Geometrothermodynamics of black holes

Abstract

The thermodynamics of black holes is reformulated within the context of the recently developed formalism of geometrothermodynamics. This reformulation is shown to be invariant with respect to Legendre transformations, and to allow several equivalent representations. Legendre invariance allows us to explain a series of contradictory results known in the literature from the use of Weinhold's and Ruppeiner's thermodynamic metrics for black holes. For the Reissner-Nordstr\"om black hole the geometry of the space of equilibrium states is curved, showing a non trivial thermodynamic interaction, and the curvature contains information about critical points and phase transitions. On the contrary, for the Kerr black hole the geometry is flat and does not explain its phase transition structure.Comment: Revised version, to be published in Gen.Rel.Grav.(Mashhoon's Festschrift

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

“…In Ref. [93], it was shown that it is possible to obtain an infinite number of Legendre invariant metrics. The simplest way to construct Legendre invariant metrics is to apply a conformal transformation.…”

Correspondence of phase transition points and singularities of thermodynamic geometry of black holes

“…In [16], the geometrothermodynamic approach is used to obtain the phase transition points. However, this theory is not able to produce the correct phase transition points.…”

“…Over the last decade, thermodynamic geometry and some of a e-mail: sa.hosseinimansoori@ph.iut.ac.ir b e-mail: b.mirza@cc.iut.ac.ir its new formulations have also been applied to black holes [10][11][12][13][14][15]. Another geometric formulation of thermodynamics was proposed by Quvedo [16]. Quvedo's method incorporates Legendre invariance in a natural way, and it allows us to derive Legendre invariant metrics in the space of equilibrium states.…”

A unified thermodynamic picture of Hořava-Lifshitz black hole in arbitrary space time

“…In order to introduce the idea of differential geometry to thermodynamics, according to [48][49][50], we have to define a (2n + 1) dimensional thermodynamic phase space T . It can be coordinated by the set Z A = {Φ, E a , I a }, where Φ represents the thermodynamic potential and E a and I a represent extensive and intensive thermodynamic variables respectively.…”

“…But these two metrics fail in explaining the thermodynamic properties and they lead to many puzzling situations. By incorporating the idea of Legendre invariance, Quevedo et al [48][49][50] …”