Geometrothermodynamics of asymptotically Anti-de Sitter black holes

Abstract

We apply the formalism of geometrothermodynamics to the case of black holes with cosmological constant in four and higher dimensions. We use a thermodynamic metric which is invariant with respect to Legendre transformations and determines the geometry of the space of equilibrium states. For all known black holes in higher dimensions, we show that the curvature scalar of the thermodynamic metric in all the cases is proportional to the heat capacity. As a consequence, phase transitions, which correspond to divergencies of the heat capacity, are represented geometrically as true curvature singularities. We interpret this as a further indication that the curvature of the thermodynamic metric is a measure of thermodynamic interaction.

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.…”

“…None of Weinhold and Ruppeiner metrics were invariant under the Legendre transformation. The first Legendre invariance metric was introduced by Quevedo [81,82], which can remove some problems of Weinhold/Ruppeiner methods. Although the Quevedo metric can solve some issues of the previous methods, it is confronted with other problems in specific systems.…”

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

“…In order to remove some failures of the Weinhold and Ruppeiner metrics, recently, another metric which is Legendre invariant has been introduced by Quevedo [83,84]. The Quevedo metric has the following form:…”

“…None of Weinhold and Ruppeiner metrics were invariant under Legendre transformation. Recently, Quevedo [83,84] removed some problems of Weinhold and Ruppeiner methods by proposing a Legendre invariant thermodynamical metric. Although Quevedo could solve some problems which previous metrics were involved with, it has been confronted with another problems in some specific systems.…”

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

“…Then, Ruppeiner proposed an alternative approach which has entropy as its thermodynamical potential [85,86]. Since Weinhold and Ruppeiner's approaches are not Legendre invariant, Quevedo introduced another approach for GTs [87,88]. Several investigations regarding thermodynamics of the black holes through these methods were done in Refs.…”