116 πG Z d d +1 x √− g (cid:20) R − d − F d − (cid:21) , (1)where F d − is a ( d − R µν = 12( d − F µ ...µ d − µ F νµ ...µ d − − d − d − g µν F , (2) ∇ µ F µµ ...µ d − = 0 . (3)The form field must satisfy Bianchi identity, d F d − = 0, in addition to Eq. (3). By adimensional reduction method, a black string solution in this system can be obtainedfrom a black hole solution in a d -dimensional dilatonic system [18]. The explicit form ofthe solution is ds = − f + dt + dr f + f − + f − dz + r d Ω d − , f ± ( r ) = 1 − (cid:16) r ± r (cid:17) d − ,F = ˜ Qε d − , ˜ Q = ± p ( d − d − r + r − ) ( d − / , (4)where d Ω d − and ε d − are the line and volume elements of a unit ( d − r = r + and r = r − correspond to an event horizon and innerhorizon, respectively. To make physical quantities finite, let the spacetime be compact in z -direction with length L . Then, mass, temperature and magnetic charge are calculatedas M = Ω d − Lr d − πG (cid:0) d − q d − (cid:1) , T = d − πr p f − (cid:12)(cid:12)(cid:12) r = r + ,Q = Ω d − Lr d − πG ( d − q ( d − / , (5)where Ω d − is the surface area of unit ( d − q is a charge (or extremal) param-eter, defined by q ≡ r − r + , (0 ≤ q < . (6)Note that the magnetic charge in Eq. (5) is normalized so that Q → M in the extremallimit, q →
1. We can calculate the specific heat for the above black string, C Q = (cid:18) ∂M∂T (cid:19) Q = Ω d − Lr d − G p − q d − [( d − − q d − ] − d − q d − . (7)We can see from Eq. (7) that there is a critical value of the charge parameter, we denoteit by q GM , above which the specific heat becomes positive, q GM ≡ d − / ( d − . (8)2he GM conjecture asserts that the GL instability does not exist for q > q GM . Wenote that this criterion of thermodynamic stability corresponds to the one in a canonicalensemble. If a magnetic charge is allowed to be re-distributed in z -direction, which isnot the case in the present spacetime, we have to take into account also the positivity ofisothermal permittivity, corresponding to working in a grandcanonical ensemble [4]. According to [16], we consider the following “maximally general ansatz”, ds = − f + e a ( r,z ) dt + e b ( r,z ) f + f − dr + f − e β [ dz − α ( r, z ) dr ] + r e c ( r,z ) d Ω d − . (9)We regard all metric functions introduced in Eq. (9), ( a, b, c, α, β ), as small perturbations.Since metric (9) has extra degrees of freedom to describe a non-uniform black string, onecan impose gauge conditions to restrict the perturbations to be physically relevant. Forexample, a conformal-type gauge in ( r, z ) plane, defined by b = β and α = 0, wasadopted in Ref. [19]. In this gauge, b can be written in terms of a and c , and we havecoupled equations for a and c (after expanding each function by suitable harmonics in z -direction). It is known that this gauge equally goes well for the charged case [11]. Onthe other hand, the “optimal gauge” proposed in Ref. [16] for the static perturbationsof neutral black strings ( q = 0 in the above solution) is to set a = β = 0 with b , c and ∂ z α taken as variables. In this gauge, one can show that both b and ∂ z α are “non-dynamical variables”, whose derivatives do not enter into EOMs and can be written interms of master variable c (see [20] for a general description of such gauge optimizationand decoupling procedure). Also in the present charged case ( q = 0), in which theEinstein equation has a source term of the gauge field, we can take the same gauge toobtain a master equation for c . It is not so trivial that this gauge goes well even in thecharged case. The point is that under ansatz (9), the gauge field F = ˜ Qǫ d − in Eq. (4)remains to be a solution, i.e., EOM (3) and the Bianchi identity are satisfied in thisform. As a result, the right hand side of Einstein equation (2) contains only c , e.g., F ∝ ˜ Q / ( re c ) d − . In other words, the perturbation of form field can be written interms of the perturbation of the metric. Thus, the gauge field neither introduces anyunknown function nor changes the structure of the Einstein equations.Now, let us normalize the coordinates by the horizon radius, y ≡ rr + , x ≡ zr + . (10)Then, we take the gauge of a = β = 0 and move to a Fourier space by X ( y, x ) = X ( y ) cos( kx ) , ( X = b, c ) ,α ( y, x ) = α ( y ) ∂ x cos( kx ) , (11)3here k is GL critical wavenumber to be determined later. Substituting (11) into Einsteinequation (2), we have three independent equations, which do not contain the derivativesof b and α . Using two of these three equations, both b and α can be written in termsof c (and its first derivative). Finally, the master equation for c is obtained, which wewill investigate in the rest part of this section. q > q GM The master equation for c in the potential form is given by (cid:2) − e − A − B ∂ y ( e A − B ∂ y ) + V ( y ) (cid:3) c = − k c , (12)where e A = y d f + f ′ +4 [2( d − f + + yf ′ + ] , e − B = f + f − , (13) V ( y ) = 2( d − f − { − d − [ d − d + 5 − ( d − d − f − ] f + } y [ d − d − f + ] − d −
2) ˜ Q f − ( d − y d − ( f + − . (14)The second term in potential (14) comes from the source term of the Einstein equation.Substituting the explicit form of f ± and ˜ Q , Eq. (14) can be written in a simpler form, V ( y ) = 2( d − (cid:2) − d − q d − (cid:3) ( y d − − q d − )[2( d − y d − − ( d − y d − . (15)From Eq. (15), one can see that the potential is regular and finite everywhere betweenthe horizon and infinity, 1 ≤ y < + ∞ . The behavior of potential in the asymptotic regionis V ( y ) ∼ y − ( d − . The most interesting property of the potential which can be seenfrom Eq. (15) is q S q GM ⇐⇒ V ( y ) S . (16)This charge dependence of the potential is visualized in Fig. 1 (a). Using this significantproperty, one can show the non-existence of GL critical mode for q > q GM as follows.Multiplying Eq. (12) by e A + B c and integrating it by parts, we have Z ∞ e A − B (cid:2) c ′ + e B ( V + k ) c (cid:3) dy − (cid:2) e A − B c c ′ (cid:3) y = ∞ y =1 = 0 . (17) The neutral limit ( q →
0) of potential (14) or (15) does not coincide with the potential for the staticperturbations of a neutral black string obtained in Ref. [16]. This stems from the fact that the potentialin [16] is not for c itself but for a linear combination of a and c . - - - H y L k r + q 0 0.5 0 0.25 0.5 0.75 1 k r + q (a) (b)Figure 1: (a) The charge dependence of potential (15) for d = 4. The qualitative feature isthe same in the other dimensions ( d ≥ y = 1 and q = 1corresponds to the extremal limit. The background black string is thermodynamically stablefor q > q GM = 1 / ( d − / ( d − , ( q GM = 1 / d = 4). The potential (colored surface withmesh) intersects with the zero plane (gray plane) on q = q GM (blue dotted line) and is positivedefinite for q > q GM . (b) Numerical plot of the charge dependence of GL critical wavenumber(red triangles) for d = 4. The curve is just to guide the eye. The critical wavenumber decreasesmonotonically as the charge increases and vanishes at q = q GM (blue dotted line). The non-existence of k ∈ ℜ for q > q GM is analytically show by Eq. (17). The boundary term in Eq. (17) vanishes. Then, it is obvious that the critical wavenumber k ∈ ℜ does not exist for V ( y ) >
0. This is the analytic evidence that the GL instabilitycannot exist for the thermodynamically stable black strings in this system.Now, the proof of the GM conjecture should be completed by showing the existenceof the critical mode for 0 ≤ q < q GM . However, it seems not so easy to solve Eq. (12)analytically even in the simplest case d = 4, and we have to resort to numerical compu-tations. Therefore, we integrate Eq. (12) numerically for 0 ≤ q < q GM with regularityconditions at the horizon and infinity. In Fig. 1 (b), we show the charge dependenceof the critical wavenumber k . We can see that k indeed decreases monotonically as q increases and vanishes almost exactly at q = q GM . See Refs. [11, 21] for the observationthat the vanishing of the critical wavenumber around q GM exhibits a power-low behaviorwith universal critical exponent 1 /
2, i.e., k ∝ | Q GM − Q | / , where Q GM is the physicalcharge corresponding to q GM . 5 Discussion
We have shown that magnetically charged black string (4) serves as an analytic evidencefor the Gubser-Mitra conjecture. To prove the conjecture in a complete form, the partialproof given in Sec. 3.2 should be followed by an analytic identification of the Gregory-Laflamme critical mode —– alternatively, just by showing the existence of the modeanalytically —– for the thermodynamically unstable strings (0 ≤ q < q GM ). The exten-sion to dynamical perturbation might be important, although we have only consideredthe marginally stable critical mode.Here, we mention that perturbation (11) can be generalized to higher-order ones,taking into account the non-linear backreactions as developed in [19]. In fact, we didconfirm a single master equation to be derived at each order of perturbation for thecharged strings considered in this paper. Technically speaking, this fact reduces thenumber of parameters to be determined numerically to one for each mode and savesthe computation time considerably. We will discuss such non-linear perturbationselsewhere, by which we could confirm the critical dimension [22, 23] and critical charges[11] with more accurate numerics/much higher-order perturbations.Besides the black objects with the translational symmetries, the correlation betweenthe instabilities is relevant to a black string in Anti-de Sitter spacetime [24, 25], havingno translational symmetry, as well as to de Sitter spacetime [26]. It would be interestingto generalize the gauge optimization in [16] to these spacetimes to understand/refine theGM conjecture.
Acknowledgments
The author would like to thank H. Kudoh for collaborating at an early stage of this work,and V. Asnin, B. Kol and N. A. Obers for useful discussions and comments. The authorthanks also to the organizers of the workshop “Einstein’s Gravity in Higher Dimensions”(The Hebrew University of Jerusalem, Feb. 2007), where this work was finalized by usefuldiscussions with participants. This work is supported by a grant from the 21st CenturyCOE Program (Holistic Research and Education Center for Physics of Self-OrganizingSystems) at Waseda University. Precisely speaking, such master equations were derived only for Kaluza-Klein ( k = 0) modes. Forhomogeneous ( k = 0) modes, which inevitably appear in the non-linear perturbations, we were not ableto obtain the master equation, which was done for the neutral black string in [16]. Instead, we havea set of coupled equations for c and β in the gauge of b = α = 0 ( a is a non-dynamical variable to bewritten in terms of c and β ). However, we have only one shooting parameter even for such coupledequations. eferences [1] R. Gregory and R. Laflamme, Black strings and p-branes are unstable , Phys. Rev.Lett. (1993) 2837–2840, [ hep-th/9301052 ].[2] R. Gregory and R. Laflamme, The instability of charged black strings andp-branes , Nucl. Phys.
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