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

It is well known that by interpreting the cosmological constant as the pressure, the anti--de Sitter 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 change phases about a critical point in the $P\text{\ensuremath{-}}V$ 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 $T\text{\ensuremath{-}}S$ and the other one is $Y\text{\ensuremath{-}}X$ 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.

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