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

## TL;DR

This paper introduces an alternative thermogeometric approach using extrinsic curvature to analyze van der Waals-like phase transitions in black holes, providing a new perspective on critical phenomena.

## Contribution

It proposes analyzing the induced metric's extrinsic curvature in thermogeometry, offering a novel method to identify phase transitions in black holes.

## Key findings

- Extrinsic curvature diverges at critical points.
- Method is applicable to any black hole with van der Waals type transitions.
- Provides an alternative to existing thermogeometric descriptions.

## 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 $P-V$, $T-S$, $Y-X$, $Q^2-\Psi$ etc. 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.