Phase transition and thermodynamic stability in extended phase space and charged Hořava–Lifshitz black holes

Abstract

For charged black holes in Hořava–Lifshitz gravity, a second order phase transition takes place in extended phase space where the cosmological constant is taken as thermodynamic pressure. We relate the second order nature of phase transition to the fact that the phase transition occurs at a sharp temperature and not over a temperature interval. Once we know the continuity of the first derivatives of the Gibbs free energy, we show that all the Ehrenfest equations are readily satisfied. We study the effect of the perturbation of the cosmological constant as well as the perturbation of the electric charge on thermodynamic stability of Hořava–Lifshitz black hole. We also use thermodynamic geometry to study phase transition in extended phase space. We investigate the behavior of scalar curvature of Weinhold, Ruppeiner, and Quevedo metric in extended phase space of charged Hořava–Lifshitz black holes. It is checked if these curvatures could reproduce the result of specific heat for the phase transition.

Appendix: Extended Ehrenfest equations

Here, we extend Ehrenfest equations to obtain nine relations between thermodynamic quantities which are true at the point in which the specific heat diverges. Our approach is based on the recently developed method of [37]. We consider the general case of charged extended black holes in an ensemble with fixed charge and pressure. In this ensemble the Gibbs free energy is defined as $G=H-TS=M-TS$. So, by using the first law (17), we have

$$\begin{aligned} dG=-SdT+VdP+\varPhi dQ. \end{aligned}$$

(51)

So

$$\begin{aligned} S=-\left( \frac{\partial G}{\partial T}\right) _{P,Q}, \qquad V=\left( \frac{\partial G}{\partial P}\right) _{T,Q}, \qquad \varPhi =\left( \frac{\partial G}{\partial Q}\right) _{T,P}. \end{aligned}$$

(52)

Maxwell relations are obtained straightforwardly

$$\begin{aligned} \left( \frac{\partial S}{\partial P}\right) _{T,Q}= & {} -\left( \frac{\partial V}{\partial T}\right) _{P,Q}, \nonumber \\ \left( \frac{\partial S}{\partial Q}\right) _{T,P}= & {} -\left( \frac{\partial \varPhi }{\partial T}\right) _{P,Q}, \nonumber \\ \left( \frac{\partial \varPhi }{\partial P}\right) _{T,Q}= & {} \left( \frac{\partial V}{\partial Q}\right) _{T,P}. \end{aligned}$$

(53)

The first derivatives of the Gibbs free energy are continuous at the divergent point of the specific heat. So, $S_{1}=S_{2}$, $V_{1}=V_{2}$, and $\varPhi _{1}=\varPhi _{2}$, where the indices 1 and 2 denote the state before and after the transition. Then, one can conclude that

$$\begin{aligned} dS_{1}=dS_{2}, \qquad dV_{1}=dV_{2}, \qquad d\varPhi _{1}=d\varPhi _{2}. \end{aligned}$$

(54)

By expressing entropy S as a function of temperature T, pressure P, and charge Q, we would have

$$\begin{aligned} dS=\frac{C_{P,Q}}{T}dT-V\alpha dP-\varPhi \alpha ^{\prime }dQ, \end{aligned}$$

(55)

where we have used the definition of the specific heat $C_{P,Q}=T(\frac{\partial S}{\partial T})_{P,Q}$, volume expansion coefficient $\alpha =-\frac{1}{V}(\frac{\partial S}{\partial P})_{T,Q}=\frac{1}{V}(\frac{\partial V}{\partial T})_{P,Q}$, and $\alpha ^{\prime }=-\frac{1}{\varPhi }(\frac{\partial S}{\partial Q})_{P,T}=\frac{1}{\varPhi }(\frac{\partial \varPhi }{\partial T})_{P,Q}$. Since temperature T, volume V, and electric potential $\varPhi$ are continuous and $dS_{1}=dS_{2}$, Eq. (55) gives

$$\begin{aligned} V(\alpha _{2}-\alpha _{1})\left( \frac{dP}{dT}\right) _{S}+\varPhi (\alpha ^{\prime }_{2}- \alpha ^{\prime }_{1})\left( \frac{dQ}{dT}\right) _{S}-\frac{(C_{P,Q})_{2}-(C_{P,Q})_{1}}{T}=0. \end{aligned}$$

(56)

The above equation can be rearranged to yield

$$\begin{aligned} \left( \frac{dP}{dT}\right) _{S}=\frac{(C_{P,Q})_{2}-(C_{P,Q})_{1}}{TV(\alpha _{2}- \alpha _{1})}-\frac{\varPhi (\alpha ^{\prime }_{2}-\alpha ^{\prime }_{1})}{V(\alpha _{2}-\alpha _{1})} \left( \frac{dQ}{dT}\right) _{S}. \end{aligned}$$

(57)

Pressure P can be expressed as a function of T, S, and Q at least in principle, so

$$\begin{aligned} dP=\left( \frac{\partial P}{\partial T}\right) _{S,Q}dT+\left( \frac{\partial P}{\partial S}\right) _{T,Q}dS +\left( \frac{\partial P}{\partial Q}\right) _{T,S}dQ. \end{aligned}$$

(58)

For $dS=0$ we would obtain

$$\begin{aligned} \left( \frac{dP}{dT}\right) _{S}=\left( \frac{\partial P}{\partial T}\right) _{S,Q}+ \left( \frac{\partial P}{\partial Q}\right) _{T,S}\left( \frac{dQ}{dT}\right) _{S}. \end{aligned}$$

(59)

Comparing this equation with (57) we find the first and second extended Ehrenfest equations

$$\begin{aligned} \left( \frac{\partial P}{\partial T}\right) _{S,Q}= & {} \frac{(C_{P,Q})_{2}-(C_{P,Q})_{1}}{TV(\alpha _{2}-\alpha _{1})}, \end{aligned}$$

(60)

$$\begin{aligned} \left( \frac{\partial P}{\partial Q}\right) _{T,S}= & {} -\frac{\varPhi (\alpha ^{\prime }_{2}- \alpha ^{\prime }_{1})}{V(\alpha _{2}-\alpha _{1})}. \end{aligned}$$

(61)

Equation (56) can also be rearranged to give

$$\begin{aligned} \left( \frac{dQ}{dT}\right) _{S}=\frac{(C_{P,Q})_{2}-(C_{P,Q})_{1}}{T\varPhi (\alpha ^{\prime }_{2}- \alpha ^{\prime }_{1})}-\frac{V(\alpha _{2}-\alpha _{1})}{\varPhi (\alpha ^{\prime }_{2}-\alpha ^{\prime }_{1})} \left( \frac{dP}{dT}\right) _{S}. \end{aligned}$$

(62)

For Q as a function of T, S, and P, one can write, for constant entropy,

$$\begin{aligned} \left( \frac{dQ}{dT}\right) _{S}=\left( \frac{\partial Q}{\partial T}\right) _{S,P}+ \left( \frac{\partial Q}{\partial P}\right) _{T,S}\left( \frac{dP}{dT}\right) _{S}. \end{aligned}$$

(63)

By comparing this equation with (62) we would find the third and fourth extended Ehrenfest equations

$$\begin{aligned} \left( \frac{\partial Q}{\partial T}\right) _{S,P}= & {} \frac{(C_{P,Q})_{2}-(C_{P,Q})_{1}}{T\varPhi (\alpha ^{\prime }_{2}-\alpha ^{\prime }_{1})}, \end{aligned}$$

(64)

$$\begin{aligned} \left( \frac{\partial Q}{\partial P}\right) _{T,S}= & {} -\frac{V(\alpha _{2}-\alpha _{1})}{\varPhi (\alpha ^{\prime }_{2}-\alpha ^{\prime }_{1})}. \end{aligned}$$

(65)

For V as a function of T, P, and Q, we have

$$\begin{aligned} dV=V\alpha dT+V\kappa dP+\varPhi \kappa ^{\prime }dQ, \end{aligned}$$

(66)

where $\kappa =-\frac{1}{V}(\frac{\partial V}{\partial P})_{T,Q}$ is the isothermal compressibility and $\kappa ^{\prime }=\frac{1}{\varPhi }(\frac{\partial V}{\partial Q})_{T,P}=\frac{1}{\varPhi }(\frac{\partial \varPhi }{\partial P})_{T,Q}$. Following the same procedure as above and $dV_{1}=dV_{2}$, we find four other extended Ehrenfest equations

$$\begin{aligned} \left( \frac{\partial P}{\partial T}\right) _{V,Q}= & {} \frac{\alpha _{2}-\alpha _{1}}{\kappa _{2}-\kappa _{1}}, \end{aligned}$$

(67)

$$\begin{aligned} \left( \frac{\partial P}{\partial Q}\right) _{T,V}= & {} \frac{\varPhi (\kappa ^{\prime }_{2}-\kappa ^{\prime }_{1})}{V(\kappa _{2}-\kappa _{1})}, \end{aligned}$$

(68)

$$\begin{aligned} \left( \frac{\partial Q}{\partial T}\right) _{V,P}= & {} -\frac{V(\alpha _{2}-\alpha _{1})}{\varPhi (\kappa ^{\prime }_{2}-\kappa ^{\prime }_{1})}, \end{aligned}$$

(69)

$$\begin{aligned} \left( \frac{\partial Q}{\partial P}\right) _{T,V}= & {} \frac{V(\kappa _{2}-\kappa _{1})}{\varPhi (\kappa ^{\prime }_{2}-\kappa ^{\prime }_{1})}. \end{aligned}$$

(70)

A general expression for $\varPhi$ as a function of T, Q, and P would give

$$\begin{aligned} d\varPhi =\varPhi \alpha ^{\prime }dT+\varPhi \kappa ^{\prime }dP+\varPhi \chi dQ, \end{aligned}$$

(71)

in which we have used $\chi =\frac{1}{\varPhi }(\frac{\partial \varPhi }{\partial Q})_{T,P}$. By the same calculations we find the following equations

$$\begin{aligned} \left( \frac{\partial P}{\partial T}\right) _{\varPhi ,Q}= & {} -\frac{\alpha ^{\prime }_{2}-\alpha ^{\prime }_{1}}{\kappa ^{\prime }_{2}-\kappa ^{\prime }_{1}}, \end{aligned}$$

(72)

$$\begin{aligned} \left( \frac{\partial P}{\partial Q}\right) _{T,\varPhi }= & {} -\frac{\chi _{2}-\chi _{1}}{\kappa ^{\prime }_{2}-\kappa ^{\prime }_{1}}, \end{aligned}$$

(73)

$$\begin{aligned} \left( \frac{\partial Q}{\partial T}\right) _{\varPhi ,P}= & {} -\frac{\alpha ^{\prime }_{2}-\alpha ^{\prime }_{1}}{\chi _{2}-\chi _{1}}, \end{aligned}$$

(74)

$$\begin{aligned} \left( \frac{\partial Q}{\partial P}\right) _{T,\varPhi }= & {} -\frac{\kappa ^{\prime }_{2}-\kappa ^{\prime }_{1}}{\chi _{2}-\chi _{1}}. \end{aligned}$$

(75)

Not all of the twelve equations we have obtained, are independent. In fact Eqs. (65), (70), and (75) are respectively the reverse of (61), (68), and (73). Therefore, a total of nine independent extended Ehrenfest equations in canonical ensemble remain.