Critical Phenomena and Reentrant Phase Transition of Asymptotically Reissner-Nordstrom Black Holes
aa r X i v : . [ g r- q c ] F e b Critical Phenomena and Reentrant Phase Transition ofAsymptotically Reissner–Nordstr¨om Black Holes
Mehrab Momennia , ∗ and Seyed Hossein Hendi , , † Department of Physics, School of Science, Shiraz University, Shiraz 71454, Iran Biruni Observatory, School of Science, Shiraz University, Shiraz 71454, Iran Canadian Quantum Research Center 204-3002 32 Ave Vernon, BC V1T 2L7 Canada
By considering a small correction to the Maxwell field, we show that the resultant black holesolutions (also known as the asymptotically Reissner–Nordstr¨om black holes) undergo the reentrantphase transition and can have a novel phase behavior. We also show that such a small nonlinearcorrection of the Reissner–Nordstr¨om black holes has high effects on the phase structure of thesolutions. It leads to a new classification in the canonical ensemble of extended phase space providingthe values of the nonlinearity parameter α being α S q /
7. We shall study these three classes andinvestigate deviations from those of the standard Reissner–Nordstr¨om solutions. Interestingly, wefind that there is the reentrant phase transition for α < q /
7, and for the case of α = 4 q / PACS numbers:
I. INTRODUCTION
It is well known that a black hole can be investigated as an ordinary thermodynamic system [1–3] with typicalentropy [4] and temperature [5] such that in most cases usually obeys the first law of thermodynamics [6]. It was alsoshown that these highly dense compact objects treat as usual thermodynamic systems enjoying the phase transitionphenomenon [7]. More interestingly, we can see the van der Waals-like (vdW-like) phase transition including chargedblack hole systems by considering the correspondence ( Q, Φ) ↔ ( P, V ) between conserved quantities and thermody-namic variables [8, 9]. Recently, in the context of black hole thermodynamics, a possible connection between thecosmological constant as a thermodynamical pressure is proposed [10, 11] which has attracted much attention. Thisrelation is defined as follows P = − Λ8 π , (1.1)in which the thermodynamical volume V is the conjugate quantity to pressure as V = (cid:18) ∂M∂P (cid:19) rep , (1.2)where ” rep ” refers to ”residual extensive parameters”. Indeed, the primary motivation of considering Λ as a thermo-dynamical pressure comes from the fact that several physical constants, such as Yukawa coupling, gauge couplingconstants, and Newton’s constant are not fixed values in some fundamental theories. In addition, in Tolman–Oppenheimer–Volkoff equation, Λ is added to pressure that shows the cosmological constant can play the role ofthe thermodynamical pressure. Besides, Λ is a slow variation parameter and has the dimension ( length ) − which isthe dimension of the pressure. Usually, a vdW-like small-large black hole (SBH-LBH) phase transition can be ob-served in thermodynamical systems including black holes whenever Λ behaves as a thermodynamical pressure. Thistype of phase transition has been studied extensively in the background spacetime of various black hole solutions (forinstance, see an incomplete list [12–25] and references therein written during recent years).The reentrant phase transition (RPT) phenomenon can be observed in an ordinary thermodynamical system when amonotonic change of any thermodynamical variable provides more than one phase transition such that the final phaseis macroscopically similar to the initial phase. There is a special range in temperature in the asymptotically Reissner–Nordstr¨om (ARN) black holes so that these solutions enjoy a large-small-large phase transition by a monotonicchange in the pressure. This interesting phase behavior has been observed in ordinary thermodynamical systems, ∗ email address: [email protected] † email address: [email protected] such as nicotine-water mixture [26], liquid crystals, binary gases, multicomponent fluids, and other different typicalthermodynamic systems [27]. In the context of black hole thermodynamics, the RPT is reported for Born-Infeldsolutions [28, 29], rotating black holes [30], asymptotically dS black holes [31], hairy black holes [32], black holesolutions in massive gravity [33], and Born–Infeld-dilaton black holes [34].In this paper, we study the thermodynamics of ARN black holes, investigate the RPT in the extended phase space,and find deviations from those of the standard Reissner–Nordstr¨om (RN) solutions. We also discuss novel phenomenonof our black hole case study and compare it with the standard RN black holes. II. REVIEW OF SOLUTIONS AND THERMODYNAMICS
In this section, we are going to briefly mention the solutions and thermodynamics of black holes in the presence ofquadratic nonlinear electrodynamics. Before proceeding, it is worthwhile to give some motivations.Nonlinear field theories are of interest in various classes of mathematical physics since most physical systems arebasically nonlinear in the nature. The nonlinear electrodynamic (NED) fields are much richer than the Maxwelltheory and in special cases they reduce to the linear Maxwell field. Different constraints of the Maxwell field, likethe radiation propagation inside specific materials [35–38] and description of the self-interaction of virtual electron-positron pairs [39–41], motivate one to consider NED theories as effective fields [42, 43]. Moreover, a well-knownoutstanding problem is that most gravitational theories predict a singularity in the center of black holes. It wasshown that by employing the NED fields, the big bang and black hole singularities can be removed [44–49]. Besides,the NEDs have important effects on the structure of the superstrongly magnetized compact objects, such as pulsarsand strange stars [50–52].The Lagrangian of Born-Infeld-type NED theories [53–56], which each one was constructed based on various moti-vations, tends to the following form for weak nonlinearity [57] L ( F ) = − F + αF + O (cid:0) α (cid:1) , (2.1)where F = F µν F µν is the Maxwell invariant, F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor, and A µ is thegauge potential. In this equation, α denotes nonlinearity parameter that is a small quantity and proportional to theinverse value of nonlinearity parameter in Born-Infeld-type NED fields. Indeed, although different models of NEDshave been constructed with various primitive aims, they contain physical and experimental importance just for theweak nonlinearity since the Maxwell field in various branches leads to near accurate or acceptable results. Thus, intransition from the Maxwell field to NEDs, considering the weak nonlinearity effects seems to be reasonable and alogical decision. In other words, we expect to obtain precise physical results with experimental agreements wheneverthe nonlinearity is considered as a correction to the Maxwell theory. In this context, regardless of constant parameterswhich are contracted in α , most NED Lagrangians reduce to Eq. (2.1) for weak nonlinearity and we shall considerthis Lagrangian as an effective matter source coupled to gravity.The mentioned motivations have led to publish some interesting and reasonable works by employing Eq. (2.1) as aneffective Lagrangian of electrodynamics [39–43, 58–67]. Heisenberg and Euler demonstrated that quantum correctionslead to nonlinear properties of vacuum [39–43, 58]. Besides, it was shown that a quartic correction of the Maxwellinvariant appears in the low-energy limit of heterotic string theory [59–67]. Therefore, considering a correction termto the Maxwell field and investigating Eq. (2.1) as an effective and suitable Lagrangian of electrodynamics instead ofthe Maxwell and other NED fields is a reasonable and logical decision.According to the mentioned motivations, we consider the topological black holes in ( n + 1)-dimensional spacetimewith perturbative nonlinear electrodynamics [57]. The ( n + 1)-dimensional line element reads ds = − f ( r ) dt + dr f ( r ) + r d Ω n − , (2.2)where f ( r ) is the metric function and d Ω n − represents the line element of ( n − n −
1) ( n − k and volume ω n − with the following explicit form d Ω n − = dθ + n − P i =2 i − Q j =1 sin θ j dθ i k = 1 dθ + sinh θ dθ + n − P i =3 i − Q j =2 sin θ j dθ i ! k = − n − P i =1 dφ i k = 0 , (2.3)The metric function of these black holes can be obtained as [57] f ( r ) = k − mr n − − r n ( n −
1) + 2 q ( n −
1) ( n − r n − − q [2 ( n −
2) ( n + 2) + ( n −
3) ( n − r n − α + O (cid:0) α (cid:1) , (2.4)in which m and q are two integration constants which are related to the total mass and total electric charge of theblack hole, and the last term indicates the effect of nonlinearity.The Hawking temperature of these black holes can obtained by using the definition of the surface gravity on theoutermost horizon, r + , T = 12 π ( n − ( n −
1) ( n − k r + − Λ r + − q r n − + 2 q r n − α ! + O (cid:0) α (cid:1) . (2.5)Moreover, as we are working in Einstein gravity, the entropy of the black holes can be calculated via the quarter ofthe event horizon area S = r n − , (2.6)which shows the entropy per unit volume ω n − . The electric potential Φ, measured at infinity as a reference withrespect to the event horizon is given byΦ = q ( n − r n − − q (3 n − r n − α + O (cid:0) α (cid:1) . (2.7)Besides, the total electric charge per unit volume ω n − , can be obtained by considering the flux of the electric fieldat infinity as Q = q π . (2.8)At the final stage of calculating the conserved and thermodynamic quantities, one can get the total mass of obtainedblack holes by using the behavior of the metric at large r . Therefore, the total mass per unit volume ω n − is given by M = ( n − m π . (2.9)Considering the entropy and electric charge as a complete set of extensive parameters, one can show that theseconserved and thermodynamic quantities satisfy the first law of thermodynamics [57] dM = T dS + Φ dQ. (2.10)It is worthwhile to mention that all the equations (2.4)-(2.10) are representing the background geometry andthermodynamics of the higher dimensional ARN black holes and they reduce to the higher dimensional standard RNsolutions in the special limit α = 0. III. REENTRANT PHASE TRANSITION
It is proved that the Schwarzschild (AdS) black holes have no vdW-like phase transition [68], while this phenomenonhas been observed in the RN black holes [12]. Thus, the electric charge of black holes usually plays a key role inobserving such a phase transition and it would be interesting to include the effects of black hole’s charge in thethermodynamic calculations. In this paper, we show how a small correction to the Maxwell field highly affectsthe phase transition structure of the RN solutions and extends the thermodynamical phase space into three newdifferent regions. The mentioned correction is motivated by nonlinear properties of vacuum generated from quantumcorrections, appearing a quartic correction of the Maxwell invariant in the low-energy limit of heterotic string theory,and physical and experimental importance of adding a weak nonlinearity to the Maxwell field.In what follows, we concentrate our attention on the spherical symmetric black holes with negative cosmologicalconstant in 4-dimensional spacetime. Calculations show that the RPT can also occur for higher dimensional solutions, r + c r + FIG. 1: r + c and ˆ r + in q − α plane. The black region on the left indicates imaginary r + and ˆ r + whereas the colorful arearepresents real r + and ˆ r + . At the border between black and colorful areas, r + and ˆ r + are equal to 2 q . but this is not the case for positive cosmological constant and/or flat or hyperbolic solutions. The negative cosmologicalconstant in the extended phase space plays the role of a positive thermodynamical pressure as follows [10, 11] P = − Λ8 π . (3.1)In this scenario, the total mass (2.9) behaves as the enthalpy of system, and the Smarr formula and first law ofthermodynamics are modified as M = 2 T S + Φ Q − V P + 2 A α ; A = (cid:18) ∂M∂α (cid:19) S,Q,P , (3.2) dM = T dS + Φ dQ + V dP + A dα, (3.3)where A is a new thermodynamical variable conjugate to α and as mentioned before, V is the thermodynamicalvolume conjugate to P as follows V = (cid:18) ∂M∂P (cid:19) S,Q,α = 13 r . (3.4)Here, we study the thermodynamics of 4-dimensional black holes in the canonical ensemble (fixed Q and α ) ofextended phase space. So, by using the temperature (2.5) for n = 3 T = − Λ r + π + 14 πr + − q πr + q α πr , (3.5)and the relation between the cosmological constant and pressure (3.1), it is straightforward to show that the equationof state, P = P ( r + , T ), is given by P = T r + − πr + q πr − q α πr . (3.6)The thermodynamic behavior of the system and its global stability are governed by the free energy, and thus, weobtain the Gibbs free energy as well. We can determine the Gibbs free energy per unit volume ω in the extendedphase space by employing the following relation G = M − T S = r + π − r P q πr + − q α πr . (3.7) T < T t T t < T < T z T z < T < T c T = T c T > T c - r + P P < P t P t < P < P z P z < P < P c P = P c P > P c T G T FIG. 2: P − r + and G − T diagrams for different regions of temperatures and pressures. In the right panel, the solid lines referto C P > C P < C P diverges at the joins of dashed and solid lines. Besides, G − T curves are shifted for clarity. The vertical black line at T , T t < T < T z , shows a discontinuity in the Gibbs free energy andindicates a zeroth-order SBH-IBH phase transition. T P critical pointSBH LBHtriple point P t P c T t T c no BHregion IBH SBH LBH P t P z T t T z FIG. 3: P − T diagram. The yellow region illustrates the no black hole area and the right panel represents a close-up of theRPT area. The blue curve shows the coexistence line of SBHs and LBHs whereas the green curve refers to the coexistenceline of IBHs and small ones. On crossing the blue (green) line, the system goes under a first (zeroth) -order phase transitionbetween SBHs and LBHs (IBHs and SBHs). On the other hand, the heat capacity help us to find the local thermal stability, and thus, we calculate it in extendedphase space at constant pressure as C P = T (cid:18) ∂S∂T (cid:19) P = r (cid:0) πP r + r − q r + 2 q α (cid:1) (cid:0) πP r − r + 3 q r − q α (cid:1) . (3.8)Here, since we are working in the canonical ensemble, C P is the heat capacity at constant P , Q , and α . Thenegativity of C P indicates unstable solutions while its positivity refers to local stability (or at least metastability).In order to study the phase transition of black holes, one can use the definition of inflection point at the criticalpoint of isothermal P − V (or equivalently P − r + ) diagram ∂P ( r + , T ) ∂r + (cid:12)(cid:12)(cid:12)(cid:12) T = ∂ P ( r + , T ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) T = 0 , (3.9)which can be used to obtain the critical horizon radius r + c and critical temperature T c . One can easily show that thisequation leads to the following equation for the critical horizon radius r − q r + 56 q α = 0 , (3.10)with at most two real positive solutions as follows r + c = s q (cid:18) q X (cid:19) + 2 X , (3.11)ˆ r + c = vuut q i (cid:0) i + √ (cid:1) q X ! + i (cid:16) i − √ (cid:17) X , (3.12)where X = (cid:18) q − q α + 12 p αq (7 α − q ) (cid:19) / . (3.13)From now, the thermodynamic behavior of these black holes depends on the values of (3.11) and (3.12) which areillustrated in Fig. 1; (a) when r + c and ˆ r + c are imaginary/complex, there is neither vdW-like phase transition norRPT (black region of Fig. 1). In addition, the behavior is not like the ideal gas and we shall discuss this region later.(b) in the colorful area of Fig. 1 that both r + c and ˆ r + c are real, the RPT is observed which we investigate it in thissection. (c) at the border of these black and colorful areas, r + c is equal to ˆ r + c and is determined by α = 4 q /
7. Inthis case, there is a critical point such that no phase transition occurs below and at this point. Besides, the SBHsand LBHs are thermodynamically distinguishable above this critical point and we will study this border in the nextsection. (d) for some higher values of q and α , ˆ r + c is imaginary/complex while r + c is real. This leads to the standardvdW-like (first-order SBH-LBH) phase transition which is investigated extensively before (for instance, see [12] forthe standard RN black holes and [69] for our black hole case study) and we do not consider it in this paper. Theother option, means real ˆ r + c and imaginary r + c , is not accessible for the system.One may note that in the absence of the nonlinearity (the case of RN black hole), r + c reduces to √ q and ˆ r + c vanishes, as it should be. Therefore, the RN black hole can only undergo the vdW-like phase transition for nonzerovalues of the electric charge. This fact uncovers the significant role of the nonlinearity parameter α on the phasetransition structure of these black holes. However, there is a constraint on choosing the values of q and α . Consideringthe last (correction) term of Eqs. (3.6) and (3.7), we should choose some values of q and α so that this term be ignorablecompared with the third term, and therefore, can be considered as a perturbation. Hence, as the simplest option,we can consider some values of q and α so that αq << r /
2. However, we can ignore this restriction since one canconsider the last term as a nonlinear term rather than just a perturbation (correction) term.As a typical example and without loss of generality, we consider q = 0 . α = 0 . T c = 12 πr + c − q πr c + 4 q απr c , (3.14) P c = T c r + c − πr c + q πr c − q α πr c . (3.15)For the fixed q = 0 . α = 0 .
1, the general behavior of the ARN black holes is shown in Fig. 2. This figureis plotted for various areas of temperature (or equivalently pressure) in P − r + (or equivalently G − T ) diagram.The dashed red lines show the negative heat capacity (3.8) and refer to unstable black holes while the solid linesstand for the positive heat capacity representing the stable (or metastable) black holes (for more discussion regardingthe relation between Gibbs free energy and heat capacity see [23]). Considering Fig. 2, we see that a critical pointlocated at P = P c in G − T diagram (at T = T c in P − r + diagram with an inflection point) and demonstrates asecond-order phase transition from SBHs to large ones. In G − T diagram, the curve looks like the Hawking-Pagephase transition for P > P c [7]. For P t < P < P c and T t < T < T c , there is an area that black holes undergothe standard first-order SBH-LBH phase transition. Besides, there are three different phases including intermediate T < T t T t < T < T z T z < T < T c T = T c T > T c T < T c T = T c T > T c - r + P RN - BH T P critical pointSBH LBHtriple point P t P c T t T c no BHregion RN critical point FIG. 4: P − r + and P − T diagrams including the RN black hole. The dotted lines in both figures show the behavior of theRN black hole. The nonlinear parameter highly affects the SBHs and this modification term creates a new black hole region asIBHs and increases the critical temperature and pressure. black holes (IBHs), SBHs, and LBHs for P ∈ ( P t , P z ). The vertical line at T = T ∈ ( T t , T z ) indicates a zeroth-orderphase transition between SBHs and IBHs which is characterized by a discontinuity in the Gibbs energy. In this areaof pressures and temperatures, black hole undergoes a first-order SBH-LBH phase transition as well. This behavioris known as the RPT. Note that IBHs are macroscopically similar to large ones, and thus, black holes enjoy thelarge-small-large phase transition in this region of pressures and temperatures. Finally, we have just LBHs for P < P t and T < T t .Figure 3 describes the coexistence lines of SBHs+LBHs (the blue line) and IBHs+SBHs (the green line) in differentscales. The blue line is located between the critical point ( T c , P c ) and the triple point ( T t , P t ) between SBHs, IBHs,and LBHs. Similarly, the green line is bounded between this triple point and point ( T z , P z ). The black holes enjoy afirst (zeroth)-order phase transition from SBHs to LBHs (IBHs to SBHs) whenever they cross the blue (green) linefrom left to right or top to bottom. Therefore, we observe the RPT behavior of the ARN black holes for a narrowrange of temperatures T ∈ ( T t , T z ) and pressures P ∈ ( P t , P z ).Now, it is worthwhile to do a comparison between the ARN black holes and the RN ones to see how this smallperturbation in the Maxwell field changes the thermodynamical behavior of the RN black holes significantly. Indeed,observing the RPT for this kind of black hole is very interesting since such a behavior cannot be seen for a large classof black holes even with more complicated generalizations in the matter field and/or gravitational sector of the fieldequation.Figure 4 shows the differences between the ARN black holes and the RN ones. From the left panel of this figure, wefind that the nonlinearity parameter reduces the pressure of SBHs significantly whereas the pressure of LBHs almostremains unchanged. Besides, the high pressure SBHs at low temperatures are not allowed to exist while this is notthe case for high temperature SBHs. These facts can be seen analytically from the equation of state (3.6) as well. ForSBHs, the nonlinear term grows significantly and reduces the pressure since it has a negative sign, and finally leadsto a negative pressure for these black holes that are not allowed to exist. But for high temperatures, the first termdominates the pressure and we have SBHs in this case. On the other hand, the correction term will be very small forthe LBHs and does not affect the pressure of these black holes.It is worthwhile to mention that the minimum accessible size for SBHs at (and below) the critical point is about r + ∼ .
8, and therefore, the ratio of the correction and Maxwell terms ( correction/M axwell ratio) is at most about ∼ .
3. Thus, the nonlinear term is small even in the worst case and never dominates the behavior of the system.In addition, from the right panel of Fig. 4, one finds that the nonlinear term creates a new region as IBHs andincreases the critical temperature and pressure. This behavior can also be understood from Eqs. (3.14) and (3.15)by considering the fact that the last term is ignorable since r c >> αq (see the left panel of Fig. 1). However, thenonlinearity parameter does not affect the SBH-LBH phase transition point significantly (the right panel of Fig. 4). T < T c T = T c T > T c T < T c T = T c T > T c - - r + P RN - BH T P RN critical pointSBH LBH P c T c H no BH region L critical point FIG. 5: P − r + and P − T diagrams for the special case α = 4 q / IV. SPECIAL CASE α = 4 q / From the previous section, we observed that the colorful region related to α < q / α = 4 q /
7, and also, the black areaof this figure determined by α > q /
7. For α = 4 q /
7, Eqs. (3.11) and (3.12) give the same results as follows r + c = ˆ r + c = 2 q, (4.1)which is a critical point since the response function (3.8) diverges at this point. In this case, the critical temperature(3.14) and critical pressure (3.15) reduce to T c = 17 πq ; P c = 3256 πq . (4.2)Here, we fix q as q = 0 . α = 4 q / r + . From the left panel, we find thatthe nonlinearity parameter converts the stable small RN black holes to unconditionally unstable SBHs and slightlyaffects the LBHs. The right panel shows the effect of α on LBHs so that the region of these black holes is extendedand the pressure is increased. In the case of the RN black holes (and also, the critical phenomena of other black holesolutions) the SBHs and LBHs are thermodynamically indistinguishable above the critical point since the coexistencecurve always terminates at the critical point whereas for our black hole case study, there is always a border between(unconditionally unstable) SBHs and large ones (blue line of right panel of Fig. 5). This distinguishable property isa novel feature observed in this special type of black holes and is due to the fact that there is no SBH-LBH phasetransition in this special case and SBHs are always unstable. Indeed, another interesting and new behavior is thatthere is no SBH-LBH phase transition at (and below) the critical point.It is worthwhile to mention that the thermodynamic behavior of the black area of Fig. 1 determined by α > q / α = 4 q / r + c and ˆ r + c are imaginary, and thus, the critical point specified by (4.1) and(4.2) is absent. Thus, in this case, the SBHs and LBHs are always distinguishable while the SBHs are unconditionallyunstable. V. CONCLUSIONS
In this paper, we have considered the cosmological constant as thermodynamical pressure and studied the thermo-dynamics of 4-dimensional ARN black holes in the canonical ensemble of extended phase space and deviations fromthose for the standard RN black holes were investigated. We interestingly found that by considering a small correctionin the Maxwell field, the thermodynamical behavior of the RN black holes changes significantly and a novel criticalphenomenon can be observed. Based on the values of the nonlinearity parameter, the phase space classified into threeregions, and thus, three kinds of behaviors have been found which one of them was the RPT and the other one wasa novel behavior in the extended phase space thermodynamics.Specially, we have seen that in addition to the standard vdW-like phase transition of the black hole case study [69]and the RN black holes [12], they can enjoy the RPT by considering this small correction in the Maxwell field. Itwas shown that this behavior happens for a narrow range of temperatures and pressures. In this range of RPT, blackholes undergo a zeroth-order IBH-SBH phase transition and first-order SBH-LBH phase transition, and this behaviorcould be seen for special values of the nonlinearity parameter α < q /
7. In comparison with the RN black holes,the nonlinearity parameter highly affected the SBHs and converted them to unstable ones. This modification termcreated a new black hole region as IBHs and increased the critical temperature and pressure as well.Moreover, it was shown that for special values of the nonlinearity parameter as α ≥ q /
7, the correction termhighly affects the thermodynamical behavior of the solutions as well. Specially, in the case of α = 4 q /
7, we observeda novel critical point such that below and at this point, the black holes had no phase transition, and above this criticalpoint, SBHs and LBHs were thermodynamically distinguishable. Besides, the stable small RN black holes convertedto unconditionally unstable SBHs. This nonlinear term extended the area of LBHs and increased the pressure of theseblack holes.As the final remark, since introducing a small correction in the Maxwell field, interestingly, had significant effectson the thermodynamical structure of the RN black holes, it would be nice to consider dynamical perturbations inthe background geometry of the ARN black holes and investigate the effects of the nonlinearity parameter on thedynamical stability and quasinormal modes, and then compare them with those of the RN solutions.
Acknowledgments
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