P-V criticality in the extended phase space of black holes in massive gravity

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+c02c2m2 in the four-dimensional case, where k characterizes the horizon curvature and c2m2 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 c3m2 or c4m2, which is third or fourth term of massive potential. When c3m2 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 the critical point and find they are the same as those in the van der Waals liquid-gas system.Comment: 11 pages, 16 figure

Lichnerowicz-Type Theorems for Self-gravitating Systems with Nonlinear Electromagnetic Fields

We consider a self-gravitating system containing a globally timelike Killing vector and a nonlinear Born-Infeld electromagnetic field and scalar fields. We prove that under certain boundary conditions (asymptotically flat/AdS) there can't be any nontrivial field configurations in the spacetime. To explore nontrivial solutions one should break any of the conditions we imposed. The case with another type of nonlinear electromagnetic field is also analyzed, and similar conclusions have been obtained under certain conditions.Comment: revtex4, 7 pages, no figure, one conclusion and an explanation for the derivation adde