Abstract
It is well-known that the non-extremal anti-de-Sitter (AdS) black holes show various van der Waals type criticalities, such as the P−V, T−S, Y−X, Q2−Ψetc. In these cases, the phase space diagram of the relevant quantities are similar in behaviour to the isotherms on P−V diagram for the usual van der Waals gas system. The existing thermogeometric descriptions of these criticalities for the black holes deal with the whole thermogeometric manifold and its curvature. However, while observing the criticality, one only notices the behaviour of the relevant phase space variables and keep all other thermodynamic quantities as fixed. Therefore it is quite natural to investigate the geometrical behaviour of the induced metric, defined by these constant macroscopic variables, corresponding to the whole manifold of the thermogeometry. Using geometrothermodynamics (GTD), we precisely address this issue. It is observed that the extrinsic curvature of this induced metric also diverges at the critical point. This provides an alternative but important aspect of the thermogeometric description of phase transition of black holes. The entire formalism in this paper is very general as it is valid for any arbitrary black hole which shows the van der Waals type phase transition.
Highlights
Introduction and MotivationThe thermodynamic structure of black holes have been found decades ago
In the AdS space, when the cosmological constant is regarded as the thermodynamic pressure [13–17], it was found that the P − V diagram of black holes in the extended phase space looks exactly similar to that of the ordinary thermodynamics [18, 19], which implies a van der Waals phase transition in the black hole thermodynamics
We show that one can construct a pair of Legendre-invariant metrics and show that the critical point can be identified where the extrinsic curvatures of those metrics diverge simultaneously
Summary
The thermodynamic structure of black holes have been found decades ago. With proper identification of the entropy [1], temperature [2, 3] and all other thermodynamic parameters, it can be shown that black hole shows several thermodynamic features that of the ordinary thermodynamics. It will be interesting to see what happens if we fix those coordinates in the thermogeometric manifold to form a hypersurface (or line) and study the behaviour of the extrinsic curvature of that particular hypersurface (or line) In this regard let us mention that, recently the property of the extrinsic curvature (of a particular surface) has been observed for Reissner-Nordstrom AdS black hole [70] in the case of Davis type phase transition, where it has been found that the extrinsic curvature of the Ruppeiner metric is divergent on the same points where the heat capacity diverges. Notations: In our paper P stands for pressure; V for volume; T for temperature; S for entropy; Yi for multiple charges, angular momentum etc. and Xi, which is conjugate to Yi, refers to the potentials due to those charges, angular velocity etc
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