Geometrical Method for Thermal Instability of Nonlinearly Charged BTZ Black Holes

Abstract

We consider three-dimensional BTZ black holes with three models of nonlinear electrodynamics as source. Calculating heat capacity, we study the stability and phase transitions of these black holes. We show that Maxwell, logarithmic, and exponential theories yield only type one phase transition which is related to the root(s) of heat capacity, whereas, for correction form of nonlinear electrodynamics, heat capacity contains two roots and one divergence point. Next, we use geometrical approach for studying classical thermodynamical behavior of the system. We show that Weinhold and Ruppeiner metrics fail to provide fruitful results and the consequences of the Quevedo approach are not completely matched to the heat capacity results. Then, we employ a new metric for solving this problem. We show that this approach is successful and all divergencies of its Ricci scalar and phase transition points coincide. We also show that there is no phase transition for uncharged BTZ black holes.

Geometrical thermodynamics and P–V criticality of the black holes with power-law Maxwell field

“…Depending on the choice of the thermodynamical potential, the components of the phase space differ (this is due to the fact that each thermodynamical quantity has its specific extensive parameters). There are several well-known methods for constructing a phase space; see Weinhold [76,77], Ruppeiner [78,79], Quevedo [81,82], and HPEM [83][84][85][86]. In order to investigate the thermodynamical properties of the system, the successful method has divergencies which exactly are matched with bound and phase transition points.…”

“…Therefore, in order to remove these problems, a new method was proposed in Refs. [83][84][85][86] which is known as the HPEM (Hendi-Panahiyan-Eslam Panah-Momennia) metric. In this paper, we want to study the thermal stability and phase transition in the context of the methods of GT and extended phase space for black holes in Einstein gravity with the PMI source in higher dimensions.…”

Charged BTZ black holes in the context of massive gravity’s rainbow

“…Recent studies in the context of the GTs approaches for the black hole thermodynamics have shown that the Ricci scalars of Weinhold, Ruppeiner and Quevedo metrics may lead to extra divergencies which are not matched with the bound points and the phase transitions [97][98][99][100]. In other words, there were cases of mismatch between divergencies of the Ricci scalar and the mentioned points (bound and phase transition points), and also existence of extra divergency unrelated to these points were reported [97][98][99][100]. In order to overcome the shortcomings of the mentioned methods (Weinhold, Ruppeiner and Quevedo metrics), the HPEM method was introduced and it was shown that the specific structure of this metric provides satisfactory results regarding GTs of different classes of the black holes.…”

“…[97][98][99][100], for more details). In other words, obtained results of these three approaches were not consistent with those extracted from other methods.…”

Phase Transition of Black Holes in Brans–Dicke Born–Infeld Gravity through Geometrical Thermodynamics

“…Although Quevedo could solve some problems which previous metrics were involved with, it has been confronted with another problems in some specific systems. To solve these problems, a new method was proposed in [85][86][87] which is known as HPEM metric. It was shown that HPEM metric is completely consistent with the results of the heat capacity in canonical ensemble in different gravitational systems.…”