Geometrothermodynamics of five dimensional black holes in Einstein–Gauss–Bonnet theory

Abstract

We investigate the thermodynamic properties of 5D static and spherically symmetric black holes in (i) Einstein-Maxwell-Gauss-Bonnet theory, (ii) EinsteinMaxwell-Gauss-Bonnet theory with negative cosmological constant, and in (iii) Einstein-Yang-Mills-Gauss-Bonnet theory. To formulate the thermodynamics of these black holes we use the Bekenstein-Hawking entropy relation and, alternatively, a modified entropy formula which follows from the first law of thermodynamics of black holes. The results of both approaches are not equivalent. Using the formalism of geometrothermodynamics, we introduce in the manifold of equilibrium states a Legendre invariant metric for each black hole and for each thermodynamic approach, and show that the thermodynamic curvature diverges at those points where the temperature vanishes and the heat capacity diverges.

The conformal metric structure of Geometrothermodynamics

“…This is an additional degree of freedom that can be used to reach diverse objectives. For instance, in the study of the thermodynamics of black holes [12], we found that the curvature singularities determine the phase transition structure and, in addition, Λ can be chosen in such a way that the limiting case of extremal black holes corresponds to curvature singularities too. Here, we have found that Λ can also be used to reach representation invariance directly from the phase space.…”

Understanding Gravitational Entropy of Black Holes: A New Proposal via Curvature Invariants

“…To put it differently, black hole entropy is a dynamical quantity (which depends on the field equations), unlike its temperature which is kinematical [61][62][63]. Two examples of modified gravity in which black hole entropy takes a different form (S = A/4) are f (R) gravity [64][65][66][67][68] and the Einstein-Gauss-Bonnet gravity [69][70][71].…”

On the ensemble dependence in black hole geometrothermodynamics

“…Unfortunately, it is not possible to express S and Q in terms of T and φ so that the Gibbs potential cannot be written explicitly. However, in the limiting case of a vanishing cosmological constant, it is possible to find explicitly the Gibbs potential and, as shown in [38], the corresponding thermodynamic metric can be computed and the invariance with respect to the total Legendre transformation can be shown at the level of the scalar curvature.…”

“…shown in [38], the corresponding thermodynamic metric can be computed and the invariance with respect to the total Legendre transformation can be shown at the level of the scalar curvature.…”