# Thermodynamic Geometry of Black Holes in the Canonical Ensemble

## TL;DR

This paper explores the thermodynamic geometry of four-dimensional RN-AdS and Kerr-AdS black holes, revealing phase transition behaviors and critical phenomena similar to liquid-gas systems using scalar curvature analysis.

## Contribution

It applies thermodynamic geometry to analyze phase transitions in black holes within the canonical ensemble, highlighting the divergence of scalar curvature at critical points.

## Key findings

- Scalar curvature characterizes liquid-gas like phase transitions.
- Divergence of scalar curvature at critical temperature.
- Thermodynamic geometry reveals critical phenomena in black holes.

## Abstract

We investigate the thermodynamics and critical phenomena for four dimensional RN-AdS and Kerr-AdS black holes in the canonical ensemble both for the normal and the extended phase space employing the framework of thermodynamic geometry. The thermodynamic scalar curvatures for these black holes characterizes the liquid-gas like first order phase transition analogous to the van der Waals fluids, through the $R$-Crossing Method. It is also shown that the thermodynamic scalar curvatures diverge as a function of the temperature at the critical point.