Critical behavior of Born Infeld AdS black holes in higher dimensions

# Critical behavior of Born Infeld AdS black holes in higher dimensions

Rabin Banerjee ,Dibakar Roychowdhury
S. N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata-700098, India
e-mail: rabin@bose.res.ine-mail: dibakar@bose.res.in
###### Abstract

Based on a canonical framework, we investigate the critical behavior of Born-Infeld AdS black holes in higher dimensions. As a special case, considering the appropriate limit, we also analyze the critical phenomena for Reissner Nordstrom AdS black holes. The critical points are marked by the divergences in the heat capacity at constant charge. The static critical exponents associated with various thermodynamic entities are computed and shown to satisfy the thermodynamic scaling laws. These scaling laws have also been found to be compatible with the static scaling hypothesis. Furthermore, we show that the values of these exponents are universal and do not depend on the spatial dimensionality of the AdS space. We also provide a suggestive way to calculate the critical exponents associated with the spatial correlation which satisfy the scaling laws of second kind.

## 1 Introduction

Gravitational physics in higher dimensions has been a topic of interest since the advances made in string theory, where most of the efforts have been made at studying theories in space time dimensions greater than . At the same time, black holes in higher dimensional theories of gravity have been found to posses much richer structures than those in four dimensions [1]. Now a days, it is generally believed that, understanding of the physics of black holes in higher dimensions is essential in order to understand a full theory of quantum gravity. Studying thermodynamic aspects of these black holes is one of the significant issues in this context.

The study of thermodynamic properties of black holes has been an intense topic of research since the discovery of the four laws of black hole mechanics by Bardeen, Carter and Hawking in the early seventies [2]. Since then the question regarding various thermodynamic aspects of black holes, specially the issue of phase transition and critical behavior in black holes remain highly debatable and worthy of further investigations. The study of phase transition as well as the critical phenomena in black holes requires an extensive analogy between the laws of black hole mechanics and that of the ordinary laws of thermodynamics. An attempt along this direction was first commenced by Davies [3] and Hut[4], where they studied the phase transition phenomena in dimensional Kerr Newman and Reissner Nordstrom black holes. The investigation of the thermodynamic behavior of black holes in AdS space was first made by Hawking and Page [5]. Their analysis was strictly confined to the dimensional Schwarzschild AdS space time. However, in recent years, thermodynamics of black holes in AdS space has attained renewed attention in the context of AdS/CFT duality where one can identify a similar thermodynamic structure in the dual conformal theory residing at the boundary [6]. A novel approach to study and classify the phase transition phenomena in black holes, based on the Ehrenfest’s scheme [7] of standard thermodynamics, has been recently initiated [8] and applied extensively to dimensional black holes in AdS space[9]-[12]. In this approach one can in fact show that black holes in dimensional AdS space can undergo a second order phase transition in order to attain thermodynamic stability [9]-[12]. Like in dimensions, black holes in higher dimensional AdS space have also been found to posses a stable thermodynamic structure [13].

In ordinary thermodynamics, a phase transition occurs whenever there is a singularity in free energy or one of its derivatives. The corresponding point of discontinuity is known as the critical point of phase transition. In Ehrenfest’s classification of phase transition, the order of the phase transition is characterized by the order of the derivative of free energy that suffers discontinuity at the critical points. For example, if the first derivative of the free energy is discontinuous then the corresponding phase transition is first order in nature. On the other hand, if the first derivatives are continuous and second derivatives are discontinuous or infinite then the transition may be referred as higher order or continuous. These type of transitions correspond to a divergent heat capacity, an infinite correlation length, and a power law decay of correlations near the critical point.

The primary aim of the theory of phase transition is to study the singular behavior of various thermodynamic entities (for example heat capacity) near the critical point. It order to do that one often expresses these singularities in terms of power laws characterized by a set of static critical exponents [14]-[15]. Generally these exponents depend on a few parameters of the system, like, (1)the spatial dimensionality () of the space in which the system is embedded, (2)the range of interactions in the system etc. To be more specific, it is observed that for systems possessing short range interactions, these exponents depend on the spatial dimensionality (). On the other hand for systems with long range interactions these exponents become independent of , which is the basic characteristic of a mean field theory and is observed for the case in usual systems. It is interesting to note that the (static) critical exponents are also found to satisfy certain thermodynamic scaling laws[14]-[15] which apply to a wide variety of thermodynamical systems, from elementary particles to turbulent fluid flow. Such studies have also been performed, albeit partially, in the context of black holes[16]-[32]. In this paper we provide a detailed analysis of these issues and show how the study of critical phenomena in black holes is integrated with corresponding studies in other areas of physics adopting a mean field approximation.

For the past two decades gravity theories with Born-Infeld action have garnered considerable attention due to its several remarkable features [33]-[40]. For example, Born-Infeld type effective actions, which arise naturally in open super strings and D branes are free from physical singularities. Also, using the Born Infeld action one can in fact study various thermodynamic features of Reissner Nordstrom AdS black holes in a suitable limit. During the last ten years many attempts have been made in order to understand the thermodynamics of Born-Infeld black holes in AdS space [41]-[46]. In spite of these efforts, several significant issues remain unanswered. Studying the critical phenomena is one of them. In order to have a deeper insight regarding the underlying phase structure for these black holes one needs to compute the critical exponents associated with the phase transition and check the validity of the scaling laws near the critical point. Although an attempt to answer these questions has been commenced very recently [47], still a systematic analysis of critical phenomena for higher dimensional Born-Infeld AdS (BI AdS) black holes is lacking in the literature. A similar remark also holds for the higher dimensional Reissner Nordstrom AdS (RN AdS) black holes [48]-[50].

In this paper, based on a canonical framework, we aim to study the critical behavior of charged black holes taking the particular example of Born-Infeld AdS (BI AdS) black holes in dimensions. Results obtained in the above case smoothly translate to that of the dimensional RN AdS case in the appropriate limit. The critical points of the phase transition are characterized by the discontinuities in heat capacity at constant charge . We compute the static critical exponents associated with this phase transition and verify the scaling laws. These laws are found to be compatible with the static scaling hypothesis for black holes [14]-[15]. Finally, taking the spatial dimensionality of the space time as , we also provide a suggestive argument in order to compute the critical exponents associated with the spatial correlation.

Before we proceed further, let us briefly mention about the organization of our paper. In section 2 we make a qualitative as well as quantitative analysis of the various thermodynamic entities for dimensional BI AdS black holes in order to have a meaningful discussion on their critical behavior in the subsequent sections. In section 3 we study various aspects of the critical phenomena both for the BI AdS and RN AdS black holes in dimensions. Finally, we draw our conclusion in section 4.

## 2 Thermodynamics of charged black holes in higher dimensional AdS space

In this section we compute the essential thermodynamic entities both for the BI AdS and RN AdS black holes that is required in order to explore the critical behavior of these black holes.

The action for the Einstein- Born-Infeld gravity in dimensions is given by, [43],

 S=∫dn+1x√−g[R−2Λ16πG+L(F)] (1)

where,

 L(F)=b24πG(1−√1+2Fb2) (2)

with . Here is the Born-Infeld parameter with the dimension of mass and is the cosmological constant. It is also to be noted that for the rest of our analysis we set Newton’s constant .

By solving the equations of motion, the Born-Infeld anti de sitter (BI AdS) solution may be found as,

 ds2=−χdt2+χ−1dr2+r2dΩ2 (3)

where,

 χ(r)=1−mrn−2+[4b2n(n−1)+1l2]r2−2√2bn(n−1)rn−3√2b2r2n−2+(n−1)(n−2)q2 +2(n−1)q2nr2n−4H[n−22n−2,12,3n−42n−2,−(n−1)(n−2)q22b2r2n−2], (4)

and is a hyper-geometric function [51].

It is interesting to note that for the metric (4) has a curvature singularity at . It is in fact possible to show that this singularity is hidden behind the event horizon(s) whose location may be obtained through the condition . Therefore the above solution (3) could be interpreted as a space time with black holes [45].

In the limit and one obtains the corresponding solution for Reissner Nordstrom (RN) AdS black holes. Clearly this is a nonlinear generalization of the RN AdS black holes. Here is related to the ADM mass () of the black hole as [43],[45],

 M=(n−1)16πωn−1m

where is the volume of the unit sphere. Identical expression could also be found for the RN AdS case [50].

Electric charge () may defined as [43],

 Q=14π∫∗FdΩ (6)

which finally yields,

 Q=√(n−1)(n−2)4π√2ωn−1q. (7)

Using (LABEL:mq) one can rewrite (4) as,

 χ(r)=1−16πM(n−1)ωn−1rn−2+r2l2+4b2r2n(n−1)⎡⎣1− ⎷1+16π2Q2b2r2(n−1)ω2n−1⎤⎦ +64π2Q2n(n−2)r2n−4ω2n−1H[n−22n−2,12,3n−42n−2,−16π2Q2b2r2n−2ω2n−1]. (8)

In order to obtain an expression for the ADM mass () of the black hole we set , which yields,

 M=(n−1)ωn−1rn−2+16π+ωn−1(n−1)rn+16πl2+b2rn+ωn−14πn⎡⎣1− ⎷1+16π2Q2b2r2(n−1)+ω2n−1⎤⎦ +4πQ2(n−1)n(n−2)ωn−1rn−2+[1−8π2Q2(n−2)b2(3n−4)r2n−2+ω2n−1]+O(1/b4) (9)

where is the radius of the outer event horizon. Here the parameter is chosen in such a way so that or, in other words our analysis is carried out in the large limit. Also, such a limit is necessary for abstracting the results for RN AdS black holes obtained by taking . Therefore all the higher order terms from onwards have been dropped out from the series expansion of . Moreover our results are valid upto an order . Using (LABEL:mq) one can express the electrostatic potential difference () between the horizon and infinity as,

 Φ = √n−12n−4qrn−2+H[n−22n−2,12,3n−42n−2,−(n−1)(n−2)q22b2r2n−2] (10) = 4πQ(n−2)ωn−1rn−2+[1−8π2Q2(n−2)b2(3n−4)r2n−2+ω2n−1]+O(1/b4)

where is the electric charge. Henceforth all our results are valid upto only.

Using (8) and (9), the Hawking temperature may be obtained as,

 T = χ′(r+)4π (11) = 14π⎡⎣n−2r++nr+l2+4b2r+n−1⎛⎝1− ⎷1+16π2Q2b2r2(n−1)+ω2n−1⎞⎠⎤⎦.

In the appropriate limit () the corresponding expression of Hawking temperature for the RN AdS black hole may be obtained as,

Using (9) and (11) the entropy of the BI AdS black hole may be found as,

 S=∫T−1(∂M∂r+)Qdr+=ωn−1rn−1+4. (13)

It is interesting to note that identical expression could also be found for the RN AdS case [50].

In order to investigate the critical phenomena, it is necessary to compute the heat capacity at constant charge (). Using (11) and (13) the specific heat (at constant charge) may be found as,

 CQ=T(∂S∂T)Q=T(∂S/∂r+)Q(∂T/∂r+)Q=I(r+,Q)R(r+,Q) (14)

where,

 I(r+,Q)=(n−1)ωn−1r3n−7+4√1+16π2Q2b2r2(n−1)+ω2n−1 ×⎡⎣(n−2)r2++nr4+l2+4b2r4+n−1⎛⎝1− ⎷1+16π2Q2b2r2(n−1)+ω2n−1⎞⎠⎤⎦ (15)

and,

 −4b2r2n−2+n−1⎛⎝1− ⎷1+16π2Q2b2r2(n−1)+ω2n−1⎞⎠. (16)

From (14) we note that in order to have a divergence in one must satisfy the following condition,

 −4b2r2n−2+n−1⎛⎝1− ⎷1+16π2Q2b2r2(n−1)+ω2n−1⎞⎠=0. (17)

Although it is quite difficult to solve (17) analytically, one can attempt to solve this equation numerically. In order to do that it is first necessary to fix the parameters ( and ) of the theory. It is generally observed that the choice of parameters is not completely arbitrary in dimension [44]. Nevertheless one can give certain plausibility arguments that give a bound on the parameter space in order to have real positive roots for (17). The boundedness of the parameter space could be achieved by demanding that a smooth extremal limit holds. In other words, we choose the parameters in such a way so that an extremal black hole could be found in the appropriate limit. Furthermore, once the choice of parameters has been determined in this manner, it is easy to show that meaningful results are not obtained in the non extremal case if one is not confined to this choice. Let us consider the extremal BI AdS black holes. Here both and vanish at the degenerate horizon (). From the above two conditions and using (9) we arrive at the following equation,

 1+(4b2n−1+nl2)r2en−2−4b2r2e(n−1)(n−2) ⎷1+16π2Q2b2r2(n−1)eω2n−1=0. (18)

For , equation (18) can be solved analytically for which results in the following bound on the parameter space of the BI AdS black hole as [44], [47],

 0.5≤bQ<∞. (19)

For , the root becomes negative and hence there exists no real positive solution for [44],[47]. A detailed analysis of the critical phenomena for the non extremal case was done by us [47] subject to the condition (19) that ensured a smooth extremal limit. We adopt a similar stance for higher dimensional black holes also.

The case

For , on the other hand, equation (18) turns out to be a cubic equation in the variable , which is indeed quite difficult to solve analytically. However, it is possible to solve equation (18) numerically for . The solutions are provided below in a tabular form (table 1) for various choice of parameters ( and ).

From the roots of equation (18) it is quite evident that for there is as such no bound on the parameter space of BI AdS black holes as far as a smooth extremal limit is concerned. It is also interesting to note that for we have only one real positive root for , whereas for we have two real positive roots for . We are now in a position to find out the roots of the equation (17) numerically for considering various values of the parameters ( and ), which are given below in table 2.

Two crucial points are to be noted at this stage- (i) (14) possesses only simple poles and (ii) there are always two real positive roots ( and ) of (17) for different choice of parameters. These two roots ( and ) correspond to the critical points for the phase transition phenomena occurring in BI AdS black holes. For our detailed analysis we choose and , corresponding to the first row in table 2. With this particular choice the critical points are found to be and , which are also depicted in various figures (1 and 2).

From figures 1 and 2 we observe that suffers discontinuities exactly at two points, namely and (discussed in the previous paragraph), which may be identified as the critical points for the phase transition phenomena in BI AdS black holes. From these figures we note that there is a sign flip in the heat capacity around , which indicates the onset of a continuous higher order transition near critical points. Using a grand canonical framework, one can in fact show that BI AdS black holes undergo a second order phase transition near the critical point [12]. From figures 1 and 2 we note that is positive for and , while it is negative for . It is the positive heat capacity that corresponds to a thermodynamically stable phase, whereas, on the other hand, negative heat capacity stands for a thermodynamically unstable phase. Also, since the black hole with larger mass possesses a larger horizon area/radius, therefore corresponds to the critical point for the transition between a lower mass (stable) black hole to an intermediate higher mass (unstable) black hole. On the other hand stands for the critical point for the transition between the intermediate unstable black hole to a higher mass (stable) black hole.

Similar features may also be observed for the RN AdS case. The expression for the heat capacity is obtained by taking the limit of (14). One obtains,

The plot of vs are given in figures 3 and 4,

The nature of the phase transition is similar to BI AdS case.

The case

For , equation (18) turns out to be a quartic equation in the variable . The corresponding roots () of the equation (18) are provided in the following tabular form (table 3). From table 3 we note that, even for one can have two real positive roots of the equation (18) in dimensions. However one should note that for the number of real positive roots again reduces to one (see, for instance, the last row in table 3).

Finally, we aim to find out the roots of the equation (17) for different choice of parameters ( and ), which essentially gives us the critical points for the phase transition in the non extremal regime.

Like in the earlier case, from table 4 we observe that (14) possesses simple poles, two of which are real positive ( and ) that may be regarded as the critical points corresponding to the phase transition phenomena in (non extremal) BI AdS black holes. Taking and , these critical points have been shown explicitly in figures (5 and 6).

Likewise, in the appropriate limit (), one can also obtain the corresponding critical points for the RN AdS case which are shown in various figures (7 and 8).

Therefore, from the above analysis it is quite suggestive that the boundedness on the parameter space, which exists in dimension, eventually disappears in higher dimensions which is also consistent with our finding of the critical points in the corresponding non extremal case for various choice of parameters.

## 3 Critical exponents and scaling laws in higher dimensions

In the usual theory of phase transitions it is customary to study the behavior of a given system close to its critical point by means of a set of critical exponents . These critical exponents determine the qualitative nature of the critical behavior of the given system in the neighborhood of the critical point. By virtue of the so called scaling laws, one can in fact see that only two of the eight critical exponents are actually independent. Based on the renormalization group approach, one can calculate these critical exponents. Systems having identical critical exponents have similar critical behavior and hence fall into the same universality class. Here we show similar conclusions may be drawn for black holes also. We find that, apart from that takes the value 2, all other critical exponents are . The result is independent of the dimensionality of space time. This shows that BI AdS black holes (and hence RN AdS black holes) in arbitrary dimensions fall in the same universality class.

In order to calculate the critical exponent () that is associated with the divergences of the heat capacity (), we first note that near the critical points () we can write

 r+=ri(1+Δ),    i=1,2 (21)

where .

As already discussed there are two distinct positive roots for the critical point ( and ). Also, any function of , in particular the temperature , may be expressed as

 T(r+)=T(ri)(1+ϵ) (22)

where . As a next step, for a fixed value of the charge (), we Taylor expand in a sufficiently small neighborhood of which yields,

 T(r+)=T(ri)+[(∂T∂r+)Q=Qc]r+=ri(r+−ri)+12⎡⎣(∂2T∂r2+)Q=Qc⎤⎦r+=ri(r+−ri)2 +higher  order  terms. (23)

Since diverges at , therefore the second term on the R.H.S. of (23) vanishes by virtue of equation (14). This fact has also been depicted in various other figures (see figures 5 and 6). Using (21) we finally obtain from (23)

 Δ=ϵ1/2D1/2i (24)

where111We use the notation .,

 Di=r2i2Ti⎡⎣(∂2T∂r2+)Q=Qc⎤⎦r+=ri=Σ(ri,Qc)4πr2n−3iTi(1+16π2Q2cb2r2n−2iω2n−1)3/2 (25)

with,

 Σ(ri,Qc)=(n−2)r2n−4i(1+16π2Q2cb2r2n−2iω2n−1)3/2+512(n−1)π4Q4cb2r2n−2iω4n−1 −32(2n−3)π2Q2cω−2n−1(1+16π2Q2cb2r2n−2iω2n−1). (26)

In the limit , the corresponding expression for RN AdS case may be obtained as,

A closer look at figures 5 and 6 reveals that in the neighborhood of we always have so that is positive. On the other hand for any point close to we have implying that is negative. We will exploit these observations to find near the critical points.

Let us first compute the value of near the critical point (where is positive). Substituting from (21) into (14) we obtain,

 CQ=I(ri(1+Δ),Qc)R(ri(1+Δ),Qc). (28)

Taylor expanding around the critical point we obtain,

 CQ≃   [Aiϵ1/2]ri=r2 (29)

where,

 Ai=D1/2iζ(ri,Qc)ξ(ri,Qc) (30)

with,

 ζ(ri,Qc)=(n−1)ωn−1r3n−7i8 ⎷1+16π2Q2cb2r2n−2iω2n−1 ×⎡⎣(n−2)r2i+nr4il2+4b2r4in−1⎛⎝1− ⎷1+16π2Q2cb2r2n−2iω2n−1⎞⎠⎤⎦ (31)

and,

 ξ(ri,Qc)=r2n−4i[(n−1)nr2il2−(n−2)2] ⎷1+16π2Q2cb2r2n−2iω2n−1 −8(n−1)π2Q2cb2r2iω2n−1(2−n+nr2il2). (32)

On the other hand, following a similar approach, the singular behavior of near (where is negative) may be expressed as,

 CQ≃   [Ai(−ϵ)1/2]ri=r1. (33)

Combining both of these facts into a single expression, we may therefore express the singular behavior of the heat capacity () near the critical points as,

 CQ ≃ Ai|ϵ|1/2 (34) = AiT1/2i|T−Ti|1/2.

Similar arguments also hold for the RN AdS case (see figures 7 and 8). In order to obtain the corresponding expression for the singular behavior of the heat capacity near the critical points, we set , which finally yields,

where,

Comparing (34) and (35) with the standard form

 CQ∼|T−Ti|−α (37)

we find .

Next, we want to calculate the critical exponent which is related to the electric potential () for a fixed value of charge as,

 Φ(r+)−Φ(ri)∼|T−Ti|β. (38)

In order to do that we Taylor expand close to the critical point which yields,

 Φ(r+)=Φ(ri)+[(∂Φ∂r+)Q=Qc]r+=ri(r+−ri)+higher  order  terms. (39)

Ignoring all the higher order terms in (39) and using (10) and (24) we finally obtain

 Φ(r+)−Φ(ri)=−⎛⎝4πQcrn−2iωn−1T1/2iD1/2i⎞⎠(1−8π2Q2cb2r2n−2iω2n−1)|T−Ti|1/2. (40)

For the RN AdS case () the corresponding expression becomes,

Comparing (40) and (41) with (38) we find .

We next calculate the critical exponent which is related to the singular behavior of the isothermal compressibility related derivative (near the critical points ) for a fixed value of charge (). This is defined as,

 K−1T∼|T−Ti|−γ (42)

In order to calculate we first note that,

 K−1T=Q(∂Φ/∂Q)T=−Q(∂Φ∂T)Q(∂T∂Q)Φ, (43)

where we have used the thermodynamic identity .

Finally, using (10) and (11) the expression for may be found as,

 K−1T=4πQ(n−2)ωn−1rn−2+(1−24π2Q2(n−2)(3n−4)ω2n−1b2r2n−2+)℘(Q,r+)R(Q,r+) (44)

where,

 ℘(Q,r+)=r2n−2+(nl2−n−2r2+) ⎷1+16π2Q2b2r2(n−1)+ω2n−1+1024π4Q4(n−2)r2n−2+b2ω4n−1(n−1)(3n−4) −4b2r2n−2+n−1⎛⎝1− ⎷1+16π2Q2b2r2(n−1)+ω2n−1⎞⎠. (45)

Taking the appropriate limit () one can easily obtain the corresponding expression of for the RN AdS black hole as,

From the above expressions it is interesting to note that both and the heat capacity () posses common singularities. It is interesting to note that a similar feature may also be observed for the Kerr Newmann black hole in asymptotically flat space [20]. It is reassuring to note that all these features are in general compatible with standard thermodynamic systems[52].

Following our previous approach, we substitute from (21) into (44) and use (24) which finally yields,

 K−1T=Bi|ϵ|1/2=BiT1/2i|T−Ti|1/2 (47)

where,

 Bi=2πQc(n−2)ωn−1rn−2i(1−24π2Q2c(n−2)(3n−4)ω2n−1b2r2n−2i)D1/2i℘(Qc,ri)ξ(Qc,ri). (48)

In the appropriate limit () the corresponding expression for the RN AdS black hole may be obtained as,