Phase transitions and critical behavior of black branes in canonical ensemble
aa r X i v : . [ h e p - t h ] J a n USTC-ICTS-10-11
Phase transitions and critical behavior of blackbranes in canonical ensemble
J. X. Lu a , Shibaji Roy b and Zhiguang Xiao a a Interdisciplinary Center for Theoretical StudyUniversity of Science and Technology of China, Hefei, Anhui 230026, China b Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta-700 064, India
Abstract
We study the thermodynamics and phase structure of asymptotically flat non-dilatonic as well as dilatonic black branes in a cavity in arbitrary dimensions ( D ).We consider the canonical ensemble and so the charge inside the cavity and thetemperature at the wall are fixed. We analyze the stability of the black braneequilibrium states and derive the phase structures. For the zero charge case wefind an analog of Hawking-Page phase transition for these black branes in arbitrarydimensions. When the charge is non-zero, we find that below a critical value of thecharge, the phase diagram has a line of first-order phase transition in a certain rangeof temperatures which ends up at a second order phase transition point (criticalpoint) as the charge attains the critical value. We calculate the critical exponentsat that critical point. Although our discussion is mainly concerned with the non-dilatonic branes, we show how it easily carries over to the dilatonic branes as well. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] Introduction
Over the years much is known about the thermodynamics and phase structure of blackholes in asymptotically AdS space. The reason is that the black holes in AdS space, unlikethose in flat space, are thermodynamically stable [1]. Further, more interests were drawnwith the advent of AdS/CFT correspondence [2, 3, 4] as the AdS black holes provide agood laboratory for testing the correspondence at finite temperature [5]. So, for example,the AdS black holes are well-known to undergo a Hawking-Page phase transition [1] andby AdS/CFT this corresponds to the confinement-deconfinement phase transition in large N gauge theory [5]. Similarly, the phase structure of the charged AdS black holes whichincludes, in the canonical ensemble, a similarity with the van der Waals-Maxwell liquid-gassystem, can also be understood from the dual field theories [6, 7].However, it was pointed out [8, 9] that the above mentioned phase structure is notunique for the asymptotically AdS black holes. In fact, very similar phase structure wasshown to arise for the suitably stabilized asymptotically flat as well as asymptotically dSblack holes [8]. As, the higher dimensional theories like string or M-theory admits higherdimensional black objects like black p -branes [10, 11, 12], it is natural to ask what kindof phase structures do they give rise to – do they have a similar phase structure as theblack holes or they have a different phase structure altogether?Motivated by this we looked into the thermodynamics and the phase structure of black p -brane solution of D -dimensional gravity coupled to ( p + 1)-form gauge field. In the be-ginning, we consider only the non-dilatonic branes, (when D = 11, they correspond toM2 and M5 branes of M-theory and when D = 10, it is the D3-brane of string theory)and then towards the end we show how the analysis carries over to the dilatonic branesas well. The solutions we consider are as usual asymptotically flat and so they are ther-modynamically unstable and an isolated black brane would radiate energy in the formof Hawking radiation. In order to restore thermodynamic stability so that equilibriumthermodynamics and the phase structure can be studied, we must consider ensemblesthat include not only the branes under consideration but also their environment. Asself-gravitating systems are spatially inhomogeneous, any specification of such ensemblesrequires not just thermodynamic quantities of interest but also the place at which theytake the specified values. In other words, we place the brane in a cavity a la York [13](see also [14, 15, 16, 17, 18]) and its extension in the charged case [19]. Concretely, wewill keep the temperature fixed at the surface of the cavity and as the black p -branes arecharged under the ( p + 1)-form gauge field, we will keep the charge inside the cavity alsofixed. This will define a canonical ensemble and we will study the phase structure of the2lack p -branes in this ensemble.After some generalities we first consider the case when the charge in the cavity isfixed to be zero. In this case we find that there is a minimum temperature below whichno black brane state exists in equilibrium inside the cavity. But above this temperaturethere exist two black brane states with different radii. The larger one is locally stableand the smaller one is unstable. The locally stable or ‘supercooled’ large black brane willeventually decay to energetically more favorable state ‘hot flat space’. There is a phasetransition temperature at which the ‘hot flat space’ and the corresponding large blackbrane can coexist . But when the temperature rises above this transition temperaturelarger black brane becomes globally stable and the ‘hot flat space’ can decay to the blackbrane. The small black brane still remains unstable. As the temperature rises more, thesize of the small black brane decreases and that of the large black brane increases and atinfinite temperature the size of the small black brane goes to zero whereas, the size ofthe large black brane coincides with the size of the cavity. This situation is analogous tothe Hawking-Page transition of AdS black holes or asymptotically flat or dS black holesin a cavity [1, 8].When the charge is non-zero but fixed inside the cavity, the phase structure becomesmore complicated. Here we find that when the charge | q | is greater than a critical value | q c | , there exists a globally stable black brane solution at every temperature in betweenzero and infinity. However, when | q | < | q c | , there is a range of temperature, where thereexist three black brane solutions with different radii. The black branes of the smallestand the largest sizes are locally stable as they correspond to the local minima of the freeenergy. On the other hand, the intermediate size black brane is unstable, corresponding tothe maximum of the free energy, and never exists. The free energies of the largest and thesmallest black branes are not the same and one is greater than the other which depends onthe temperature of the system. However, there exists a transition temperature, for a givencharge | q | < | q c | , where the two free energies become the same. This is the temperaturewhere two black brane states the smallest and the largest can coexist and can make atransition freely from one phase to the other just like the van der Waals-Maxwell liquid-gas phase transition. Since this transition involves an entropy change, therefore, it is afirst order phase transition. This structure was also noticed in AdS [6, 7], asymptoticallyflat and dS black holes [9, 8] in canonical ensemble. When the temperature, in the range One expects this to be a topological first order phase transition since this involves a topology changeas well as an entropy change. When the temperature is of the order of Planck scale, classical gravity description will break downand one must consider quantum gravity.
3e mentioned, is less (more) than the transition temperature, the smallest black branehas lower (higher) free energy than the largest black brane. So, in one case the smallestblack brane is globally stable and in other the largest black brane is globally stable. Thereis a first order phase transition line which depends on the charge | q | < | q c | and the linegets shrunk as we increase the charge ending up at a second order phase transition point(critical point) when | q | = | q c | . We calculate the critical exponents at this critical pointand found that they have very similar structure as in the black hole cases with a universalcritical exponent [8, 6] for the specific heat as − /
3. Since the stability analysis for theblack branes, in general, is quite complicated and extracting exact values of the parametersmay not always be easy, we will illustrate the behavior by numerical calculations exceptfor some special cases specified later.This paper is organized as follows. In section 2, we give the non-dilatonic black p -brane solution in space-time dimensions D and derive the action from which we studythe stability and analyze the phase structure. The details of the general stability analysisis discussed in section 3. The phase structure for the case of zero charge in the cavityis considered in section 4. Section 5 discusses the case of non-zero charge. The criticalexponents are given in section 6 and we conclude in section 7. The dilatonic brane casesare also mentioned and discussed in appropriate places in the respective section and howthe whole analysis of the paper carries over to the dilatonic branes is discussed in theAppendix. p -brane solution and the action The black p -brane solution was originally constructed as a solution to the ten dimensionalsupergravity containing a metric, a dilaton and a ( p + 1)-form gauge field [10]. Thiswas generalized to arbitrary dimensions in [11]. These solutions are given in Lorentziansignature, but for the purpose of studying the thermodynamics [13] we will write theblack p -brane solution (without the dilaton field as we will be studying the non-dilatonicbranes, the dilatonic branes will be considered in the Appendix) in Euclidean signature4s (see for example [22]), ds = ∆ + ∆ − dD − − dt + ∆ ˜ dD − − d − X i =1 ( dx i ) + (∆ + ∆ − ) − dρ + ρ d Ω d +1 A [ p +1] = − i "(cid:18) r − r + (cid:19) ˜ d/ − (cid:18) r − r + ρ (cid:19) ˜ d/ dt ∧ dx · · · ∧ dx p F [ p +2] ≡ dA [ p +1] = − i ˜ d ( r − r + ) ˜ d/ ρ ˜ d +1 dρ ∧ dt ∧ dx · · · ∧ dx p (1)In the above we have defined, ∆ ± = 1 − (cid:18) r ± ρ (cid:19) ˜ d (2)where r ± are the two parameters characterizing the solution which are related to the massand the charge of the black brane. The metric in (1) has an isometry S × SO( d − × SO( ˜ d + 2) and therefore represents a ( d − ≡ p -brane in Euclidean signature. The totalspace-time dimension is D = d + ˜ d + 2, where the space transverse to the p -brane has thedimensionality ˜ d + 2. A p -brane couples to the ( p + 1)-form gauge field whose form andits field-strength are given in (1). It is clear from the Lorentzian form of the above metricthat when r − = 0, it reduces to D -dimesional Schwarzschild solution which has an eventhorizon at ρ = r + , whereas, at ρ = r − , there is a curvature singularity. So, the metric in(1) represents a black p -brane only for r + > r − , with r + = r − being its extremal limit.Note that in the above we have defined the gauge potential with a constant shift, following[19], in such a way that it vanishes on the horizon so that it is well-defined on the localinertial frame. For the metric in (1) to be well defined without a conical singularity at ρ = r + , the Euclidean time in the metric is periodic with a periodicity β ∗ = 4 πr + ˜ d − r ˜ d − r ˜ d + ! d − , (3)which is the inverse of temperature at ρ = ∞ . The local β = ∆ / ∆ − d D − − β ∗ , (4)which is the inverse of local temperature at ρ when in thermal contact with environmentat the same temperature. For the canonical ensemble we have fixed ρ at the wall of cavitydenoted by ρ B , fixed local temperature at ρ B , fixed local brane volume V p = ∆ ˜ d ( d − D − − V ∗ p ,with V ∗ p = R d p x and fixed charge defined as Q d = i √ κ Z ∗ F [ p +2] = Ω ˜ d +1 √ κ ˜ d ( r + r − ) ˜ d/ . (5)5n (5) κ = √ πG D , where G D is the D -dimensional Newton’s constant, ∗ F [ p +2] denotesthe Hodge dual for the ( p + 2) form-field given in (1). Also Ω n denotes the volume of aunit n -sphere. With these data we will evaluate the action.The relevant action for the gravity coupled to a ( p + 1)-form gauge field in a manifold M of dimension D with the Euclidean signature has the form, I E = I E ( g ) + I E ( F ) (6)where the first term is the purely gravitational action, I E ( g ) = − κ Z M d D x √ gR + 1 κ Z ∂M d D − x √ γ ( K − K ) (7)consisting of the usual Einstein-Hilbert term and the Gibbons-Hawking boundary term[23] where ∂M denotes the boundary of the manifold M . In the above K is the trace ofthe extrinsic curvature K µν defined as, K µν = −
12 ( ∇ µ n ν + ∇ ν n µ ) , and so , K = −∇ µ n µ (8)where n µ is a space-like vector normal to the boundary and is normalized as n µ n µ = 1.Also γ αβ is the boundary metric with α, β the indices of the boundary coordinates and γ is its determinant. K is the subtraction term which serves as an infra-red regulator sothat a finite result can be obtained [1, 13, 19, 3] . This is calculated by embedding thesame surface in the flat space.The second term is due to the form-field F [ p +2] and its expression in the canonicalensemble has the form, I E ( F ) = 12 κ d + 1)! Z M d D x √ gF d +1] − κ d ! Z ∂M d D − x √ γn µ F µµ ...µ d A µ ...µ d (9)Note that for canonical ensemble the charge in the cavity is fixed. For a given gravitationalconfiguration in Euclidean signature, the corresponding thermodynamical partition func-tion in the saddle point (or the zero-loop) approximation can be obtained as Z ≈ e − I E , In the charged case, though this flat space is a solution of the corresponding equations of motion, itis not a thermal state and serves only as an infra-red regulator. One way to see the rationale behind isto first calculate the corresponding action in the grand canonical ensemble for which the flat space is agood reference state subtractor. Then the Euclidean action in the canonical ensemble can be obtainedfrom the calculated Euclidean action in the grand canonical ensemble by a thermodynamical Legendretransformation, noting that the respective action is related to β times the corresponding ensemble poten-tial to leading order. The canonical ensemble action so obtained is also finite and is actually equivalentto that by the subtraction procedure mentioned in the text. I E is evaluated for the given configuration (see for example [20]) and we will eval-uate it in the black p -brane configuration given in (1). On the other hand, we also have Z = e − βF , where β is the inverse temperature of the ensemble and F is the correspondingfree energy (for canonical ensemble this is Helmholtz free energy). Therefore, we have inthis approximation F = I E /β . This is the relevant quantity we are going to evaluate forstudying the stability of the black branes and its phase structure.To evaluate I E given in (6), we can use the equation of motion of the metric andobtain, R = ˜ d − d D − d + 1)! F d +1] (10)Substituting (10) in the action we rewrite I E as, I E = d κ ( D − d + 1)! Z M d D x √ gF d +1] + 1 κ Z ∂M d D − x √ γ ( K − K ) − κ d ! Z ∂M d D − x √ γn µ F µµ ...µ d A µ ...µ d (11)In the above we have used d + ˜ d = D − d ˜ d = 2( D −
2) for thenon-dilatonic branes which is a solution of supergravity with maximal supersymmetry,the type we are considering. We will evaluate each term in (11) separately. For thatpurpose we need the forms of the normal vector n µ , the trace of the extrinsic curvatures K and K which can be calculated from the metric in (1) and are given as, n µ = (∆ + ∆ − ) / δ µρ K = −∇ µ n µ = − √ g ∂ µ ( √ gn µ )= − ˜ d + 1 ρ (∆ + ∆ − ) / − ˜ d ρ ˜ d +1 "(cid:18) ∆ − ∆ + (cid:19) / r ˜ d + + (cid:18) ∆ + ∆ − (cid:19) / r ˜ d − K = − ˜ d + 1 ρ (12)Now from the metric and the form-field given in (1), it is straightforward to evaluate theaction (11) as, I E = − βV p Ω ˜ d +1 κ ρ ˜ dB " (cid:18) ∆ + ∆ − (cid:19) / + ˜ d (cid:18) ∆ − ∆ + (cid:19) / + ˜ d (∆ + ∆ − ) / −
2( ˜ d + 1) ρ = ρ B (13)Note in the above that ρ is fixed on the cavity as ρ B . Now since we have I E = βF , with F = E − T S , the Helmholtz free energy in the canonical ensemble, so this implies that7 E = βE − S , with E the energy in the cavity enclosed and S , the entropy for the system.Following Braden et. al. [19] for black hole, we expect that the entropy in the presentcase (which is just that of the black brane under consideration) should be independent ofthe choice of the location of the cavity. However, this is not manifest in (13). For this wecan rewrite (13) as, I E = − βV p Ω ˜ d +1 κ ρ ˜ dB " ( ˜ d + 2) (cid:18) ∆ + ∆ − (cid:19) / + ˜ d (∆ + ∆ − ) / −
2( ˜ d + 1) ρ = ρ B − π V ∗ p Ω ˜ d +1 κ r ˜ d +1+ − r ˜ d − r ˜ d + ! + d , (14)where we have used the expression of β ∗ given in (3). The last term in (14) is preciselythe entropy of the non-dilatonic black p -brane S = 4 π V ∗ p Ω ˜ d +1 κ r ˜ d +1+ − r ˜ d − r ˜ d + ! + d = 4 πV p Ω ˜ d +1 κ r ˜ d +1+ ∆ − − d − − r ˜ d − r ˜ d + ! + d , (15)where we have used V p = ∆ ˜ d ( d − D − − V ∗ p , and is independent of the location of the cavity. Wecan also read off the energy for the cavity as E ( ρ B ) = − V p Ω ˜ d +1 κ ρ ˜ dB " ( ˜ d + 2) (cid:18) ∆ + ∆ − (cid:19) / + ˜ d (∆ + ∆ − ) / −
2( ˜ d + 1) ρ = ρ B , (16)which approaches the ADM mass at ρ B → ∞ . In casting the Euclidean action in the formof (14), we actually treated the fixed β at the wall of cavity, the inverse of temperatureof the environment, to be independent of r + for the time being. Then the internal energy E and the entropy S are functions of r + only when ρ B , V p and Q d are all fixed as in thisensemble. The stability of the black p -branes can now be discussed from the action (14)when the second line of (15) for entropy is employed to which we turn in the next section. p -branes We have mentioned that in the canonical ensemble the charge inside the cavity given in(5) is fixed. This implies that r − is not an independent parameter and can be expressed8n terms of r + as, r − = √ κQ d Ω ˜ d +1 ˜ d ! / ˜ d r + = ( Q ∗ d ) r + (17)where we have defined Q ∗ d = [( √ κQ d ) / (Ω ˜ d +1 ˜ d )] / ˜ d . With these, ∆ − will be given as,∆ − = 1 − (cid:18) ( Q ∗ d ) r + ρ (cid:19) ˜ d (18)We are now considering the canonical ensemble for which the quantities β , V p , ρ B and Q d are all fixed (this implies that β ∗ , V ∗ p are not fixed), and therefore the only parameterwhich can vary is r + . This implies that the entropy will change with r + , therefore if wevary the action (14) with respect to r + and set that to zero (that is, at the stationarypoint), we must be able to recover β (since the temperature is a conjugate variable tothe entropy) and this in turn will give a non-trivial check that the form of action (14)we have obtained is indeed correct. For this we will replace V ∗ p by V p using the relation, V p = ∆ + d D − − V ∗ p = ∆ + d − V ∗ p where we have used d ˜ d = 2( D −
2) in the second equalityand so, (14) takes the form, I E = − βV p Ω ˜ d +1 κ ρ ˜ dB " ( ˜ d + 2) (cid:18) ∆ + ∆ − (cid:19) / + ˜ d (∆ + ∆ − ) / −
2( ˜ d + 1) − π V p ∆ − − d − Ω ˜ d +1 κ r ˜ d +1+ − r ˜ d − r ˜ d + ! + ~ d . (19)In the expression of ∆ ± here and below the variable ρ must be replaced by constant ρ B the size of the cavity. From ∂I E /∂r + = 0, we obtain after some simplification, β ˜ d − πr + ∆ / ∆ − d − − r ˜ d − r ˜ d + ! d − " ˜ d + 2 + ˜ d − ˜ d + 22∆ − ! − r ˜ d − r ˜ d + ! = 0 . (20)Since the second factor in the l.h.s. of (20) is greater than zero, we must have β = 4 πr + ˜ d ∆ / ∆ − d − − r ˜ d − r ˜ d + ! d − = ∆ / ∆ − d D − − β ∗ . (21)This is precisely the correct form of β given in (4), which we obtain from the metric in(1). We will use the relation (21) to discuss the stability and the phase structure of black p -branes. For this purpose let us rewrite β explicitly as, β = 4 πr + ˜ d − Q ∗ d d r d + ! d − − r ˜ d + ρ ˜ dB ! / − Q ∗ d d r ˜ d + ρ ˜ dB ! − d , (22)9here we have used (17). Now following refs. [19, 8, 9], we define x = (cid:18) r + ρ B (cid:19) ˜ d ≤ , ¯ b = β πρ B , q = (cid:18) Q ∗ d ρ B (cid:19) ˜ d , (23)where the dimensionless parameters ¯ b , q are fixed but x is the only parameter whichcan change. Note that the parameter ¯ b is related to the inverse of temperature of theenvironment, q is related to the charge and x is related to the horizon size. Also, as( Q ∗ d ) /r = r − /r + < x > q . In terms of these parameters the above equation of state(or thermal equilibrium condition) for β , (22), can be rewritten as,¯ b = b q ( x ) (24)where b q ( x ) = 1˜ d x / ˜ d (1 − x ) / (cid:16) − q x (cid:17) − d (cid:16) − q x (cid:17) d . (25)In the above (25) we have used∆ + = 1 − x, ∆ − = 1 − q x , − r ˜ d − r ˜ d + = 1 − q x . (26)If we define the reduced Euclidean action as˜ I E ≡ κ I E πρ ˜ d +1 B V p Ω ˜ d +1 = − ¯ b ( ˜ d + 2) − x − q x ! / + ˜ d (1 − x ) / (cid:18) − q x (cid:19) / −
2( ˜ d + 1) − x d − q x − q x ! + d , (27)one can show ∂ ˜ I E ∂x = f q ( x ) (cid:2) ¯ b − b q ( x ) (cid:3) (28)where b q ( x ) is as given above in (25) and f q ( x ) = (1 − x ) − / (cid:18) − q x (cid:19) − / " ˜ d + 2 − ˜ d + 22 − q x − q x ! + ˜ d (cid:18) − q x (cid:19) > . (29) Since the charge Q d as well as Q ∗ d denotes the corresponding absolute value, so q also denotes theabsolute value. Therefore, from now on, we use q to denote the absolute value of the reduced charge. b, q, ˜ d and the variable x . The only relevant dimensionalityis ˜ d which remains unchanged under the so-called double-dimensional reductions [21]. Inother words, the black branes related by this reduction will have the same stability andphase structure at least in the approximation employed. For example, the D = 11 M2branes and the D = 10 fundamental strings share the same above mentioned propertiesand so do the other branes related by this kind of reductions (see, for example, [22]).Now from (28) we find that (at the stationary point) ∂b q ( x ) ∂x > ⇒ ∂ ˜ I E ∂x < ,∂b q ( x ) ∂x < ⇒ ∂ ˜ I E ∂x > , (30)Therefore, the system will be stable (at least locally) when ∂b q ∂x <
0, corresponding to theminimum of the Euclidean action or the free energy. On the other hand, if ∂b q ∂x >
0, thecorresponding Euclidean action is maximum and the free energy is also maximum andthe system will be unstable. Since the phase structure for the black p -branes are quitedifferent for the chargeless case and charged case, we will discuss them separately in thenext two sections. In this section we will discuss the stability and the phase structure for the chargelessblack p -brane. So, we will put q = 0 in all the expressions we obtained in section 3. Theexpression for the reduced action now takes the form,˜ I E = −
2( ˜ d + 1)¯ b h (1 − x ) / − i − x d , (31)whence we have ∂ ˜ I E ∂x = f ( x ) (cid:2) ¯ b − b ( x ) (cid:3) . (32)Where b ( x ) and f ( x ) are given as, b ( x ) = 1˜ d x / ˜ d (1 − x ) / ,f ( x ) = ( ˜ d + 1)(1 − x ) − / > . (33)11Sfrag replacements 0 x x max x g x x ¯ bb max b Figure 1: The typical behavior of b ( x ) vs x in the chargeless case.The equation of state is ¯ b = b ( x ) . (34)At the stationary point of the action, we have ∂ ˜ I E ∂x = − f (¯ x ) ∂b ( x ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) x =¯ x , (35)where ¯ x is a solution of the equation (34). Since b ( x ) = 0 at x = 0 , b ( x ) > < x <
1, so b ( x ) has a maximum in between which can be determined from ∂b ( x ) /∂x = 0 and has the value, b max = 1 p d (cid:18) d + 2 (cid:19) + d ⇒ T min = p d πρ B ˜ d + 22 ! + d (36)at x max = 2˜ d + 2 ⇒ r + max = (cid:18) d + 2 (cid:19) / ˜ d ρ B (37)So, there exists a maximum b max (or minimum temperature T min ) above (or below) whichthe system can not be in a black brane phase. Now since ∂ ˜ I E /∂x > I E = 0 at x = 0 (from (31)), therefore the system favors the ‘hot flat space’. Note thatfor the four dimensional black holes D = 4, and ˜ d = d = 1, and so, T min = 3 √ / (8 πρ B )and r + max = (2 / ρ B match exactly with the values found in [13, 9]. We will have thesame T min and r + max for black strings in D = 5, black membranes in D = 6, up to blackD6 branes in D = 10 since these branes are related to the four dimensional black hole viathe double-dimensional reductions. 12or ¯ b smaller than b max , we can have two solutions from the equation of state (34),but only the large solution x ( > x max ) will be locally stable given the relation (35). Thisbehavior of b ( x ) versus x is depicted in Figure 1. However, this does not necessarilyimply that the system is in the black brane phase. Only when the local stability becomesa global one, then the system is indeed in the black brane phase. The corresponding x can be determined from requiring the action at the stationary point be negative. Theaction (31) can be expressed using (34) as,˜ I E = − ( ˜ d + 2)¯ by y − ˜ d ˜ d + 2 ! ( y − , (38)where we have defined y = √ − ¯ x (39)with x = ¯ x . So, the necessary condition for the global stability can be seen from (38) tobe y < ˜ d ˜ d + 2 . (40)This gives ¯ x > x g = 4( ˜ d + 1)( ˜ d + 2) > x max . (41)Now we find b g ( x = x g ) = 1˜ d + 2
4( ˜ d + 1)( ˜ d + 2) ! / ˜ d ⇒ T g = ( ˜ d + 2)4 πρ B ( ˜ d + 2)
4( ˜ d + 1) ! / ˜ d . (42)So only when 0 < ¯ b < b g (in this case x g < x < b g < ¯ b < b max (in this case x max < x < x g ), the system, thoughlocally stable, will eventually tunnel to the ‘hot flat space’ at the same temperature.However, at ¯ b = b g , ˜ I E = 0 and so, both the black brane phase with ¯ x = x g and the‘hot flat space’ phase are possible. In other words, this is the place at which the twophases can coexist and the corresponding temperature is the phase transition one. Thisphase transition is both a topological and a first order one since both the topology andthe entropy of the two phases are different before and after the phase transition. Notethat when x →
1, ¯ b → b/y → / ˜ d , so the action isstill finite and is ˜ I E = −
1, implying that the system is stable. For the four dimensionalblack hole we find that the temperature ( T g ) where the large black brane becomes globallystable is 27 / (32 πρ B ) which matches with the value found in [13, 9]. Also it is clear fromFigure 1 that, as b ( x ) decreases or the temperature increases the size of the small black13rane decreases and that of the large black brane increases and eventually when b → T → ∞ , the size of the small black brane goes to zero and the size of the large blackbrane approaches the size of the cavity. The phase transition we found in the present caseis analogous to the Hawking-Page transition for the AdS black holes. In this section we will study the stability and phase structure of black p -brane in themore general case where the charge enclosed by the cavity is non-zero and fixed. Insection 3, while discussing the generalities for the stability of black p -brane we found thatthe system will be stable when ∂b q ( x ) /∂x < ∂b q ( x ) /∂x > q c , such that when q > q c , the system willbe globally stable as ∂b q ( x ) /∂x is always less than zero (see Figure 3 below), but when q < q c , ∂b q ( x ) /∂x > x lies between 1 and q . Since b q ( x ) → ∞ as x → q and b q ( x ) → x → q < q c , b q ( x ) does not decrease monotonically (asseen from Figure 2) and ∂b q ( x ) /∂x > q < x <
1. In other words,we should have a minimum of b q ( x ) ( b min ) occurring at x = x min and a maximum of b q ( x ) ( b max ) occurring at x = x max . When b min < b q ( x ) < b max and x min < x < x max , ∂b q ( x ) /∂x > b q ( x ) > b max or b q ( x ) < b min , ∂b q ( x ) /∂x < b with b min < ¯ b < b max , there will bethree solutions to the equation of state (24). If we denote the three solutions as x , x and x with x < x < x , then x and x correspond to the local minima of the freeenergy and x corresponds to the maximum. Among the two minima, one expects thatthe one with the lower free energy will be globally stable and a transition will occur fromthe state of higher free energy to the lower free energy. So, it is important to find whichof the two black branes have lower free energy. To determine this we write from (28),˜ I E ( x ) − ˜ I E ( x ) = S ( x , x ) − ¯ S ( x , x ) , (43)where S ( x , x ) = Z x x dxf q ( x ) (cid:2) ¯ b − b q ( x ) (cid:3) ≥ , (44)¯ S ( x , x ) = Z x x dxf q ( x ) (cid:2) b q ( x ) − ¯ b (cid:3) ≥ , (45)14Sfrag replacements b q b max ¯ bb min q x min x max xx x x Figure 2: The typical behavior of b q ( x ) vs x when there is a phase transition ( q < q c ).with b q ( x ) as given in (25). Note that for ¯ b = b min , the points x and x coincide and so S ( x , x ) = 0 and ¯ S ( x , x ) takes a maximum value. Similarly, for ¯ b = b max , the points x and x coincide and therefore, ¯ S ( x , x ) = 0 and S ( x , x ) takes a maximum value. Thusboth the function S ( x , x ) and ¯ S ( x , x ) change continuously from 0 to their maximumvalue as ¯ b is varied from b min to b max . So, in between there must exist a fixed value of ¯ b which we call ¯ b t (inverse of which is related to a phase transition temperature) for eachgiven charge q < q c , for which S ( x , x ) = ¯ S ( x , x ) and therefore the Euclidean actionor the free energies of the two stable black brane configurations of sizes x and x arethe same. In other words, this is a phase transition temperature where the two blackbrane phases coexist and make a transition freely from one phase to the other, much likea van der Waals-Maxwell liquid-gas phase transition. This was also noticed earlier forasymptotically AdS, dS and flat black holes in canonical ensemble [6, 9, 8].Now since at b max , ¯ S ( x , x ) = 0 and S ( x , x ) is maximum and at ¯ b t , they are thesame, so, if ¯ b lies in between i.e., b max > ¯ b > ¯ b t , S ( x , x ) > ¯ S ( x , x ), or ˜ I E ( x ) > ˜ I E ( x ).So, in this case the small black brane phase is globally stable. Similarly, since at b min , S ( x , x ) = 0 and ¯ S ( x , x ) is maximum and at ¯ b t , they are the same, so, if ¯ b lies inbetween i.e., b min < ¯ b < ¯ b t , ¯ S ( x , x ) > S ( x , x ), or ˜ I E ( x ) > ˜ I E ( x ). So, in this caselarge black brane phase is globally stable. Thus we conclude that if the temperature isabove the transition value, but below T max , large black brane phase is globally stable andif it is below the transition value, but above T min , the small black brane phase is globallystable. So, there will be a phase transition from the small black brane to large black braneor vice versa depending on whether the temperature of the black brane is below or abovethe transition temperature. 15Sfrag replacements b q ¯ b q ¯ x x Figure 3: b q ( x ) decreases monotonically with x when q < x < q > q c ).It is clear from the expression of entropy given in the last term of (14) that for given q < q c , the entropy will depend on the parameter r + or x and so, the entropy will changeduring the phase transition we just mentioned. Therefore, this is a first order phasetransition for which the entropy has a discontinuity. In fact there is a first order phasetransition line when we move the charge q < q c towards q = q c and the line gets shrunkending up at a second order phase transition point (critical point) for q = q c , which occursat x = x c , and where the entropy discontinuity disappears.Having understood the qualitative features of the equilibria and the phase structureof black p -branes with non-zero charge inside the cavity, we will try to understand thestructure in a more quantitative way and then corroborate our observations by numericalcalculations how the various situations for q > q c , q = q c and q < q c described here arise inthree different values of ˜ d , for examples, for the non-dilatonic branes (we will consider only˜ d = 3 in D = 11, which corresponds to M5-brane, ˜ d = 6 in D = 11, which corresponds toM2-brane and ˜ d = 4 in D = 10, which corresponds to D3-brane) though these calculationswork for the corresponding dilatonic branes related via the double-dimensional reductionsin other dimensions as well.For understanding the stability, as we mentioned, the quantity to look at is ∂b q ( x ) /∂x .From (25) we find, ∂b q ( x ) ∂x = − x / ˜ d n (1 + ˜ d ) x − [1 + (2 + ˜ d ) q ] x − q ( ˜ d − x + q [ ˜ d − d q ] x − ˜ dq o ˜ d x (1 − x ) / (cid:16) − q x (cid:17) ˜ d D − (cid:16) − q x (cid:17) d D − . (46)16he position of the extremality will be determined from the vanishing of the numeratorof the above equation (46), i.e., d ! x − " d ! q x − q ˜ d − ! x + q " ˜ d − d q x − ˜ dq = 0 , (47)This is a quartic equation and has four roots in general. We will make some observationabout the roots of this equation which will support the various structures we describedqualitatively in this section and then give some numerical solution of this equation insome special cases. First note that the discriminant of the above equation (47) whichtells us about the roots has the form,∆( q, ˜ d ) = − ( q − q h(cid:0)
4( ˜ d − − d (4 + ˜ d ) q (cid:1) −
108 ˜ d (2 + ˜ d − ˜ d ) q (1 − q ) i (48)The discriminant will vanish within 0 < q <
1, if the last factor within the square bracketin (48) vanishes in that range. This will be determined by the intersection point of thefirst term, i.e., a cubic curve with the second term, i.e., a parabola. The parabola meetsthe q -axis at q = 0 and at q = 1 and remains positive in this range. On the other handthe cubic curve takes a positive value of 4 ( ˜ d − for ˜ d > q = 0, monotonicallydecreases and meets the q -axis at q = 4( ˜ d − / (3 ˜ d (4 + ˜ d )) < d > . Therefore,there is a unique crossing point q c of the cubic curve and the parabola in the range0 < q < < q c < q = 4( ˜ d − / (3 ˜ d (4 + ˜ d )) <
1. This shows the existence of aunique critical point at q = q c , where ∂b q ( x ) /∂x vanishes. Note that since this is a singleextremum, as b q ( x ) varies from ∞ to 0, this can not be a maximum or minimum, but isan inflection point where ∂ b q ( x ) /∂x also vanishes. This feature is reflected in Figure 4below. Note that for ˜ d = 1, q = 0 and the intersection now occurs at q = 0 which isnot in the range of 0 < q <
1, therefore we don’t expect a critical point and further apossible phase transition to occur. This will be checked explicitly in a subsection of thissection later.When q > q c , we have ∆( q, ˜ d ) <
0, so (47) (since this is a quartic equation, there arefour roots of this equation in general), must have a pair of complex conjugate roots. Alsosince the ratio of the coefficient of x term and the constant term (it is − ˜ dq / (1 + ˜ d/ q < x <
1. However,for ˜ d >
2, as the l.h.s. of equation (47) is positive at both x = q (it is ( − d )( − q ) q )and x = 1 (it is ˜ d ( − q ) / We will discuss ˜ d = 1 case separately in a subsection of this section. < x <
1, must be even . So, there can not be any root of (47) in the range q < x < q > q c and this is consistent with the typical behavior given in Figure 3.When q < q c , we have ∆( q, ˜ d ) > q < x <
1, must be even (two or none). The finding that x c falling in theregion q < x < q = q c , for each of the cases considered in the following, implies thatindeed there exist two roots in the region q < x < q < q c . This is also consistentwith the typical behavior given in Figure 2.Now we give some numerical calculation to illustrate the above picture. When ∆( q, ˜ d ) =0, one expects that q in (48) has only one real positive root q c , For this critical q c , weshould have x min = x max in the region q c < x <
1. Let us examine carefully if this isindeed true. Let us first consider M5-brane, i.e., ˜ d = 3 first. Now ∆( q c , ˜ d = 3) = 0 givesus, −
512 + 27648 y − y + 250047 y = 0 , (49)where we have defined y = q c . One can check that indeed y has only one positive realsolution given by y = 4 (cid:18) − (cid:16) − √ (cid:17) / + 2 / (cid:0) − √ (cid:1) / (cid:19) ≈ . , (50)whence we get q c = 0 . . (51)Substituting this in (47) we obtain − . . x − . x − . x + 52 x = 0 , (52)which indeed gives a unique solution in the region q c < x < x c = 0 . x = − . x = 0 . b c as, b c = x / c (1 − x c ) / − q c x c ) / (1 − q c x c ) / = 0 . . (54)18Sfrag replacements b q b c q x c x Figure 4: The typical behavior of b q ( x ) vs x when there is a turning point for which x min = x max = x c ( q = q c ).From (47), one expects that there exist a minimum of b q ( x ), occurring at x = x min anda maximum, occurring at x = x max , when q < q c , corresponding to Figure 2 (∆( q, ˜ d ) > x min and x max should not exist when q > q c corresponding to Figure3 (∆( q, ˜ d ) < q = 0 . > q c which is slightly larger thanthe critical value, we find indeed that there exist no real solution in the region q < x < q = 0 . < q c which is slightly smaller than the critical value, wefind that indeed there exist two solutions in the region q < x <
1, one is x min = 0 . x max = 0 . d = 6, we find exactly the same behavior. In this case ∆( q c , ˜ d = 6) = 0gives (see (48)) − y − y + 5832000 y = 0 , (55)where again we have defined y = q c . The only real positive solution of (55) gives, y = 321 − (cid:16) − √ (cid:17) / + 2 7 / (cid:0) − √ (cid:1) / , ≈ . , (56)which gives q c = 0 . . (57) We will discuss ˜ d = 2 in a separate subsection of this section. x c = 0 . , (58)from the corresponding equation (47) − . . x − . x − . x + 4 x = 0 , (59)in the region of q c < x <
1. We also have b c = 0 . . (60)The other cases q > q c and q < q c can be discussed similarly as above.For D3-brane, ˜ d = 4, ∆( q c , ˜ d = 4) = 0 gives − y − y + 884736 y = 0 , (61)whose unique positive real solution is y = 73 − (cid:16) − √ (cid:17) / + 5 / (cid:0) − √ (cid:1) / , ≈ . , (62)which gives q c = 0 . . (63)With this critical q c , we have the unique x c = 0 . q c < x < − . . x − . x − . x + 3 x = 0 . (65)Now b c = 0 . . (66)Once again, the other cases q > q c and q < q c can be similarly discussed.We have given the numerical results for ˜ d = 3 , , and 4 since they are related tothe M5, M2 and D3 branes. As indicated earlier and with the discussion given in theAppendix for dilatonic branes, any brane, non-dilatonic or dilatonic, related to each ofthese branes by the so-called double-dimensional reductions will share the same properties,20ince the only dimensionality entering this discussion is ˜ d and it remains the same underthis reduction.Note that since we have 1 ≤ ˜ d ≤
7, for completeness and to show that the critical value x c always falls in the range q c < x c <
1, we give the results also for ˜ d = 5 and 7 ( ˜ d = 1 , d = 5 are ( q c = 0 . , x c =0 . , b c = 0 . d = 7 they are ( q c = 0 . , x c = 0 . , b c =0 . ≤ ˜ d ≤ d q c x c b c q c < x c <
1, where q c is related to the absolute value of critical charge and b c is related tothe inverse critical temperature as defined earlier. We observe that the critical quantitiesall decrease as ˜ d increases. ˜ d = 2 The general equilibria and the phase structure that we discussed in this section does notapply to ˜ d = 2 , d = 2 case. We find from(25) that for ˜ d = 2 b q ( x ) = x / (1 − x ) / (cid:16) − q x (cid:17) / . (67)The corresponding equation (from (47)) for finding the extrema for ˜ d = 2 is2 x − (cid:0) q (cid:1) x + q (cid:0) q (cid:1) x − q = 0 (68)which can be factorized as, ( x − q )( x − x + )( x − x − ) = 0 (69)21Sfrag replacements b q ¯ b q ¯ x xq > q c b c ¯ b q ¯ x q = q c b max ¯ b q x s x + x l q < q c Figure 5: The typical behaviors of b q ( x ) vs x for ˜ d = 2.where, x ± = 14 (cid:16) q ± p ˜∆ (cid:17) (70)with ˜∆ = (1 − q )(1 − q ). The x = q solutions are irrelevant here and x ± can be realonly if ˜∆ ≥ < q ≤ /
3. Note that the discriminant for the present case from(48) has the form, ∆( q,
2) = 4 q (1 − q ) (1 − q ) (71)This gives the same requirement as ˜∆ for the reality of the solutions. Comparing withour earlier discussion of ∆( q, ˜ d ), we note that here there is no parabola and the cubiccurve meets q axis at q = 1 /
3. So, this is also the critical point q c = q = 1 / b q ( x = q ) = √ q/ ∞ (for non-zero charge), the value for ˜ d > b (1) = 0 remains the same.Now let us see the phase structure in detail (see Figure 5). When 1 > q > q c = 1 / q < x <
1, since now ˜∆ < ∂b q ( x ) ∂x = − b q ( x )2 (cid:20) x − x (1 + 3 q ) + 2 q x (1 − x )( x − q ) (cid:21) < . (72)So, for now, when 0 < ¯ b < b q ( q ), we have a stable black brane . For ¯ b > b q ( q ), there isno stable black brane and we don’t have a description available for such phase.For q = q c = 1 /
3, ˜∆ vanishes and the two roots x ± are equal and has the value 1 / q c . So, b q ( x = x c ) = b c is a fake critical point which is not accessible since this is also anextremal point. This point is only marginally stable so long as the thermodynamics is Unless we have a phase like ‘hot flat space’, but now carrying a charge, whose free energy can be thelowest to be the globally stable phase, this black brane phase is globally stable. x beyond this value, b q ( x ) monotonically decreases andgoes to zero at x = 1 and in this range ∂b q /∂x <
0. Thus we find that in the range b c = 1 / (2 √ > ¯ b > b q (1) = 0, there exists a stable black brane phase. However, if¯ b > / (2 √
3) there are no black brane phase or such description is not available . In termsof temperature, if the temperature T is below √ / (2 πρ B ), there are no black brane phase,however, if the temperature is in the range √ / (2 πρ B ) < T < ∞ , there is a stable blackbrane phase.For q < q c = 1 /
3, ˜∆ >
0, so in this case we have two real solutions of ∂b q ( x ) /∂x = 0given by (70). One can check that x − < q and q < x + < x + is the relevantsolution which gives a maximum of b q ( x ) at x = x + as b max = (cid:16) q + p (1 − q )(1 − q ) (cid:17) / (cid:16) p − q − p − q (cid:17) √ > b q ( q ) = √ q . (73)Note that for q < x < x + , ∂b q ( x ) /∂x > x + < x <
1, it is less than zero.Therefore in the range 0 < ¯ b < b q ( q ) = √ q/
2, there is only one black brane phase with1 > x > x + which is stable (we assume that there exists no other stable phase) and inthe range b q ( q ) < ¯ b < b max , there are two black brane phases in which the smaller one isunstable and the larger one is stable. For ¯ b > b max there is no black brane phase or sucha description is not available. ˜ d = 1 For ˜ d = 1, we find from (25) the form of the parameter b q ( x ) as b q ( x ) = x (1 − x ) / (cid:16) − q x (cid:17) − q x . (74)From b q ( x = q ) = b q ( x = 1) = 0 and the fact that b q ( x ) > q < x <
1, there mustexist one and only one extremum which corresponds to a maximum, denoted as b max (seeFigure 6). This is due to the absence of a critical point as discussed earlier for the presentcase. Also as discussed in the previous section, since now q = 0, so∆( q,
1) = (15) ( q − q (cid:20) q − q + 16125 (cid:21) < , (75)for 0 < q <
1. So there must exist a pair of complex conjugate solutions for the followingextremal equation of b q ( x ) which is a special case of (47) for ˜ d = 1,32 x − (cid:18) q (cid:19) x + 32 q x + 32 q x − q = 0 . (76)23Sfrag replacements b q b max ¯ b q x s x max x l x Figure 6: The typical behavior of b q ( x ) vs x for ˜ d = 1.Furthermore, as the product of four roots of the above equation is less than zero, thisimplies that the other two roots must be one negative and one positive. Given that thereexists a maximum, this positive root must be the one we expect and should fall in theregion q < x max <
1. Let us take a special case of q = 0 .
50 as an example. Now the foursolutions of (76) are x = − . , x = 0 . − . i, x = 0 .
33 + 0 . i, x = 0 . , (77)which are exactly as expected with only x = 0 . > q as the location of maximum of b q ( x ) for this q = 0 .
50. The situation here is similar to the chargeless case but we havehere q < x < b q ( x ) vs x . In addition, we do not have the corresponding transition to‘hot flat space’, i.e., the Hawking-Page transition, since the charge is fixed for the presentsystem.For a given 0 < ¯ b < b max , there are two blackbrane solutions: the small black braneis unstable while the large black brane is at least locally stable. Unless some new phasecarrying the same charge is found with lower free energy, we can not justify whether sucha large brane is globally stable or locally stable.This therefore completes our analysis of the equilibria and the phase structure ofblack p -branes in the case of non-zero charge. We have seen that the ˜ d = 2 serves asa borderline which distinguishes the ˜ d = 1 case from the ˜ d > x set by the charge. In addition, there doesn’tappear to exist the obvious analog of Hawking-Page type transition. Apart from this, thephase structure remains basically the same as the chargeless case. For the latter case,however, the introduction of charge does significantly change the stability as well as the24hase structure from the chargeless case as described in detail in this section. One ofstriking features is the appearance of a critical charge which determines both the stabilitybehavior and the phase structure of the underlying system. When the charge is less thanthe critical charge, there exists a first order phase transition line which ends at a secondorder phase transition point which is the critical point at the critical charge. We wouldlike to remark that although the details of the phase structure is quite different for ˜ d > We observed in section 5 for the case of charged black branes, that when the charge isbelow certain critical value, q < q c , there exists two stable black brane states in certainrange of temperature. Let us denote the sizes of the two black branes as x s and x ℓ , wherethe former denotes the small black brane and the latter is the large black brane. There isa transition temperature where the free energies of the two black branes are the same andat this temperature these two black brane phases coexist. This transition temperature,which is completely determined by the given charge q < q c , therefore, forming a first orderphase transition line, can be described by T t ( q ) (with the subscript ‘t’ denoting it as aphase transition temperature). As the charge increases, this phase transition line endsup to a second order phase transition point (critical point) at q = q c . We would like tocalculate the critical exponents at this critical point. Expanding b q ( x ) around the criticalpoint x c we have, b q − b c = 13! ∂ b q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x c ( x − x c ) + · · · . (78)Note that at the critical point the first and the second order derivatives of b q ( x ) withrespect to x are zero. From (25) we find, ∂ b q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x c = − x d c n (cid:16) ˜ d (cid:17) x c − h (cid:16) ˜ d (cid:17) q c i − q c (cid:16) ˜ d − (cid:17)o ˜ d x c (1 − x c ) (cid:16) − q c x c (cid:17) ˜ d −
22 ˜ d +1 (cid:16) − q c x c (cid:17) d +1 , (79)where we have used ∂b q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x c = 0 , ∂ b q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x c = 0 . (80)We mentioned in section 2, that the form of the entropy can be read off from the lastterm of the Euclidean action (14). If we now define the reduced entropy for fixed ρ B and25 p as, ˜ S = 2 κ S/ (Ω ˜ d +1 V p ρ ˜ d +1 B ), then from (27) we read off the form of reduced entropyas, ˜ S = 4 πx ˜ d +1˜ d (cid:18) − q x (cid:19) − − d (cid:18) − q x (cid:19) + d . (81)We now expand the entropy around the critical point as,˜ S − ˜ S c = ∂ ˜ S∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x c ( x − x c ) + · · · , (82)where from (81), ∂ ˜ S∂x = 2 πx d h
2( ˜ d + 1) x − (3 ˜ d + 4) q x + 2 q x + q ˜ d i ˜ dx (cid:16) − q x (cid:17) + d (cid:16) − q x (cid:17) ˜ d −
22 ˜ d . (83)We would like to have the expansion of entropy not around the x c , rather around thereduced critical temperature τ c = 1 /b c . Note that near the critical temperature, τ − τ c = 1 b q − b c = − b c ∂ b q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x c ( x − x c ) + · · · . (84)where we have used (78). So, using (84) we can write (82) in terms of the reducedtemperature as, ˜ S − ˜ S c = ∂ ˜ S∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x c b c − ∂ b q ∂x (cid:12)(cid:12)(cid:12) x = x c / ( τ − τ c ) / + · · · . (85)The reduced specific heat therefore can be calculated as,˜ c v = T ∂ ˜ S∂T = τ ∂ ˜ S∂τ = 13 ∂ ˜ S∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x c − ∂ b q ∂x (cid:12)(cid:12)(cid:12) x = x c b c / ( τ − τ c ) − / + · · · . (86)Therefore, the critical exponent α of ˜ c v is − / d ≥ b q − b c = 13! ∂ b q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x c ( x − x c ) + · · · = − . x − x c ) + · · · . (87)26 S − ˜ S c = 12 . x − x c ) + · · · . (88)We have now τ − τ c = 47 . x − x c ) + · · · , (89)˜ S − ˜ S c = 3 . τ − τ c ) / + · · · , (90)and ˜ c v = 5 . τ − τ c ) − / + · · · . (91)Note that the critical exponent for the specific heat has a universal value − / To conclude, in this paper we have studied in detail the equilibrium states and the phasestructures of the asymptotically flat non-dilatonic and the dilatonic black branes in acavity in arbitrary space-time dimensions D . Although we mostly concentrated on thenon-dilatonic branes, the whole discussion applies also to the dilatonic branes as we givethe details of how the analysis can be carried over in the Appendix. We consideredonly the canonical ensemble and so the charge inside the cavity and the temperature atthe wall of the cavity were held fixed. We employed the Euclidean action formalism tocompute the thermodynamics and the phase structure of the black branes. There is amarked difference in the phase structure when the charge enclosed in the cavity is zeroand non-zero. When the charge is non-zero, there is also a qualitative difference in thephase structure when the ˜ d >
2, ˜ d = 2 and ˜ d <
2. We discussed them each separately.For zero charge we found an analog of Hawking-Page transition even for these higherdimensional black objects in asymptotically flat background. So, we found for the zerocharge case that there exists a minimum temperature given in eq.(36), below which thereis no black brane phase and here the system will be in ‘hot flat space’ phase. But abovethis temperature there exists two black brane phase with different radii. The smallerone is unstable and the larger one is locally stable. When the temperature lies betweenthe minimum value (36) and the value given in (42), the locally stable black brane willeventually tunnel into ‘hot flat space’ since the latter configuration in this region haslower free energy. There is a phase transition temperature given by (42) at which theblack brane of size x g given in (41) and the ‘hot flat space’ can coexist. But above this27emperature, the larger black brane becomes globally stable and the system can remainin this black brane phase. However, the smaller black brane still remains unstable. Aswe increase the temperature the size of the smaller black brane becomes smaller and thatof the larger black brane becomes larger. As the temperature tends to infinity the size ofthe smaller black brane tends to zero and the larger black brane approaches the size ofthe cavity.When the charge enclosed in the cavity is fixed but non-zero, we found that thereexists a critical charge q c at or above which there is a single globally stable black branephase in the absence of an analog of ‘hot flat space’ or some other unknown configurationswith favorable free energy for the charged system. When the charge is below this criticalvalue, then there exists a certain range of temperature denoted by T min and T max belowand above which there is a single globally stable black brane phase. But within this rangethere are three black brane phase with different radii. The largest and the smallest ofwhich are locally stable as they correspond to the local minima of the free energy, but theintermediate one is unstable as it corresponds to the maximum of free energy. We foundthat the values of the free energies of the black brane phases depend on the temperature.In fact there exists a unique transition temperature at which the free energies of the largestand the smallest black brane become equal and so these two phases can coexist at thistemperature and can make a phase transition freely from one phase to the other much likethe van der Waals-Maxwell liquid gas phase transition. Above this transition temperaturethe larger black brane is globally stable and below this temperature the smaller black braneis globally stable. So, there is a first order phase transition from smaller black brane tolarger black brane or vice versa above or below the transition temperature as the entropyof the system changes in this transition. In fact when the charge increases from q < q c towards q = q c , there is a first order phase transition line which eventually ends up in asecond order phase transition point (critical point) at q = q c . At this critical point wehave calculated the critical exponents and found that the critical exponent of specific heathas a universal value − /
3. We have elaborated this phase structure both analyticallyand numerically to illustrate the various situations. We found that this general phasestructure is valid only for ˜ d >
2, where ˜ d is related to the dimensionality of the black p -brane by ˜ d = D − p −
3. ˜ d = 2 , d = 1 case is very similar in structure to the zero charge case except here b q vs x curve starts from x = q instead of x = 0 as in zero charge case. Also here there isno analog of ‘hot flat space’ since the system has non-zero charge. For ˜ d = 2, we foundthat there exists a critical charge q c above which there is a single globally stable blackbrane phase when the temperature of the system is above a certain value, but below this28alue of the temperature there is no black brane phase and we do not have a suitabledescription for this phase. When the charge becomes the critical value q c , again thereexists a temperature T = √ / (2 π ¯ ρ B ) above which there is a single globally stable blackbrane phase and below this temperature there is no description available for the chargedsystem. When the charge is below q c , there is a certain range of temperature where thereare two black brane phases, the smaller one is unstable and the larger one is globallystable. Above this range, there is a single globally stable black brane phase and belowthe minimum temperature we do not have a description available. This whole analysisworks for both the non-dilatonic and dilatonic branes.The only dimensionality which is relevant to the stability and phase structure is ˜ d and this implies that the branes related via the so-called double-dimensional reductionshave the same stability and phase structure at least in the leading order approximationadopted. For example, this implies that the D = 11 M2 brane, the D = 10 fundamentalstring and the D = 9 0-brane all have the same stability and phase structure, so do the D = 11 M5, the D = 10 D4, the D = 9 3-brane, upto the D = 6 0-brane, and so on. Wealso observe that the critical quantities ( q c , x c , b c ) all decrease when ˜ d increases from 2 to7. Acknowledgements:
JXL would like to thank the participants of the advanced workshop “Dark Energy andFundamental Theory”, supported by the Special Fund for Theoretical Physics from theNSF of China with grant no: 10947203, for stimulating discussions. He acknowledgessupport by grants from the Chinese Academy of Sciences, a grant from 973 Program withgrant No: 2007CB815401 and a grant from the NSF of China with Grant No : 10975129.29 ppendix
Here we will consider the dilatonic black p -brane solutions in D space-time dimensions.For studying thermodynamics we give their form in the Euclidean signature as, ds = ∆ + ∆ − dD − − dt + ∆ ˜ dD − − d − X i =1 ( dx i ) + ∆ − ∆ a
22 ˜ d − − dρ + ρ ∆ a
22 ˜ d − Ω d +1 ,A [ p +1] = − ie aφ / "(cid:18) r − r + (cid:19) ˜ d/ − (cid:18) r − r + ρ (cid:19) ˜ d/ dt ∧ dx ∧ . . . ∧ dx p ,F [ p +2] ≡ dA [ p +1] = − ie aφ / ˜ d ( r − r + ) ˜ d/ ρ ˜ d +1 dρ ∧ dt ∧ dx ∧ . . . ∧ dx p ,e φ − φ ) = ∆ a − , (92)where ∆ ± are as defined before in section 2. φ is the dilaton and φ is its asymptoticvalue and related to the string coupling as g s = e φ . a is the dilaton coupling given by , a = 4 − d ˜ dD − β ∗ given as, β ∗ = 4 πr + ˜ d − r ˜ d − r ˜ d + ! d − (94)This is the inverse temperature at ρ = ∞ . The local β is given as, β = ∆ / ∆ − d d + ˜ d ) − β ∗ (95)which is the inverse of local temperature at ρ . The black p -brane will be placed in a cavitywith its wall at ρ = ρ B . It is clear from the metric in (92) that the physical radius of thecavity is ¯ ρ B = ∆ a
24 ˜ d − ρ B , (96)while ρ B is merely the coordinate radius. It is this ¯ ρ B which we should fix in the followingdiscussion and not ρ B (as in the non-dilatonic case). Also we fix the dilaton on theboundary which is the requirement of obtaining its standard equation of motion fromthe action given later. In other words, we fix the dilaton at ¯ ρ B , which indicates that Note that the form of a is fixed by supersymmetry, in the sense that these are solutions of super-gravity with maximal supersymmetry. r ± = ∆ a
24 ˜ d − r ± (97)and ¯ r ± are the proper parameters which we should use in the present context. In termsof the ‘barred’ variables ∆ ± remain the same as before,∆ ± = 1 − r ˜ d ± ρ ˜ dB = 1 − ¯ r ˜ d ± ¯ ρ ˜ dB (98)For the canonical ensemble we have fixed local temperature at the wall of the cavity, fixedlocal brane volume V p = ∆ ˜ d ( d − D − − V ∗ p and fixed charge defined as, Q d = i √ κ Z e − a ( d ) φ ∗ F [ p +2] = Ω ˜ d +1 √ κ e − aφ / ˜ d ( r + r − ) ˜ d/ = Ω ˜ d +1 ˜ d √ κ e − a ¯ φ/ (¯ r + ¯ r − ) ˜ d/ (99)where in the last line we have expressed the asymptotic value of the dilaton by the fixeddilaton ¯ φ ≡ φ ( ¯ ρ B ) at the wall of the cavity from the relation given in (92) and thenexpressed r ± by ¯ r ± from (97).With these data we will now evaluate the action. The relevant action for the gravitycoupled to the dilaton and a ( p + 1)-form gauge field with the Euclidean signature hasthe form I E = I E ( g ) + I E ( φ ) + I E ( F ) (100)where, I E ( g ) is the gravitational part of the action, I E ( φ ) is the action for the dilaton and I E ( F ) is the action for the form-field and are given as, I E ( g ) = − κ Z M d D x √ g R + 1 κ Z ∂M d D − x √ γ ( K − K ) ,I E ( φ ) = − κ Z M d D x √ g (cid:18) −
12 ( ∂φ ) (cid:19) ,I E ( F ) = 12 κ d + 1)! Z M d D x √ g e − a ( d ) φ F d +1 − κ d ! Z ∂M d D − x √ γ n µ e − a ( d ) φ F µµ µ ··· µ d A µ µ ··· µ d , (101)The various quantities in the above actions have already been defined in section 2. The31quations of motion following from the action (100) have the forms, R µν − g µν R = 12 ∂ µ φ∂ ν φ −
14 ( ∂φ ) g µν + 12 1 d ! e − a ( d ) φ (cid:18) F µµ µ ··· mu d F ν µ µ ··· µ d − d + 1) g µν F (cid:19) , (102) (cid:3) φ = − a ( d )2( d + 1)! e − a ( d ) φ F d +1] (103) ∇ µ ( e − a ( d ) φ F µ ··· µ d +1 ) = 0 (104)Using the equation of motion, the action can be reduced to: I E = d D − κ ( d + 1)! Z M d D x √ ge − a ( d ) φ F d +1 + 1 κ Z ∂M d D − x √ γ ( K − K ) − κ d ! Z ∂M d D − x √ γ n µ e − a ( d ) φ F µµ µ ··· µ d A µ µ ··· µ d . (105)From the metric in (92) we have n µ = ∆ / ∆ − a
24 ˜ d − δ µρ , (106)The extrinsic curvature for the p -brane can be calculated as before, K = −∇ µ n µ = − √ g ∂ µ ( √ gn µ )= − ∆ + ∆ − a
24 ˜ d − ρ − a d − + ˜ d a ! ∆ − − ! (107)The extrinsic curvature K can be calculated as, K = − ( ˜ d + 1)∆ − a
24 ˜ d − ρ = − ˜ d + 1¯ ρ (108)where we have defined ¯ ρ = ∆ a
24 ˜ d − ρ and note that with this redefined ρ , K takes exactlythe same form as in the non-dilatonic case.Now calculating each term in the action (105) separately as before we obtain, I E = − βV p Ω ˜ d +1 κ ¯ ρ ˜ dB " (cid:18) ∆ + ∆ − (cid:19) / + ˜ d (cid:18) ∆ − ∆ + (cid:19) / + ˜ d (∆ + ∆ − ) / −
2( ˜ d + 1) . (109)Note that in the above action everything is expressed in terms of the ‘barred’ parametersdefined earlier instead of the original parameters. Comparing the reduced action (109)32ith the corresponding reduced action for the non-dilatonic branes (13) we find that theyhave exactly the same form in terms of the redefined parameters. Once we have this formof the action (109), we can rewrite it as before in the form I E = βE − S as, I E = − βV p Ω ˜ d +1 κ ¯ ρ ˜ dB " ( ˜ d + 2) (cid:18) ∆ + ∆ − (cid:19) / + ˜ d (∆ + ∆ − ) / −
2( ˜ d + 1) ¯ ρ =¯ ρ B − π V ∗ p Ω ˜ d +1 κ r ˜ d +1+ − r ˜ d − r ˜ d + ! + d , (110)where we have used β ∗ = 4 πr + ˜ d − r ˜ d − r ˜ d + ! d − , (111)We can thus identify the entropy S = 4 πV ∗ p Ω ˜ d +1 κ r ˜ d +1+ − r ˜ d − r ˜ d + ! + d (112)and the energy of the cavity as, E = − V p Ω ˜ d +1 κ ¯ ρ ˜ dB " ( ˜ d + 2) (cid:18) ∆ + ∆ − (cid:19) / + ˜ d (∆ + ∆ − ) / −
2( ˜ d + 1) (113)Note that the entropy has exactly the same form as that of the non-dilatonic brane andwe find that the energy approaches the ADM mass at ¯ ρ B → ∞ as expected. We wouldlike to remark that all the quantities like energy, entropy and temperature in the cavity allhave the invariant forms in terms of ‘unbarred’ (non-dilatonic case) and ‘barred’ (dilatoniccase) coordinates. Energy expression given in (113) has already the same form as can becompared with (16). Entropy given in (112) can be written as, S = 4 πV ∗ p Ω ˜ d +1 κ r ˜ d +1+ − r ˜ d − r ˜ d + ! + d = 4 πV p Ω ˜ d +1 κ ¯ r ˜ d +1+ ∆ − − d − − ¯ r ˜ d − ¯ r ˜ d + ! + d (114)Comparing (114) with (15) we find that indeed they have exactly the same form. Similarlywe have from (95) and (94) the expression for inverse temperature as, β = 4 π ¯ r + ˜ d ∆ + ∆ − d − − ¯ r ˜ d − ¯ r ˜ d + ! d − (115)33omparing (115) with (21) we again find that they have exactly the same form. Thistherefore indicates that the thermodynamical quantities of the non-dilatonic branes es-sentially have the same structure as the dilatonic branes with the new physical parameters.Now using the expression of charge (99) we can write ¯ r − in terms of ¯ r + as,¯ r − = √ κQ d Ω ˜ d +1 ˜ d e a ¯ φ/ ! d r + = ( Q ∗ d ) ¯ r + (116)Where we have defined Q ∗ d = [( √ κQ d e a ¯ φ/ ) / (Ω ˜ d +1 ˜ d )] / ˜ d . Note that since the charge Q d is fixed inside the cavity and the dilaton ¯ φ is fixed at the wall of the cavity Q ∗ d is alsofixed. Therefore, ¯ r − is not an independent parameter, but is dependent on ¯ r + as given in(116). Now using (116) we can write (115) as, β = 4 π ¯ r + ˜ d − Q ∗ d d ¯ r ˜2 ˜ d + ! d − − ¯ r ˜ d + ¯ ρ ˜ dB ! / − Q ∗ d d ¯ r ˜ d + ¯ ρ ˜ dB ! d , (117)Now as before we can define x = (cid:18) ¯ r + ¯ ρ B (cid:19) ˜ d ≤ , ¯ b = β π ¯ ρ B , q = (cid:18) Q ∗ d ¯ ρ B (cid:19) ˜ d (118)In terms of these parameters (117) takes the form,¯ b = 1˜ d x / ˜ d (1 − x ) / (cid:18) − q x (cid:19) d − (cid:18) − q x (cid:19) − d ≡ b q ( x ) (119)Comparing with (24) and (25) we find that this equation (119) has exactly the same formas for the non-dilatonic branes. The expression of b q ( x ) was crucial for our analysis for theequilibria and stability structure of the black p -branes. Since the dilatonic branes have thesame expression for b q ( x ) as the non-dilatonic branes, the phase structure for the dilatonicbranes would be exactly the same as the non-dilatonic branes. The reduced Euclideanaction ˜ I E can be seen from (110) to take exactly the same form as the non-dilatonic branesgiven in (27), (28) and (29) in terms of the new parameters (118).So far, we have not addressed the issue regarding the validity of using the effectiveaction in describing the phase structure of black p -branes throughout the parameter spaceconsidered . Here we will address this for both non-dilatonic and dilatonic branes to-gether. For non-dilatonic branes, we need to keep the curvature of black brane spacetimeuniformly weak throughout the parameter space. For dilatonic branes, in addition, we We thank the anonymous referee for raising this concern. e φ = g s ∆ a/ − ,R = ˜ d " a + ∆ − (cid:18) ¯ r − ¯ ρ (cid:19) d ¯ r + ¯ r − ¯ ρ + d − ˜ dD − (cid:18) ¯ r + ¯ r − ¯ ρ (cid:19) ˜ d +1 r − ¯ r + , (120)where g s = e φ is the asymptotic string coupling. Note that for the scalar curvature eachterm in the square bracket is less than unity since ¯ ρ ≥ ¯ r + > ¯ r − , ∆ + / ∆ − < a / < d − ˜ d ) / ( D − <
1. In other words, the square bracket contributes at most a factorof order unity to the curvature. So in order to keep the curvature uniformly weak, weneed to have l R ∼ l ¯ r − ¯ r + = l " Ω ˜ d +1 √ κQ d e aφ / ∆ a / − / ˜ d ≪ , (121)where l is the relevant length scale under consideration, for example, it is the Planck scale l p in eleven dimensions or the string scale l s in ten dimensions. Note that the chargequantization gives √ κQ d e aφ / / Ω ˜ d +1 ∼ N l ˜ d with the integer N labeling the number ofbranes. So the uniformly weak curvature condition is N ∆ a / − ≫ . (122)For M branes, a = 0 and a weak curvature is the only requirement which can be satisfiedwhen N ≫
1. For D3 branes, we need in addition a small g s . For those branes, it doesn’tappear that there is any constraint on the parameter space we considered. For dilatonicbranes, the condition (122) for weak curvature can easily be satisfied for non-extremalbranes, i.e., r + > r − , for given large enough N since ∆ − is finite for ρ ≥ r + . Now theeffective string coupling as given in (120) can remain small for small g s when a > g s is chosen to be small enough for a <
0. For this case, theparameter x used in the text falls in the range q < x ≤
1. If we consider extremal branes,i.e., r + = r − , we then have to limit to the range ρ > r + = r − so that the curvatureremains small. This can also give a small effective string coupling even when a < q ≤ x <
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