# Thermodynamic geometry of a black hole surrounded by perfect fluid in Rastall theory

## TL;DR

This paper investigates the thermodynamic properties and geometric structure of a black hole surrounded by perfect fluid in Rastall gravity, analyzing stability and geometric singularities using various thermodynamic metrics.

## Contribution

It provides a detailed analysis of thermodynamic geometry in Rastall gravity, highlighting the effectiveness of HPEM metric in capturing physical singularities.

## Key findings

- HPEM metric's singularities match heat capacity zeros and divergences
- Ruppeiner and HPEM metrics' curvature singularities coincide with thermodynamic phase transitions
- Weinhold and Quevedo formalisms do not yield clear physical insights

## Abstract

In this paper, we study thermodynamics and thermodynamic geometry of a black hole surrounded by the perfect fluid in Rastall theory. In particular, we calculate the physical quantity like mass, temperature and heat capacity of the system for two different cases. From the resulting heat capacity, we emphasize stability of the system. Following Weinhold, Ruppiner, Quevedo and HPEM formalism, thermodynamic geometry of this black hole in Rastall gravity is also analyzed. We find that the singular points of the curvature scalar of Ruppeiner and HPEM metrics entirely coincides with zero points of the heat capacity. But there is another divergence of HPEM metric which coincides with the singular points of heat capacity, so we can extract more information of HPEM metric compared with Ruppeiner metric. However, we are unable to find any physical data about the system from the Weinhold and Quevedo formalism.