Thermodynamics of Schwarzschild-AdS black hole in non-commutative geometry

TL;DR

This paper explores the thermodynamics of Schwarzschild-AdS black holes in non-commutative geometry, revealing phase transitions akin to Van der Waals fluids, and discusses violations of the first law and the effects of non-commutative parameters.

Contribution

It introduces a corrected Schwarzschild-AdS black hole model in non-commutative geometry and analyzes its thermodynamic phase transitions and violations of the first law.

Findings

Black holes exhibit Van der Waals-like phase transitions.

Violation of the first law causes discontinuities in Gibbs free energy.

Phase transition disappears as non-commutative parameter increases.

Abstract

In this paper, we study the thermodynamics of Schwarzschild-anti-de Sitter black holes within the framework of non-commutative geometry. By solving the Einstein's equations, we derive the corrected Schwarzschild-AdS black hole with Lorentzian distribution and analyze the thermodynamics. Our results confirm that if the energy-momentum tensor outside the event horizon is related to the mass of the black hole, the conventional first law of thermodynamics will be violated. The study of criticality reveals that the black hole undergoes a small black hole-large black hole phase transition similar to that of the Van der Waals system, with a critical point and a critical ratio slightly smaller than that of the Van der Waals fluid. As the non-commutative parameter increases, the phase transition process shortens, leading to a critical point, and ultimately to the disappearance of the phase transition. The violation of the conventional first law results in a discontinuity of the Gibbs free energy during the phase transition, indicating the occurrence of zeroth-order phase transition. Moreover, we investigate the Joule-Thomson expansion, obtaining the minimum inversion temperature and the minimum inversion mass.