Phase Transition of RN-AdS Black Hole with Fixed Electric Charge and Topological Charge

Phase transition of RN-AdS black hole is investigated from a new perspective. Not only the cosmological constant is treated as pressure, but also the spatial curvature of black hole is treated as topological charge. We obtain the extended thermodynamic first law from which the mass is naturally viewed as enthalpy instead of energy. In canonical ensemble with fixed topological charge and electric charge, interesting van der Waals system's oscillatory behavior in $T-S$, $P-V$ graphes and swallow tail behavior in $G-T$, $G-P$ graphes are observed. By applying the Maxwell equal area law and by analysing the gibbs free energy, we obtain analytical phase transition coexistence curves ($T-P$) which are consistent with each other.

theory. The author found the topological charge naturally arisen in holography. What is more, together with all other known charges ( electric charge, mass, entropy), they satisfy an extended first law and the Gibbs-Duhem-like relation as a completeness. In our paper, we will treat both the cosmological constant and the spatial curvature as variables, then following one of their methods to derive the extended first law, from which one can see the cosmological constant is naturally viewed as pressure and the mass as enthalpy. Based on the extended first law, the black hole's phase transition property will be investigated in canonical ensemble with fixed electric charge and topological charge. This paper is organized as follows. In Sec.II, following the method in Ref. [59], we will derive the extended first law in d dimensional space-time. In Sec.III, by analysing the specific heat, the phase transition of AdS black hole in 4 dimensional space-time is studied and the critical point is determined. In Sec.IV, the van der Waals like oscillatory behavior are observed in both T − S and P − V graphes, then the Maxwell equal area law is used to obtain the phase transition coexistence curve. In Sec.V, the van der Waals like swallow tail behavior are observed in G − T and G − P graphes, then we will obtain the phase transition coexistence curve by analysing the gibbs free energy. Finally, we summarize and discuss our results in Sec.VI.

II. THE EXTENDED THERMODYNAMIC FIRST LAW
The d dimensional space-time AdS black hole solutions with maximal symmetry in the Einstein-Maxwell theory are where m, q, l are related to the ADM mass M , electric charge Q, and cosmological constant Λ by and Ω (k) d−2 is the volume of the "unit" sphere, plane or hyperbola, k stands for the spatial curvature of the black hole. Under suitable compactifications for k ≤ 0, we assume that the volume of the unit space is a constant Ω d−2 = Ω (k=1) d−2 hereafter [58,59]. Following Ref. [59], the first law can be derived. Considering an equipotential surface f (r) = c with fixed c (here set c = 0), we variate both sides of the equation and obtain Noting we obtain Multiplying both sides with an constant factor (d−2)Ω d−2

16π
, the above equation becomes where T = ∂rf 4π is the temperature, q r d−3 is the electric potential. If we introduce a new "charge" as in Ref. [58,59] = , and the black hole mass is naturally viewed as enthalpy instead of energy. Finally, the extended first law is obtained as

III. THE SPECIFIC HEAT AND PHASE TRANSITION
Hereafter, the investigation will be limited in d = 4 dimensional space-time and in canonical ensemble with fixed electric charge and topological charge, leaving the other situations for further study. First of all, we would like to analyse the behavior of the specific heat and the related possible phase transition phenomena. The first law can be rewritten in terms of So the isobaric specific heat can be written as Since we are in canonical ensemble, C P,Q, can be abbreviated as C P . From the denominator, we can conclude (1) when P < 2 1536π 3 Q 2 , C P has two diverge points at which denote a phase transition.
(2)when P = P c = 2 1536π 3 Q 2 , the two diverge points of C P merge into one at which is the phase transition critical point. The temperature is and there is no phase transition. All the physic quantities can be rescaled by those at the critical point. Defining the isobaric specific heat becomes The behavior of the rescaled specific heatCP for the cases P < P c ,P = P c ,P > P c are shown in Fig.1. The curve of specific heat for P < P c has two divergent points which divide the region into three parts. Both the large radius region and the small radius region are thermodynamically stable with positive specific heat, while the medium radius region is unstable with negative specific heat. So there is a phase transition take place between small black hole and large black hole. The curve of specific heat for P = P c has only one divergent point and always positive which denotes that c is the phase transition critical point. While the curve of specific heat for P > P c has no divergent point and always positive, implying the black holes are stable and no phase transition will take place. This behavior of specific heat is very similar to that of the liquid-gas var der Waals system.
In the last section, we have determined the critical point and found a phase transition when P ≤ P c . In this section and in the next section, we will derive the analytical phase transition coexistence curve by using different methods. The temperature and entropy are which can be rescaled by Eq. (14) to be Thus we obtainT From the above equation, we can plot the curveT (S) for differentP in Fig.2. One can see that for pressureP ≤ 1.0, temperatureT (S) curves show an interesting var der Waals system's oscillatory behavior which denotes a phase transition. The oscillatory part needs to be replaced by an isobar (denote asT * ) such that the areas above and below it are equal to each other. This treatment is called Maxwell's equal area law. In what follows, we will analytically determine this isobarT * for differentP .
The Maxwell's equal area law is manifest as At points (S 1 ,T * ),(S 2 ,T * ), we have two equations The above three equations can be solved as The last equationT * (P ) is the rescaled phase transition coexistence curve. Then we can make a backward rescale to obtain the phase transition coexistence curve, The phase diagram is four dimensional (T, P, Q, ) The phase transition coexistence curves are plotted in Fig.3  The pressure and volume are which can be rescaled to be Thus we obtainP From the above equation, we can plot the curveP (Ṽ ) for differentT in Fig.4. One can see that for temperatureT ≤ 1.0, pressureP (Ṽ ) curves show an interesting var der Waals system's oscillatory behavior which denotes a phase transition. The oscillatory part needs to be replaced by an isobar (denote asP * ) such that the areas above and below it are equal to each other. This treatment is called Maxwell's equal area law. The analytical phase transition curve is derived as follows. The Maxwell's equal area law is manifest as At points (Ṽ 1 ,P * ),(Ṽ 2 ,P * ), we have two equations The above three equations can be solved as The last equationP * (T ) is the rescaled phase transition coexistence curve, and it can be rewritten asT which is exactly same with Eq. In Sec.II, we see that the cosmological constant is naturally viewed as dynamical pressure and the black hole mass as enthalpy. Thus the gibbs free energy is and its differential form in canonical ensemble can be obtained from Eq.(9) which denotes the gibbs free energy is a function of temperature and pressure.
Substituting black hole mass, temperature and entropy into Eq.(30), then making a rescaling by the quantities at the critical point, we will obtain From the above equation, we can plotG−T curves for differentP , and plotG−P curves for differentT in Fig.5. One can see that both panels display a var der Waals system's swallow The above equations can be solved as The last equation is exactly same with Eq. (21). So the phase transition coexistence curves obtained by analysing the gibbs free energy in G−T graph and G−P graph , or by applying the Maxwell equal area law in T − S graph and P − V graph are consistent with each other.
Treating the cosmological constant as variable [13,14] and the spatial curvature as topological charge [58,59], thermodynamics of electrically charged Reissner-Nordström AdS black holes are investigated. Firstly, by variation of the equipotential equation on horizon, the extended thermodynamic first law is obtained. From the extended first law, a conjugate potential correspondent to the topological charge is arisen. Meanwhile, if the black hole volume is defined as V = Ω d−2 d−1 r d−1 , then its conjugate pressure is naturally assigned as the cosmological constant and the black hole mass as enthalpy.
Secondly, in 4 dimensional space-time and canonical ensemble with fixed electric charge and topological charge, the isobaric specific heat C P is calculated and the corresponding divergence solutions are derived. The two solutions merge into one denoting the critical point where P c = 2 /(1536π 3 Q 2 ), r c = 2 6π/ Q. When P < P c , the curve of specific heat has two divergent points and is divided into three regions. The specific heat are positive for both the large radius region and the small radius region which are thermodynamically stable, while it is negative for the medium radius region which is unstable. When P > P c , the specific heat is always positive implying the black holes are stable and no phase transition will take place.
Thirdly, rescaling the quantities by those at the critical point, the behavior of temperature inT −S graph and the behavior of pressure inP −Ṽ graph are studied. They exhibit the interesting van de Waals gas-liquid system's behavior. WhenP > 1,T > 1, the curves vary monotonically and no phase transition will take place. WhenP < 1,T < 1, the curves display an oscillatory behavior which signals phase transition. The oscillatory part is replaced by an isobar according to the Maxwell's equal area law and the analytical phase transition coexistence curves (rescaled) are obtained which are consistent with each other.
Then by making a backward rescale, the explicit phase transition coexistence curve is derive in Eq. (22) and the phase diagrams are show in Fig.3.
Fourthly, a van der Waals system's swallow tail behavior are observed in theG −T graph andG −P graph whenP < 1,T < 1. The swallow tail's cross point is the phase transition point. By analytically solving the constraint equations, the rescaled phase transition coexistence curve is obtained which is consistent with those derived inT −S graph andP −Ṽ graph.
From the above detailed study in canonical ensemble, the analogy of RN-AdS black hole as van der Waals system have been examined when the spatial curvature is treated as topological charge and the cosmological constant is treated as pressure. Both the systems share the same oscillatory behavior and swallow tail behavior. Comparing with the case when the spatial curvature is fixed [14,15], our phase transition diagram in Fig.3 is four dimensional (T, P, Q, ), which is more abundant with an extra parameter -the topological charge.
A further investigation in grand canonical ensemble is outside the scope of this paper, but it is surely a very interesting future research. The influence of this topological charge on black hole thermodynamics in other gravity theories ( such as the Lovelock, Gauss-Bonnet theory, f(R) theory ) and different dimensional space-time also deserve to be disclosed in the future research. Another interesting future research line is comparison of the influence between the electric charge and topological charge.
In the end, we would like to point out that the black hole thermodynamics discussed in this paper is based on the first law derived from the equipotential surface f (r) = c with c = 0. A more conventional way to compute thermodynamics quantities is the Euclidean formalism, where the free energy is computed firstly, then the other quantities follow from it. The difference between these two formalisms remains unknown which deserve to be investigated in future.