Maxwell’s Equal Area Law and the Hawking-Page Phase Transition

We study the phases of a Schwarzschild black hole in the Anti-deSitter background geometry. Exploiting fluid/gravity duality, we construct the Maxwell equal area isotherm？？ in the temperature-entropy plane, in order to eliminate negative heat capacity BHs. The construction we present here is reminiscent of the isobar cut in the pressure-volume plane which eliminates unphysical part of the Van der Walls curves below the critical temperature. Our construction also modifies the Hawking-Page phase transition. Stable BHs are formed at the temperature , while pure radiation persists for . turns out to be below the standard Hawking-Page temperature and there are no unstable BHs as in the usual scenario. Also, we show that, in order to reproduce the correct BH entropy , one has to write a black hole equation of state, that is, , in terms of the geometrical volume . 1. Introduction Black holes (BHs) are among the most intriguing solutions of Einstein equations. Their geometric description is fully provided by the theory of general relativity and is discussed in many excellent textbooks. However, this is only half of the story. Since the original works by Bekenstein and Hawking, some new aspects of the BH behavior emerged once quantum field theory is coupled to a BH background geometry. Even if this is only a “semiclassical” quantum gravity formulation, the outcome has profoundly changed the prospective of the BH behavior. A stellar mass, classical, BH is characterized by the unique feature of being a perfect absorber with a vanishing luminosity. From a thermodynamical point of view, a classical BH is a zero temperature black body. However, nuclear size BHs, interacting with quantized matter, are almost perfect black bodies as they emit black body radiation at a characteristic nonvanishing temperature! Moreover, BHs are assigned a thermodynamical property identified with entropy. Thus, there are two complementary descriptions of BH physics: one in terms of pure space-time geometry and the other in terms of thermodynamics. The two descriptions are related to each other through the so-called “first law” of BH (thermo)dynamics as follows: where = total mass energy, = Hawking temperature, = entropy, = the Coulomb potential on the horizon, = electric charge, = angular velocity of the horizon, and = angular momentum. The first law (1) is the basis of the thermodynamical description of the BH as a “fluid” where is the variation of total energy split into variation of “internal,” “electrostatic,” and “rotational” pieces, which are then given a thermodynamical meaning. By