Thermodynamical metrics and black hole phase transitions

Abstract

An important phase transition in black hole thermodynamics is associated with the divergence of the specific heat with fixed charge and angular momenta, yet one can demonstrate that neither Ruppeiner's entropy metric nor Weinhold's energy metric reveals this phase transition. In this paper, we introduce a new thermodynamical metric based on the Hessian matrix of several free energy. We demonstrate, by studying various charged and rotating black holes, that the divergence of the specific heat corresponds to the curvature singularity of this new metric. We further investigate metrics on all thermodynamical potentials generated by Legendre transformations and study correspondences between curvature singularities and phase transition signals. We show in general that for a system with n-pairs of intensive/extensive variables, all thermodynamical potential metrics can be embedded into a flat (n, n)-dimensional space. We also generalize the Ruppeiner metrics and they are all conformal to the metrics constructed from the relevant thermodynamical potentials.

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

“…Hence, the behavior of Weinhold's metrics is sensitive to phase transitions, needless of any other auxiliary structure whatsoever (cf. Bravetti et al [18] and Liu et al [26]). …”

Black hole enthalpy and an entropy inequality for the thermodynamic volume

“…The relevant thermodynamic quantities are given by 6) where the outer horizon is located at r = r + , the largest root of f (r + ) = 0.…”

“…where m and a i are the "mass" and the N rotation parameters appearing in the Kerr-AdS metrics, the summations and products are taken over 1 ≤ i ≤ N, the horizon radius is determined by the relation 6) and the Ξ i are given by Ξ i = 1 − g 2 a 2 i . The quantity A D−2 is the volume of the unit-radius (D − 2)-sphere, and is given by…”

“…(Some further discussion of varying the cosmological constant in the black hole thermodynamical context has recently been given in [6,7]. )…”

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

“…Therefore, the singularity points of the scalar curvature for both Helmholtz free-energy function and conjugate potential occur exactly at the same phase transition points of C Q (28). In other words, the line element of the free energy M(Q, T ) is associated with the line element of the conjugate potential M(S, Φ) [24]. We have…”