Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature

Abstract

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner's metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold's approach. The corresponding Ruppeiner's metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.

Massive charged BTZ black holes in asymptotically (a)dS spacetimes

“…So far, four different geometrical methods were introduced; Weinhold [131,132], Ruppeiner [133,134], Quevedo [135,136] and HPEM [137,138]. Several studies in context of black holes with consideration of such methods have been done [139][140][141][142][143][144]. In addition, it was pointed out that one can employ geometrical thermodynamics to obtain phase transition points in superconductors [145].…”

New perspective for black hole thermodynamics in Gauss–Bonnet–Born–Infeld massive gravity

“…Now, by using the total mass of black holes (20) with entropy (19) and electric charge (17), one can construct the phase space and calculate its Ricci scalar. Due to economical reasons, we will not present the Ricci scalar obtained but rather present its results in the following diagrams (Figs.…”

“…There are several methods regarding thermodynamical geometry which are; Weinhold [7,8], Ruppeiner [9,10], Quevedo [11,12] and HPEM [13][14][15]. The geometrical thermodynamics has been employed in the context of different types of black holes [16][17][18][19][20][21][22]. In addition, this method was also used to study the phase transition of superconductors [23].…”

Geometrothermodynamics of phantom AdS black holes

“…where for the sake of simplicity we have rescaled the entropy as S/π → S. According to classical black hole thermodynamics [14], the fundamental equation should be given by a homogeneous function of the extensive variables [14,15]. On the other hand, the mathematical definition of a homogeneous function of n variables, say (E a ), a = 1, .…”