Thermogeometric description of the van der Waals like phase transition in AdS black holes

Abstract

It is well known that interpreting the cosmological constant as the pressure, the AdS black holes behave as the van der Waals thermodynamic system. In this case, like a phase transition from vapor to liquid in a usual van der Waals system, black holes also changes phases about a critical point in the P -V picture, where P is the pressure and V is the thermodynamic volume. Here, we give a geometrical description of this phase transition. Defining the relevant Legendre invariant thermogeometrics corresponding to the two criticality conditions, which determine the critical values of respective thermodynamical entities, we show that the critical point refers to the divergence of the Ricci scalars calculated from these metrics. The similar descriptions are also provided for the other two pictures of the van der Waals like phase transition: one is T -S and the other one is Y -X where T , S, X and Y are temperature, entropy, generalized force and generalized displacement; i.e. potential corresponding to external charge, respectively. The whole discussion is very general as no specific black hole metric is being used.

Thermogeometric study of van der Waals like phase transition in black holes: An alternative approach

“…In the following section we explicitly present the form of thermo-geometric metric for different types of van der Waals criticalities (such as the P − V criticality, T − S criticality, Y − X criticality, Q 2 − Ψ criticality etc.). Remember that these have been reported earlier in [29,65] using the above prescription. So we shall just mention the metrics without any details.…”

“…In the subsequent section, we study all the van der Waals type phase transition case by case: firstly the P − V criticality and then T − S, Y − X and Q 2 − Ψ criticalities follow. In each type criticality, we show that one can construct a pair of Legendre-invariant metrics (in fact, those metrics were obtained in our earlier work [65] and we use those for the present analysis) and show that the critical point can be identified where the extrinsic curvatures of those metrics diverge simultaneously. Thereafter, we provide the conclusion of our work.…”

“…Although the analysis apparently is intriguing, its main limitation was the lack of a proper canonical definition of P and V variables. Later on, when the cosmological constant was taken as a variable and included in the extended version of first law [13][14][15][16][17] (also see our work [65], where the first law in the extended phase space has been obtained from the conserved Noether current following Wald's method), the canonical definition of P and V was provided and was shown that the van der Waals P − V criticality exists in black hole spacetime [18,19,74,75] and at the critical point, two independent conditions coincide i.e.…”

“…Recently, accounting the corrections of the thermodynamic parameters, the correction in geometrothermodynamics has been shown for f (R) gravity [64]. In one of our earlier works [65], we have constructed the Legendre invariant theromogeometrical manifolds for the van der Waals type phase transitions in AdS black holes. It was shown that the corresponding Ricci scalar diverges at the critical point.…”

Q − Φ criticality and microstructure of charged AdS black holes in f(R) gravity

“…In particular, promoted by the AdS/CFT correspondence [26,27], the investigation of black hole phase transitions in AdS spacetimes has received popular attention [28][29][30][31][32][33][34][35][36][37]. The pioneering study of the transitions can be traced back to the well-known Hawking-Page phase transition [38], which describes a first-order phase transition between Schwarzschild AdS black holes and the thermal AdS space.…”