The Thermodynamic Relationship between the RN-AdS Black Holes and the RN Black Hole in Canonical Ensemble

Abstract

In this paper, by analyzing the thermodynamic properties of charged AdS black hole and asymptotically flat space-time charged black hole in the vicinity of the critical point, we establish the correspondence between the thermodynamic parameters of asymptotically flat space-time and nonasymptotically flat space-time, based on the equality of black hole horizon area in the two different space-time. The relationship between the cavity radius (which is introduced in the study of asymptotically flat space-time charged black holes) and the cosmological constant (which is introduced in the study of nonasymptotically flat space-time) is determined. The establishment of the correspondence between the thermodynamics parameters in two different space-time is beneficial to the mutual promotion of different time-space black hole research, which is helpful to understand the thermodynamics and quantumproperties of black hole in space-time.

A note on circular geodesics and phase transitions of black holes

“…Analogous proposals for black holes in flat space time are emerging, where an artificial concentric cavity of a certain radius, say, r B is added, whose specific value is to be determined. There are also proposals, that the holographic features of black holes in AdS are possibly due to the natural confining box, rather than the specific details of the theory [55,56,58]. It has been noted in these works, that having the black hole in a box leads to a Hawking -Page transition and there also exists a second order phase transition, exactly as in AdS.…”

“…(1.1), we now see that it is possible to obtain a relation among the thermodynamic parameters of flat space black holes in a box, by studying the divergence of specific heat. Basic set up of charged black holes in flat spacetime in a box of radius r B are discussed in detail in [55,56,58] and here, we only need to recall few important results. The metric for charged black holes in flat spacetime is given by…”

“…To make contact with the study of circular geodesics, it is useful to also fix the charge in the cavity and study the system in the canonical ensemble. The action is known to be [28,55,56,58]:…”

“….7) is constrained by the requirement that the black hole does not encounter a naked singularity [11] and is plotted in figure-(1). Further, the point in parameter space where the charged black holes in asymptotically flat spacetime in a box, become critical is [28,55,56,58]: .7) and (3.8) allow us to predict that it might be possible to see the values taken by thermodynamic parameters at the critical point, by studying the classification of circular orbits of test particles in the background of charged black holes in flat spacetime, in a box. Such a computation would require putting appropriate boundary conditions for physical quantities and geodesics, such that they fall off appropriately at the boundary of the cavity, possibly using the methods being developed in [60,61].…”

Thermodynamically stable phases of asymptotically flat Lovelock black holes

“…The Reissner-Nordström black hole has one stable branch with no phase transitions in the Gibbs free energy, and the Kerr black hole has similar properties [27]. Furthermore, very few investigations in Lovelock gravity have focused on the thermodynamics of asymptotically flat black holes [28,29], with studies limited to second-order (Gauss-Bonnet) and third-order Lovelock theories [30], where a stable phase was shown to exist for a range of horizon radii between some minimal and maximal values.…”