Abstract
Employing the thermodynamic geometry approach, we explore phase transition of four dimensional spinning black holes in an anti-de Sitter (AdS) spaces and found the following novel results. (i) Contrary to the charged AdS black hole, thermodynamic curvature of the spinning AdS black hole diverges at the critical point, without needing normalization.(ii) There is a certain region with small entropy in the space of parameters for which the thermodynamic curvature is positive and the repulsive interaction dominates. Such behavior exists even when the pressure is extremely large. (iii) The dominant interactions in the microstructure of extremal spinning AdS black holes are strongly repulsive, which is similar to an ideal gas of fermions at zero temperature. (iv) The maximum of thermodynamic curvature, $ \left\vert R\right\vert $, is equal to $C_{{}_{P}}$ maximum values for the Van der Waals fluid in the supercritical region. While for the black hole, they are close to each other near the critical point.
Highlights
Thermodynamic fluctuation provides a unique framework for the geometrical description of thermodynamical systems in equilibrium
We have presented simple exact analytical expressions for the critical quantities of the Kerr-anti–de Sitter (AdS) black holes and constructed the phase diagram in the pressure-entropy parameter space, where the small black hole and large black hole phases are separated by a first order phase transition region below the critical point
Starting from the Ruppeiner geometry in an entropy representation, we have derived the thermodynamic metric for the Kerr-AdS black holes in the pressureentropy coordinates, which is valid for any ordinary thermodynamic system
Summary
Thermodynamic fluctuation provides a unique framework for the geometrical description of thermodynamical systems in equilibrium. It has been confirmed that one can extend the thermodynamic phase space by treating the cosmological constant as the thermodynamic pressure, P 1⁄4 −Λ=ð8πÞ, in an extended phase space, with its conjugate variable as volume [12–17] In this regard, continuous and discontinuous phase transitions between small and large charged AdS black holes have been realized [18], which are analogous to the Van der Waals liquid-gas phase transition and belong to the same universality class [18,19]. Two new normalized thermodynamic curvatures for a charged AdS black hole have been proposed, which diverge at the critical point of the phase transition [35–37] These thermodynamic curvatures are constructed via the heat capacity at constant volume [35,36] and adiabatic compressibility [37] and have the same behavior for large black holes. From the thermodynamic fluctuation metric in the entropy representation, we obtain a Ruppeiner line element of rotating AdS black holes in the pressureentropy coordinates, where it is valid for the ordinary thermodynamic systems, such as the simple Van der Waals fluid. In the Appendix, we calculate the thermodynamic curvature of the Van der Waals system using the Ruppeiner metric in (P-S) coordinates
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