# A general framework to study the extremal phase transition of black holes

## TL;DR

This paper presents a universal framework for analyzing the extremal phase transition of black holes, applicable to any spacetime, and confirms that the Ricci scalar diverges at extremal points in geometrothermodynamics.

## Contribution

It introduces a general formulation for black hole phase transitions that is independent of specific spacetime details and confirms the divergence of the Ricci scalar at extremal points.

## Key findings

- Phase transition occurs only in the microcanonical ensemble.
- Critical exponents match previous metric-specific results.
- Ricci scalar diverges at extremal points in geometrothermodynamics.

## Abstract

We investigate the universality of some features for the extremal phase transition of black holes and unify all the approaches which have been applied in different spacetimes. Unlike the other existing approaches where the information of the spacetime and its dimension is directly used to get various results, we provide a general formulation in which those results are obtained for any arbitrary black hole spacetime having an extremal limit. Calculating the second order moments of fluctuations of some thermodynamic quantities we show that, the phase transition occurs only in the microcanonical ensemble. Without considering any specific black hole we calculate the values of critical exponents for this type of phase transition. These are shown to be in agreement with the values obtained earlier for metric specified cases. Finally we extend our analysis to the geometrothermodynamics (henceforth GTD) formulation. We show that for any black hole, if there is an extremal point, the Ricci scalar for the Ruppeiner metric must diverge at that point.