P−Vcriticality in the extended phase space of black holes in massive gravity

Abstract

We study the P -V criticality and phase transition in the extended phase space of charged anti-de Sitter black holes in canonical ensemble of ghost-free massive gravity, where the cosmological constant is viewed as a dynamical pressure of the black hole system. We give the generalized thermodynamic first law and the Smarr relation with massive gravity correction. We find that not only when the horizon topology is spherical but also in the Ricci flat or hyperbolic case, there appear the P -V criticality and phase transition up to the combination k+c 2 0 c2m 2 in the four-dimensional case, where k characterizes the horizon curvature and c2m 2 is the coefficient of the second term of massive potential associated with the graviton mass. The positivity of such combination indicate the van der Waals-like phase transition. When the spacetime dimension is larger then four, the Maxwell charge there seems unnecessary for the appearance of critical behavior, but a infinite repulsion effect needed, which can also be realized through negative valued c3m 2 or c4m 2 , which is third or fourth term of massive potential. When c3m 2 is positive, a Hawking-Page like black hole to vacuum phase transition is shown in the five-dimensional chargeless case. For the van der Waals like phase transition in four and five spacetime dimensions, we calculate the critical exponents near critical point and find they are the same as those in the van der Waals liquid-gas system.

Massive charged BTZ black holes in asymptotically (a)dS spacetimes

“…Also, the existence of Van der Waals like behavior in extended phase space has been addressed in ref. [62]. Moreover, the generalizations of such theory to include nonlinear electrodynamics, higher derivative gravity and gravity's rainbow have been done [63][64][65][66].…”

The generalization of charged AdS black hole specific volume and number density

“…In fact, the same form (Eq. (12)) can also be found in different black hole systems [8,9], and we can also obtain ρ c = 3/8 withα = 1/α.…”

“…For charged AdS black hole, it was found that the transition is of the Van der Waals (vdW) type in the canonical ensemble [4,5]. Recently, by treating the cosmological constant as the pressure, P = −Λ/8π [6,7], the complete analogy between the vdW fluid and charged AdS black hole was found, which is a great success in studying phase transition of black holes and investigations in different black hole systems have been carried out [8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

“…For more novel results see [7]. This work is a great success in studying phase transition of black holes and investigations in different black hole systems have been carried out [8][9][10][11][12][13][14][15][16].…”

Entropy relations and bounds of horizons in modified gravity

“…Thermodynamics of black holes in massive gravity are studied in [55,56], here we list the temperature, entropy and thermodynamics volume in each horizon…”