Criticality and extended phase space thermodynamics of AdS black holes in higher curvature massive gravity
Abstract
Considering de Rham–Gabadadze–Tolley theory of massive gravity coupled with (ghost free) higher curvature terms arisen from the Lovelock Lagrangian, we obtain charged-AdS black hole solutions in diverse dimensions. We compute thermodynamic quantities in the extended phase space by considering the variations of the negative cosmological constant, Lovelock coefficients (\(\alpha _{i}\)) and massive couplings (\(c_{i}\)). We also prove that such variations are necessary in order to satisfy the extended first law of thermodynamics as well as associated Smarr formula. In addition, by performing a comprehensive thermal stability analysis for the topological black hole solutions, we show that in which regions thermally stable phases exist. Calculations show the results are radically different from those in the Einstein gravity. We find that the phase structure and critical behavior of topological AdS black holes are drastically restricted by the geometry of the event horizon. We also show that the phase structure of AdS black holes with non-compact (hyperbolic) horizon could give birth to three critical points corresponds to a reverse van der Waals behavior for phase transition which is accompanied with two distinct van der Waals phase transitions. For black holes with the spherical horizon, the van der Waals, reentrant and analogue of solid/liquid/gas phase transitions are observed.
1 Introduction
Einstein’s General Relativity (GR, also known as Einstein gravity) has been astonishingly regarded as the most successful description of gravitation and a well supported by numerous experiments since it was proposed [1, 2, 3] (see also these reviews [4, 5]). Theoretically, inconsistency appears when GR is supposed to be reconciled with the laws of quantum physics for producing the theory of quantum gravity. From the experimental side, Einstein gravity has a problem with the accelerated expansion of the universe in the large scale structure since it needs an unknown source of energy (the so-called dark energy) captured by the cosmological constant [6, 7, 8, 9]. In this regard, various attempts have been made to find an alternative such that it modifies Einstein gravity in the large scales (IR limit). Massive gravity is one of the alternatives that modifies Einstein gravity by giving the graviton a mass and provides a possible explanation for the accelerated expansion of the universe without the requirement of dark energy component [10, 11, 12]. Assuming that gravitons are dispersed in vacuum like massive particles, gravitational waves’ observation of the coalescence for a pair of stellar-mass black holes (GW170104) has bounded the graviton mass to \({m_g} \le 7.7 \times {10^{ - 23}}\,eV/{c^2}\) [13]. On the other hand, depending on the exact model of massive gravity, the graviton mass is typically bounded to be a few times the Hubble parameter today, i.e., \({m_g} \le {10^{ - 30}-10^{ - 33}}\,eV/{c^2}\), in which for graviton mass region \( m_{g}\ll 10^{ - 33}\,eV/{c^2}\) , its observable effects would be undetectable [14] (for more details on different mass bounds see [15]). Massive gravitons, if they exist, are yet to be found; but, according to the recent data of LIGO, such an assumption is experimentally logical and therefore deserves to be explored theoretically [10, 11, 12, 13, 14, 15].
Depending on what features of GR is accepted unchanged, various theories of gravity have been created. Modification of GR is characterized by a deformation parameter such as Lovelock coefficients \(\alpha _{i}\)’s in Lovelock gravity (which determines the strength of higher curvature terms) or graviton mass parameter in massive gravity models. Based on the nature of the deformation parameter, the original theory (GR) can be recovered by taking some limits (e.g. the zero limit of graviton mass parameter must recover GR and its associated outcomes). In order to have a generalized and well-defined theory, we should take care of ghosts. Although the first linear version of the massive theory (i.e., the Fierz–Pauli model [16]) is ghost-free, it does not lead to the linearized GR as the graviton mass goes to zero which is known as the vDVZ discontinuity [17, 18]. Vainshtein discovered that such discontinuity appears as a consequence of working with the linearized theory of GR [19], and by employing the Stueckelberg trick it can be found that all degrees of freedom introduced by the graviton mass do not decouple in the zero limit of graviton mass [20]. On the other hand, Boulware and Deser showed some specific nonlinear models of massive gravity suffer from ghost instabilities however they could restore continuity with GR [21]. Eventually, the de Rham–Gabadadze–Tolley (dRGT) theory of fully nonlinear massive gravity resolved the ghost problem in four dimensions by adding higher order graviton self-interactions with appropriately tuned coefficients [22, 23, 24, 25]. The higher dimensional extension of the massive (bi)gravity has been discussed in [26, 27], which confirms the absence of ghost fields using the Cayley- Hamilton theorem. Interestingly, in massive gravity framework, spherically symmetric black hole solutions were found in [28, 29] and in the limit of vanishing graviton mass they go smoothly to the Schwarzschild and Reissner-Nordström (RN) black holes. Furthermore, asymptotically flat black hole solutions were found in [30], but the curvature diverges near the horizon of these solutions. In this regard, black hole solutions with non-singular horizon were introduced in [31] with the identification of the unitary gauge to the coordinate system in which black hole has no horizon (for more details see [31]). The other interesting solutions related to the cosmology, gravitational waves and (time dependent) black holes were found in [10, 32, 33, 34, 35, 36, 37, 38] which will not discuss in this paper. Of interesting case for us is Vegh’s black hole solution [39] in which the general covariance preserves in the “t” and “r” coordinates, but, is broken in the other spatial dimensions. In [40], this solution was generalized to topological black holes in higher dimensions. Inspired by the interesting features of these solutions, black hole solutions of massive gravity coupled to the higher curvature terms, dilaton and nonlinear electromagnetic fields were constructed and studied in detail [41, 42, 43, 44, 45, 46, 47].
Of interesting case for theoretical physicists is thermodynamic properties of black holes in comparison with ordinary systems in nature. In fact, black hole mechanics obeys the same laws as the laws of thermodynamics [76], and many investigations have confirmed this statement for more complicated black hole spacetimes in modified gravities. In addition, a wealth of results in the context of black hole thermodynamics have been presented which show black holes in Einstein gravity can imitate some thermodynamic properties of ordinary systems such as the van der Waals (vdW) phase transition which represents a liquid/gas (first order) phase transition [77], the reentrant phase transition (RPT) in multicomponent fluid systems [78, 79] and the triple point in solid/liquid/gas phase transition. The phase space of Schwarzschild-anti de Sitter (Schwarzschild-AdS) black holes admits the so-called Hawking-Page phase transition [80] which is interpreted as a confinement/deconfinement transition in the dual boundary gauge theory (SYM plasma) [81]. Remarkably, RN-AdS and Kerr–Newman-AdS black holes possess a first order phase transition which closely resembles the well-known vdW phase transition in fluids [82, 83, 84]. Interestingly, Born–Infeld-AdS black holes, as a nonlinear version of RN-AdS ones, display a phase structure which relates the mass (M) and the charge (Q) of the black holes similar to the solid–liquid–gas phase diagram [85]. These considerations were done in the presence of the cosmological constant as a fixed parameter and recently are referred in the literature as non-extended phase space. In fact, as stated in [86], these mathematical analogies are confusing since some black hole intensive (extensive) quantities have to be identified with an irrelevant extensive (intensive) quantities in the fluid system, for example, the identification between the fluid temperature and the charge of the black hole is puzzling.
In thermodynamic systems, some quantities are thermodynamic variables and the others are fixed parameters which cannot vary. Only experiment can determine that a quantity (parameter) can vary or hold fixed. From the theoretical perspective, one can assume the variation of a fixed parameter of a theory and then see its consequences. In this regard, the later mismatch between extensive and intensive quantities of the black hole and fluid systems can be solved if one treats the cosmological constant (\(\Lambda \)) as a thermodynamic variable, i.e., pressure [86]. This idea (which first established in [87] and then developed in [88, 89, 90, 91, 92]) leads to an extension of the phase space thermodynamics and the exact analogy between quantities of a black hole and liquid-gas systems at the critical point. For example, a transition occurs in \(P-T\) plane for the both of RN-AdS (small/large) black hole and liquid-gas systems. In addition, the variation of \(\Lambda \) in the first law of black hole thermodynamics solves the inconsistency between the Smarr formula and the traditional form of first law since in the presence of a fixed cosmological constant the scaling argument [87] is no longer valid. This motivates consideration of the first law of black hole thermodynamics with varying \(\Lambda \) which is referred to as the extended phase space thermodynamics in physics community. Regarding the extended phase space thermodynamics, RPT has been observed for Born–Infeld-AdS and singly spinning Kerr-AdS black holes in the context of Einstein gravity [93, 94]. In a black hole system, it is interpreted as large/small/large black hole phase transition. Moreover, the analogue of solid/liquid/gas phase transition was found for doubly spinning Kerr-AdS black holes which is interpreted as small/intermediate/large black hole transition [95, 96].
The objective of this paper is to construct the higher curvature massive gravity in order to study the effects of higher order curvatures on the black hole solutions of massive gravity and investigate the associated criticality and thermodynamics in the extended phase space. Indeed, some thermodynamic features of black holes, e.g. universal ratio, may depend on the specific choice of the gravitational theory. Therefore it is so important to understand the effect of modified gravities. We select the Lovelock gravity up to third order (referred to as TOL gravity) as the higher curvature framework for our investigations. When the Lovelock massive theory of gravity (LM gravity) is constructed, in principle, the parameters \(\alpha \) (Lovelock coefficient) and m (graviton mass) are considered as deformations of GR, and by taking the limits \(m\rightarrow 0\) and \(\alpha \rightarrow 0\), GR is naturally recovered. According to scaling argument, any dimensionful parameter in a given theory has a thermodynamic interpretation and as a result, the Smarr formula and the first law of black hole thermodynamics must be modified. According to this fact, thermodynamically, more interesting phenomena can take place in a more complicated theory of gravity such as Lovelock and massive gravities which have a finite number of dimensionful parameters. One can observe that modified gravities such as massive and Lovelock theories exhibit a rich black hole phase space structure with respect to those counterparts in Einstein gravity. The existence of higher order curvatures based on the third order Lovelock (TOL) gravity can lead to critical behavior and phase transition for AdS black holes with hyperbolic horizon topology [97, 98, 99, 100] in contrast to Einstein gravity which only spherically symmetric AdS black holes admits phase transitions. Remarkably, hyperbolic vacuum black holes in Lovelock gravity expose non-standard critical exponents at a special isolated critical point which are different from those of vdW ones [101]. Until writing this paper, a wealth of evidence has been indicating that all the black hole solutions in Einstein gravity in the presence of any matter field have the same critical exponents as the vdW fluid [86, 93, 102, 103]. Interestingly, a “\(\lambda \)-line” phase transition occurs for a class of AdS-hairy black holes with hyperbolic horizon in Lovelock gravity where a real scalar field is conformally coupled to gravity [104]. In addition, for charged black branes, the inclusion of higher curvature gravities based on a generalized quasi-topological class could lead to phase transition and critical behavior with the standard critical exponents [105]. These indications reveal the rich phase space structure of Lovelock gravity’s black holes. On the other hand, in the massive gravity framework, phase transition and critical behavior could take place for all kinds of topological black holes [106]. In this regard, the vdW and RPTs were found for AdS black holes [107, 108], and in the presence of Born–Infeld (BI) nonlinear electromagnetic fields, the triple point emerges and the corresponding large/intermediate/small transition could take place [109].
Taking these considerations seriously, in this paper, we mainly focus on the critical behavior and phase transitions of AdS black hole solutions in the Lovelock massive (LM) gravity. By constructing this model, besides its novel phase structure, we could be able to figure out what characteristic features of Lovelock and massive gravities persist or ruin. Thus, we have organized this paper as follows: first, in Sect. 2.1 , we give a brief review of thermodynamics in extended phase space, stability analysis and phase transition for AdS black holes in the context of Einstein gravity. After, in Sect. 3, we construct the LM gravity by introducing the action and associated field equations and then present a new class of charged-AdS black hole solutions in arbitrary dimensions. By computing thermodynamic quantities, we prove the traditional first law of black hole thermodynamics is satisfied. After that, in Sect. 4.1, we perform a thermal stability analysis for the obtained black hole solutions in the canonical ensemble. Furthermore, we reconsider the first law of thermodynamics in the extended phase space and then study \(P-V\) criticality and phase transition(s) of black holes to complete our discussion. Finally, in Sect. 5, we finish our paper with some concluding remarks.
2 General formalism: thermodynamics, stability and phase transition for AdS black holes
In this section, we will develop the basic framework that we need to study the critical behavior and thermodynamic properties of AdS black holes in the next sections. Some useful issues will be briefly reviewed such as gravitational partition function, black hole thermodynamics, local thermodynamic stability, phase transition and critical behavior of black holes. Throughout this paper, we use the geometric units, \({G_{N}}=\hbar =c={k_\mathrm{{B}}}=1\). In these units, \([\mathrm {Energy}]=[\mathrm {Mass}]=[\mathrm {Length}]^{-1}=[\mathrm {Time}]^{-1}\), and therefore there is only one dimensionful unit. Moreover, our convention of the metric signature is \((-,+,+,+,\ldots )\).
2.1 Basic set up: partition function and action
2.2 Black hole thermodynamics
Standard thermodynamics | Black hole variables |
---|---|
Internal energy | M (mass) |
Temperature | \(T=\kappa /2\pi \) (surface gravity) |
Entropy | \(S=A/4\) (horizon area) |
2.3 Thermal stability of black holes
Now, we perform a thermal stability analysis for RN-AdS black holes. To determine thermally stable regions for black holes, first, one has to find in which regions the associated temperature is positive. That could depend on the topology of event horizons. In Fig. 1, typical behaviors of temperature are depicted for different horizon topologies (\(k=1,0,-1\)) in four and higher dimensions. Clearly, there is always a bound point for the radius of the event horizon (\(r_{b}\)) in which the temperature of the black hole is positive for \(r_{+}>r_{b}\). The topological type of event horizon (\(k=1,0,-1\)) can change the value of \(r_{b}\). The more value for k results into the lower value for \(r_{b}\). In addition, in the region \(r_{+}>r_{b}\) the mass of the black holes (M) is always positive.
Spherical black holes (\(k=1\)): In this case, the heat capacity has only one positive root (\(r_{b}\)) in which \(C_{Q}\) is negative definite for regions \(r_{+}<r_{b}\). For regions \(r_{+}>r_{b}\), depending on the values of q and spacetime dimensions (d), three possibilities may happen: (i) \(C_{Q}\) is an increasing function of \(r_{+}\), thus, in regions \(r_{+}>r_{b}\), the heat capacity will be positive and black holes are thermally stable. (ii) \(C_{Q}\) may have two divergence points for the RN-AdS black holes, where between divergence points the heat capacity is negative definite (unstable black hole region), thus a thermal phase transition can happen. (iii) \(C_{Q}\) has one divergence point which is positive around such divergency. Such a single divergence point, with positive \(C_{Q}\) around it, may indicate critical behavior of the system. In Fig. 2, the possibilities of items (i) and (ii) are depicted. We refer to the first and second divergence points as \(r_{m}\) and \(r_{u}\) respectively (\(r_{b}<r_{m}<r_{u}\)). In regions \(r_{b}<r_{+}<r_{m}\) and \(r_{+}>r_{u}\), RN-AdS black holes are thermally stable (a phase transition could occur between these two thermally stable regions) and for the other regions are unstable.
Ricci flat black holes (\(k=0\)): In this case, the heat capacity has only one positive root, \(r_{b}\), in which \(C_{Q}\) is negative definite for regions \(r_{+}<r_{b}\) and positive for \(r_{+}>r_{b}\). According to Eq. 2.28, the heat capacity does not diverge for finite values of \(r_{+}\) since the denominator of the heat capacity cannot have any root (see the left panel of Fig. 3). As a result, Ricci flat black holes are thermally stable for regions \(r_{+}>r_{b}\) and no phase transition takes place.
Hyperbolic black holes (\(k=-1\)): In this case, as well as Ricci flat black holes (\(k=0\)), there is only one root (\(r_{b}\)) for the heat capacity. Again, hyperbolic black holes are thermally stable in regions \(r_{+}>r_{b}\) since the heat capacity is always positive and no phase transition takes place (see the middle and right panels of Fig. 3).
2.4 Phase transition for RN-AdS black holes
In this section, we review the essence of critical behavior and phase transition in AdS black holes. In recent years, the interesting analogy between liquid/gas and small/large black hole phase transitions has attracted the attention of many authors. In this regard, the exact analogy between liquid-gas system (vdW fluid) and charged-AdS black holes was first completed by [86] in the context of Einstein–Maxwell gravity. In fact, RN-AdS black holes exhibit first-order phase transition with the same critical exponents as the vdW system. We will generalize the results in [86] for higher dimensions and various event horizon topologies and will show that \(P-V\) criticality only exists for spherically symmetric black holes.
vdW fluid | RN-AdS black hole |
---|---|
Temperature | T |
Pressure | \(P = - \Lambda /8\pi \) |
Volume | \({r_ + } = {(3\,V/4\pi )^{1/3}}\) |
3 Lovelock massive (LM) gravity with Maxwell field
3.1 Action and field equations
3.2 LM charged-AdS black holes
We are more interested in studying AdS black holes since these types of black objects admit dual interpretation and also possess certain phase transition(s) in the extended phase space. Hereafter, we assume only AdS black holes.
3.3 Thermodynamics of LM charged-AdS black holes
3.4 Extended phase space thermodynamics
In this section, by treating the negative cosmological constant as a thermodynamic pressure, we reconsider the first law of black hole thermodynamics Eq. (3.30) and makes it consistent with the Smarr formula. In order to extend the first law of thermodynamics for LM gravity, we also regard the massive couplings \(c_{i}\) and Lovelock coefficient \(\alpha \) as thermodynamic variables.
3.5 Thermal stability of LM charged-AdS black holes
Now, we discuss positivity of the heat capacity with respect to the event horizon topology (k) case by case and in detail.
In addition, there is another possibility for the case of one divergence point in the heat capacity, which indicates the Hawking-Page phase transition. According to Fig. 16, around such divergency, the heat capacity is negative and positive corresponding to the area on the right and the area on the left, respectively. In all cases, besides a divergence point (\(r_{m}\)), \(C_{P,X_{i}}\) may have one or two positive roots (referred to as \(r_{b_{1}}\) and \(r_{b_{2}}\)) depending on the chosen parameters. When there exist two roots, the divergence point is located between the roots. This situation is different from the previous case of hyperbolic black holes (see the items i and ii related to hyperbolic black holes). Assuming that there are two roots (\(r_{b_{1}}<r_{b_{2}}\)), an unstable region is observed for \(r_{m}<r_{b_{2}}\) in the heat capacity’s plots of Fig. 16. Consequently, thermally stable regions correspond with \(r_{b_{1}}<r_{+}<r_{m}\) and \(r_{+}>r_{b_{2}}\).
4 \(P-V\) criticality of LM charged-AdS black holes
4.1 Critical behavior and vdW phase transition
Let’s compare the LM equation of state (4.4) with the equation of state of the RN-AdS black holes (2.29). In the right hand side of the RN-AdS equation of state, the first and the last terms are always positive and the second term can be positive, zero or negative. In fact, signs of the presented terms in the equation of state could ensure the critical behavior for a given black hole solution. We introduce the signature of the equation of state of RN-AdS black holes as \(P(+,\pm ,+)\). We saw that there does not exist \(P-V\) (\(P-r_{+}\)) criticality for black holes with Ricci flat and hyperbolic horizons. In order to have \(P-V\) (\(P-r_{+}\)) criticality two positive terms and one negative term are needed, i.e., an equation of state with signature \(P(+,-,+)\) which is the case with spherical horizon (\(k=+1\)). Therefore, at least, two positive terms and one negative term in the equation of state can possibly ensure the critical behavior and phase transition for a given black hole. Regarding this, the equation of state of LM charged-AdS black holes (4.4) with signature \(P(\pm ,\pm ,\pm ,\pm ,+,\pm ,+)\) predicts critical behavior and phase transition for Ricci flat and hyperbolic black holes as well as spherically symmetric black holes depending on the massive coupling coefficients (\(c_{i}\)), Lovelock coefficient (\(\alpha \)) and topological factor (k).
We are looking for the inflection point of isothermal \(P-r_{+}\) diagrams, the subcritical isobar of \(T-r_{+}\) plots, and the characteristic swallow-tail form of \(G-T\) diagrams for the obtained black hole solutions according to Sect. 2.4. These pieces of evidence guarantee the existence of phase transition and indicate the vdW like behavior for the LM AdS black holes. In our considerations, we suppose that all the massive coupling coefficients are simultaneously positive (\(c_{1}=c_{2}=c_{3}=c_{4}=+ 1\)) or negative (\(c_{1}=c_{2}=c_{3}=c_{4}=- 1\)) and keep track of the effect of higher order curvature terms of the Lovelock Lagrangian on the outcomes of massive gravity. Later, we do not impose this assumption and will summarize the results of arbitrary signs for the massive couplings (\(c_{i}=\pm 1\)).
Topological black holes: \(d=7\), \(q=1\), \(m=1\), \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\) and \(\alpha =0.01\) (for \(k=\pm 1\))
k | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
+1 | 22.8678 | 0.69318 | 19.7148 | 0.80404 |
0 | 25.0668 | 0.68738 | 21.2994 | 0.80897 |
-1 | 27.8071 | 0.68117 | 23.3249 | 0.81207 |
Considering the obtained critical values in Table 1 for the pressure, horizon radius and temperature, one can plot their corresponding phase diagrams. In the left panels of Figs. 17, 18, 19, the characteristic behavior of pressure as a function of event horizon radius (\(r_{+}\)) is depicted for the topological black holes. Compared to \(P-r_{+}\) diagram of vdW fluid or RN-AdS black holes (see Fig. 4), it is seen that the associated \(P-r_{+}\) diagrams for LM AdS black holes qualitatively behave like vdW fluid. Therefore, critical radius (inflection point) can be found for Ricci flat or hyperbolic black holes as well as spherically symmetric black holes. For all \(P-r_{+}\) diagrams, the temperature of isotherms decreases from top to bottom. For \(T>T_{c}\), the isotherms correspond to the ideal gas with a single phase. For \(T<T_{c}\), a two-phases behavior is seen, and in comparison with the liquid/gas system, there exists (first order) small/large black hole phase transition for topological black holes. It is notable that, based on Maxwell’s equal-area law, the (unphysical) oscillatory part of each isotherm is replaced by a line of constant pressure.
Now, we are looking for the characteristic swallow-tail form of \(G-T\) diagrams. The Gibbs free energy is found by computing on-shell action (\({{\mathcal {I}}}_G\)) or using the Legendre transformation, \(G=M-TS\). Analytical calculation of the Gibbs free energy is too lengthy to write here and, therefore, we leave out the analytical result for reasons of economy. We have plotted the Gibbs free energy as a function of temperature for various pressures in right panels of Figs. 17, 18, 19. As seen, obviously, \(G-T\) diagrams indicate the characteristic swallow-tail behavior for all types of topological black holes. This behavior demonstrates a first-order phase transition in the black hole systems.
To be more specific, we analyze the equations of state and phase transitions for topological black holes case by case and in details. We focus on the effects of Lovelock coefficient (\(\alpha \)), graviton mass parameter m, spacetime dimension (d) and topological factor (k) and present related tables (see Tables 2, 3, 4, 5, 6, 7, 8, 9 and 10). In this regard, we reveal a peculiar phase transition and critical behavior for hyperbolic black holes in the LM gravity in higher dimensions of spacetime (i.e., \(d\geqslant 7\)).
Spherical horizon(\(k=+1\)):
Spherical black holes: \(k=1\), \(q=1\), \(m=1\), \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\) and \(\alpha =1\)
d | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
7 | 0.53489 | 1.17791 | 1.03718 | 0.60747 |
8 | 1.96244 | 1.03318 | 2.19721 | 0.92278 |
9 | 5.07146 | 0.97562 | 4.05437 | 1.22037 |
10 | 10.7318 | 0.94705 | 6.76637 | 1.50208 |
11 | 19.9878 | 0.93112 | 10.4942 | 1.77347 |
Spherical black holes: \(k=1\), \(d=7\), \(q=1\), \(m=1\), and \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\)
\(\alpha \) | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
0.00000 | 25.9104 | 0.68616 | 22.2263 | 0.79989 |
0.01000 | 22.8678 | 0.69318 | 19.7148 | 0.80404 |
0.10000 | 10.4454 | 0.74202 | 9.39628 | 0.82488 |
1.00000 | 0.53489 | 1.17791 | 1.03718 | 0.60747 |
10.0000 | 0.00341 | 9.05845 | 0.15170 | 0.20372 |
100.000 | 0.00004 | 78.3049 | 0.08775 | 0.03896 |
100000 | \(4.7154 \times 10^{-11}\) | 75003.7 | 0.07959 | 0.00004 |
\(\rightarrow \infty \) | \(\rightarrow 0\) | \(\rightarrow \infty \) | \(\rightarrow 0\) | \(\rightarrow 0\) |
Spherical black holes: \(k=1\), \(d=7\), \(q=1\), \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\) and \(\alpha =1\)
m | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
0.000000 | 0.026994 | 2.247111 | 0.142259 | 0.426396 |
0.001000 | 0.026994 | 2.247103 | 0.142260 | 0.426395 |
0.010000 | 0.027016 | 2.246273 | 0.142327 | 0.426375 |
0.100000 | 0.029227 | 2.165797 | 0.149086 | 0.424592 |
1.000000 | 0.534893 | 1.177908 | 1.037184 | 0.607467 |
10.00000 | 60.18878 | 1.051550 | 93.72279 | 0.675305 |
Ricci flat black holes: \(k=0\), \(q=1\), \(m=1\) and \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\)
d | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
7 | 25.0668 | 0.68738 | 21.2994 | 0.80897 |
8 | 73.0874 | 0.69501 | 50.5821 | 1.00423 |
9 | 159.769 | 0.71105 | 95.0377 | 1.19535 |
10 | 297.840 | 0.72793 | 156.551 | 1.38489 |
11 | 501.676 | 0.74375 | 237.102 | 1.57368 |
Ricci flat black holes: \(k=0\), \(d=7\), \(q=1\) and \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\)
m | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
0.00000 | – | – | – | – |
0.01000 | – | – | – | – |
0.05000 | 0.00203 | 1.78699 | 0.00497 | 0.73219 |
0.10000 | 0.017231 | 1.43748 | 0.03266 | 0.75834 |
0.50000 | 2.72792 | 0.86004 | 2.93899 | 0.79827 |
1.00000 | 25.0668 | 0.68738 | 21.2994 | 0.80897 |
5.00000 | 4599.43 | 0.40662 | 2267.57 | 0.82476 |
10.0000 | 44299.1 | 0.32383 | 17304.4 | 0.82476 |
Ricci flat horizon (\(k=0\)):
(Hot) Hyperbolic black holes: \(k=-1\), \(q=1\), \(m=1\), \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\) and \(\alpha =0.01\)
d | \(P_{c_{1}}\) | \(r_{c_{1}}\) | \(T_{c_{1}}\) | \(\frac{{{P_{c_{1}}}{r_{c_{1}}}}}{{{T_{c_{1}}}}}\) |
---|---|---|---|---|
7 | \(1.201689804\times 10^7\) | 0.199182022693580 | \(3.308081979\times 10^6\) | 0.7235461736 |
8 | \(3.540682432\times 10^8\) | 0.203488734706851 | \(8.162818360\times 10^7\) | 0.8826473362 |
9 | \(9.794234509\times 10^9\) | 0.206528200802543 | \(1.941047787\times 10^9\) | 1.042110166 |
10 | \(2.589915504\times 10^{11}\) | 0.208777110589713 | \(4.499390023\times 10^{10}\) | 1.201751955 |
11 | \(6.619241728\times 10^{12}\) | 0.210505697368947 | \(1.023424155\times 10^{12}\) | 1.361496198 |
Hyperbolic horizon (\(k=-1\)):
(Cold) Hyperbolic black holes: \(k=-1\), \(q=1\), \(m=1\), \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\) and \(\alpha =0.01\)
d | \(P_{c_{2}}\) | \(r_{c_{2}}\) | \(T_{c_{2}}\) | \(\frac{{{P_{c_{2}}}{r_{c_{2}}}}}{{{T_{c_{2}}}}}\) |
---|---|---|---|---|
7 | 27.8071 | 0.68117 | 23.3249 | 0.81207 |
8 | 81.8748 | 0.68966 | 56.3992 | 1.00118 |
9 | 178.635 | 0.70655 | 106.200 | 1.18847 |
10 | 331.629 | 0.72410 | 174.622 | 1.37516 |
11 | 556.211 | 0.74044 | 263.733 | 1.56159 |
Hyperbolic black holes: \(k=-1\), \(d=7\), \(q=1\), \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\) and \(\alpha =0.01\)
m | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
0.000000 | \(1.203736376\times 10^{7}\) | 0.1991562990 | \(3.313433416\times 10^{6}\) | 0.7235144082 |
0.010000 | \(1.203737744\times 10^{7}\) | 0.199156282030114 | \(3.313437004\times 10^{6}\) | 0.7235143849 |
0.100000 | \(1.203715916\times 10^{7}\) | 0.199156555956048 | \(3.313379914\times 10^{6}\) | 0.7235147264 |
0.500000 | \(1.203224868\times 10^{7}\) | 0.199162723328388 | \(3.312095942\times 10^{6}\) | 0.7235223426 |
Hyperbolic black holes: \(k=-1\), \(d=7\), \(q=1\), \(m=1\), and \(c_{0}=c_{1}=c_{2}=c_{3}=c_{4}= 1\)
\(\alpha \) | \(P_{c}\) | \(r_{c}\) | \(T_{c}\) | \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) |
---|---|---|---|---|
0.00000 | 24.2262 | 0.68863 | 20.3741 | 0.81883 |
0.00100 | 24.5447 | 0.68792 | 20.6369 | 0.81818 |
0.01000 | 27.8071 | 0.68117 | 23.3249 | 0.81207 |
0.03000 | 39.1688 | 0.66318 | 32.6101 | 0.79656 |
0.05000 | 66.6893 | 0.63617 | 54.6276 | 0.77664 |
0.07000 | – | – | – | – |
In Figs. 20 and 21, we have depicted the typical behavior of \(P-r_{+}\), \(T-r+\) and \(G-T\) curves in the vicinity of the first and second (physical) critical points corresponding to \(r_{c_{1}}\) and \(r_{c_{2}}\) respectively. Also, the associated critical data for temperature, pressure and event horizon radius are presented in Tables 7 and 8. In Fig. 20, for the first critical point (corresponding to the smaller horizon radius, denoted by \(P_{c_{1}}\), \(r_{c_{_{1}}}\) and \(T_{c_{1}}\)), we observe the characteristic swallow-tail behavior in \(G-T\) diagrams for pressures in the range \(P>P_{c_{1}}\) in contrast to the vdW phase transition which only takes place for \(P<P_{c}\). From this diagram, it can be inferred that a first order phase transition occurs for \(P>P_{c_{1}}\) since the first derivative of the Gibbs free energy at the critical point is undefined. In \(P-r_{+}\) diagrams, interestingly, the (unphysical) oscillating part of isotherms takes place for \(T>T_{c_{1}}\) which means the existence of two-phases behavior, and the oscillating part is replaced by an isobar according to Maxwell’s equal area law. For region \(T<T_{c_{1}}\) in \(P-r_{+}\) diagrams, the one phase behavior corresponding to ideal gas is observed. Moreover, \(T-r_{+}\) plots reveal that oscillating part of critical isobars occurs for the pressures in the range \(P>P_{c_{1}}\). Comparing with the vdW phase transition, this critical behavior is completely reverse. This evidence shows hyperbolic black holes could potentially experience the reverse vdW like behavior for (first order) phase transition at high temperature and pressure which is a remarkable result. Further, in the next part of this section, we will uncover the theory dependency’s origin of the ”reverse behavior” in LM gravity with a detailed discussion. As far as we know, there is no reverse vdW phase transition in usual thermodynamic systems.
In Fig. 21, the qualitative behavior of the hyperbolic (cold) black hole at the second critical point (\(r_{c_{2}}\)) is displayed. At this critical point, we observe the standard vdW phase transition which explained in details before. Interestingly, numerical calculations, which are presented in Tables 7 and 8, show that the critical pressure, horizon radius, temperature and the universal ratio (\(\frac{P_{c_{i}} r_{c_{i}}}{T_{c_{i}}}\)) are increasing functions of spacetime dimension (d) at both critical points.
In addition, according to numerical calculations which are presented in Table 10, there is an upper limit for the value of Lovelock parameter, \(\alpha _{u}\), in which no phase transition could happen for \(\alpha >\alpha _{u}\). This statement does not hold for LM AdS black holes with spherical horizon which is inferred from Table 3. In Table 10, we have shown only those branches corresponding to standard vdW phase transitions and the associated larger critical radii. As \(\alpha \rightarrow \alpha _{u}\), the two critical points associated with vdW and reverse-vdW phase transitions move towards each other, and eventually, when \(\alpha =\alpha _{u}\) both phase transitions with the associated critical radii disappear. This shows that inclusion of higher curvature terms (based on Lovelock Lagrangian) affects drastically the criticality of AdS black holes with the hyperbolic horizon in massive gravity; by tuning the Lovelock coefficient (\(\alpha \)) the first-order phase transition can be produced or ruined (see Table 10). Moreover, the universal ratio \(\frac{{{P_c}{r_c}}}{{{T_c}}}\) is a decreasing function of \(\alpha \) in the range \(0<\alpha <\alpha _{u}\).
4.2 Lovelock gravity: the phase transition revisited
In the case of hyperbolic black holes, there always exists only one critical radius (\(r_{c}=\sqrt{\alpha }\)) in all spacetime dimensions (\(d\geqslant 7\)) according to Eq. (4.12). As stated in Sect. 3.5, the temperature of hyperbolic black holes blows up at the point \({r_{ + }}={r_ i} = \sqrt{\alpha }\) which is referred to as the thermodynamic singularity [100] since all isotherms cross at \(r_{c}=\sqrt{\alpha }={d_2}v_{c} /4\). Also, the heat capacity is zero at this point. The critical point corresponding to this thermodynamic singularity is called the isolated critical point and is regarded as the first example of a critical point with non-standard critical exponents as \(\alpha =0\), \(\beta =1\), \(\gamma =2\) and \(\delta =3\) [100, 101] which are different from those of vdW fluid with \(\alpha =0\), \(\beta =1/2\), \(\gamma =1\) and \(\delta =3\). Since critical exponents determine the behavior of thermodynamic quantities near the critical point, so various critical exponents imply different behaviors in phase diagrams. As a result, in the case of hyperbolic black holes, a stretched swallow-tail behavior (also referred to as cusp-like behavior) in \(G-T\) diagram is observed for pressures in the range \(P>P_{c}\), in which \(P_{c}\) is the pressure at the thermodynamic singularity. When the U(1) charge is considered, one additional critical point might emerge. Regarding the charged case, if the value of U(1) charge is more than a lower value (\(Q>Q_{b}\)), there exist two critical radii which the smaller one is always unphysical (the corresponding black hole has a negative value for the temperature) and the larger one can be physical. For the small enough charges (i.e., \(0<Q<Q_{b}\)), those critical points are created near the isolated critical point, and the both of them could be physical (\(r_{c_{1}}\) and \(r_{c_{2}}\)). We observe that by increasing the value of the electric charge (Q) the distance between the thermodynamic coordinates of those points is increased in the extended phase space. As a matter of fact, one can grow up the thermodynamic quantities associated to the critical point by increasing the U(1) charge and produce a first order phase transition at high temperature and pressure for hyperbolic black holes. Here, an interesting phenomenon emerges and persists in higher dimensions (\(d\ge 7\)). Our investigations show that the reverse vdW phase transition is a characteristic feature of Lovelock AdS black holes with hyperbolic horizon corresponding to those larger critical points (this strange behavior is already pointed out in [97, 98, 100]). As seen, Fig. 23 exposes the origin of the reverse vdW like behavior which has been found in Sect. 4.1 for hyperbolic black holes in the LM gravity. In Fig. 23, we observe the existence of inflection point in the isothermal \(P-r_{+}\) diagrams, the subcritical isobar of \(T-r_{+}\) plots, and the characteristic swallow-tail form of \(G-T\) diagrams, but, in contrast to the behavior of vdW fluid, in the opposite way. Therefore, we conclude higher order curvature terms based on the Lovelock Lagrangian are responsible for the reverse vdW phase transition.
4.3 Massive gravity: the phase transition revisited
According to Eqs. (4.15), (4.16), (4.17) and (4.18), when all the massive coupling coefficients are simultaneously positive (negative) definite, there exists one critical radius and can be determined using Eq. (4.17). In order to have two critical points, one should consider some specific signs for the massive couplings (\(c_{i}\)) based on Eq. (4.18). When the combination \((k + {m^2}c_0^2{c_2})\) is positive, two critical points can be found assuming that \(c_{3}<0\) and \(c_{4}>0\), and when \((k + {m^2}c_0^2{c_2})<0\), one has to assume \(c_{3}>0\) and \(c_{4}<0\). In the both cases, the massive coupling coefficients must satisfy the constraint \(3{d_4}{m^2}c_0^2c_3^2 > 8{d_5}{c_4}(k + {m^2}c_0^2{c_2})\).
4.4 LM gravity: RPTs and triple points
This section is devoted to studying the possibility of the appearance of the RPT and triple point in the phase structure of the LM AdS black holes. In the previous section, we indicated that under certain conditions the equations of state of topological black holes in pure massive gravity (without nontrivial electromagnetic fields like BI electrodynamics) may have up to two critical points and thus exhibit vdW and RPTs which the latter corresponds to three-phases behavior. In Sect. 4.2, we showed Lovelock (un)charged-AdS black holes with spherical horizon can exhibit vdW and RPTs. Also, for the Lovelock (un)charged-AdS black holes with the hyperbolic horizon, the reverse vdW like behavior is observed which can be accompanied by a (normal) vdW phase transition. Consequently, since there are many thermodynamic variables in the extended phase space of the LM AdS black holes, one expects these black holes may enjoy a vast range of thermodynamic behaviors which found in the other gravitational theories similar to those in usual thermodynamics. Here we intend to examine these possibilities.
Investigation shows the LM AdS black holes with spherical horizon may have up to three physical critical points for charged and uncharged cases. In order to observe the reentrant behavior of phase transition for spherical black holes, we have adjusted the massive coupling coefficients to produce two critical points (referred as \(r_{c_{1}}\) and \(r_{c_{2}}\)) according to Eq. (2.35) and plotted \(P-r_{+}\), \(T-r_{+}\) and \(G-T\) diagrams in Fig. 24. As a result, a virtual triple point (\(T_{tr}\), \(P_{tr}\)) and another critical point (\(T_z\),\(P_z\)) emerge in the phase space. In addition, the behavior of the heat capacity associated to the reentrant and small/intermediate/large phase transitions are depicted in Figs. 11 and 12 near the (virtual) triple point. As seen, by monotonic decreasing the temperature, the black hole system undergoes a RPT for the certain range of pressure (\(P_{tr}<P<P_z\)). By another tuning, we arrive at one triple point (\(T_{tr}\), \(P_{tr}\)) and two physical critical points (associated with \(r_{c_{1}}\) and \(r_{c_{2}}\)). This situation is depicted in Fig. 25 in which the Gibbs free energy is displayed near the critical points for various pressures. It should be mentioned that there is a lower value for the U(1) charge (\(Q_{b}\)), where for \(Q>Q_{b}\), one of the critical points disappears. Hence, in the case of charged-AdS black holes, the analogue of the triple point and solid/liquid/gas phase transition can be found only for small enough values of the electric charge, Q.
The phase structure of hyperbolic black holes is really rich and drastically different from those of spherical and Ricci flat horizons. In both charged and uncharged cases, three (physical) critical points can be found for hyperbolic black holes. Interestingly, the analogue of triple point does not exist in the phase structure of these black holes. In fact, besides the existence of the two critical points corresponding with two distinct first order transition, we arrive at an additional reverse vdW phase transition associated to the third critical point in the phase space. This situation is illustrated in Fig. 26 which is a generic feature of this model and persists in all dimensions. This is the first example of such phase structure which is not possible for spherical and Ricci flat black holes.
We could not find any evidence related to the existence of four critical points in this model. The existence of four critical points may be potentially possible when the phase space of the spherically symmetric AdS black holes in LM gravity is enriched by adding nonlinear U(1) gauge fields in the theory.
5 Concluding remarks
The effects of massive and Lovelock gravities are encoded in the deformation parameters m and \(\alpha \), respectively. In Lovelock massive (LM) gravity, one can simply recover the outcomes of Einstein (by \(\alpha , m\longrightarrow 0\)), Lovelock (by \(m\longrightarrow 0\)) and massive (by \(\alpha \longrightarrow 0\)) theories of gravity. Considering LM gravity, in this paper, we introduced topological black hole solutions and then analyzed thermodynamic properties and critical behavior of AdS black holes in the extended phase space. The asymptotic behavior of the black hole solutions may be (A)dS or flat, and by computing the thermodynamic quantities, we have shown they satisfy the first law of thermodynamics.
Next, by treating the cosmological constant as a thermodynamic variable (pressure), we extended the thermodynamic phase space and proved the massive coupling and Lovelock coefficients, as well as cosmological constant, are required for consistency of the extended first law of thermodynamics with the Smarr formula. In addition, we examined thermal stability in the extended phase space thermodynamics of the LM AdS black holes in the canonical ensemble (where the quantities P, Q, \(c_{i}\) and \(\alpha \) are held fixed), and showed the qualitative behavior of heat capacity for AdS black holes with different horizon topologies. In this regard, we mainly focused on the topology of event horizons and showed in what regions the topological black holes are thermally stable.
In LM gravity, critical behavior and phase transition occur for all types of AdS black holes (with spherical, Ricci flat and hyperbolic topologies for event horizon) in contrast to Einstein gravity which only admits phase transition for spherically symmetric ones. For Ricci flat black holes, phase transition originated only from the interacting terms of massive gravitons and the effect of higher curvature terms vanishes. Interestingly, we found that there is a lower value for the graviton mass parameter, referred to as \(m_{b}\), in which no phase transition takes place in the region \(m<m_{b}\). This is one of the remarkable characteristics of massive gravity. For hyperbolic black holes, two radically different first order transitions are observed: (i) a (normal) vdW like behavior, and (ii) reverse vdW like behavior. The reverse behavior of vdW phase transition completely comes from the higher curvature terms of Lovelock Lagrangian which is not seen in Gauss–Bonnet gravity (as indicated in [100, 141], Gauss–Bonnet black holes with hyperbolic horizon do not admit physical phase transition). The reverse behavior predicts that hyperbolic black holes could experience first order phase transition at high temperature and pressure, which is a novel effect. Moreover, it was shown that the inclusion of higher curvature terms (based on Lovelock Lagrangian) affects the criticality. In fact, for LM AdS black holes with hyperbolic horizon topology, depending on the chosen parameters, there is always an upper limit for the value of Lovelock coefficient (\(\alpha _{u}\)) in which no phase transition could happen for \(\alpha >\alpha _{u}\). In the case of spherical black holes, this statement no longer holds and we observe criticality in the range \(0<\alpha <\infty \). Considering tables, we found that the universal ratio, i.e. \(\frac{{{P_c}{v_c}}}{{{T_c}}}\), is a function of spacetime dimensions (d), topological factor (k), graviton mass parameter (m) and strength of higher curvature terms (captured with Lovelock coefficient, \(\alpha \)).
In addition, the vdW, reentrant and analogue of solid/liquid/gas phase transitions were found in the extended phase space of (un)charged-AdS black holes with the spherical horizon. But, in the case of hyperbolic black holes, reentrant and small/intermediate/large phase transitions were not found. Indeed, the reverse vdW phase transition in the phase space of hyperbolic black holes is accompanied with one or two distinct (standard) vdW phase transitions. To our knowledge, this is the first example of such a phase structure. These pieces of evidence show that the generic features of different theories of gravitation can be summed into a unique model to produce more complex structures for the thermodynamic phase space of black holes.
Notes
Acknowledgements
We thank an anonymous referee for useful suggestions. We also acknowledge M. Momennia for reading the manuscript and useful comments. AD would like to thank S. Zarepour for useful discussions and providing Mathematica programming codes. S.H.H would like to thank the hospitality of the Institute of Physics, University of Oldenburg during his short visit. We wish to thank Shiraz University Research Council. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha, Iran.
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