On Modified First Law of Black hole Thermodynamics in The Non-Commutative Gauge Theory

TL;DR

This paper explores how non-commutative geometry modifies the thermodynamics of Schwarzschild black holes, revealing phase transitions and the role of non-commutative parameters as thermodynamic variables.

Contribution

It introduces a modified first law of black hole thermodynamics incorporating a non-commutative potential, highlighting new phase transition behaviors due to non-commutativity.

Findings

Non-commutative black holes exhibit phase transitions.

The non-commutative parameter influences thermodynamic critical points.

The non-commutative potential affects black hole evaporation at final stages.

Abstract

In this paper, we investigated the thermodynamic properties of Schwarzschild black hole (SBH) in the non-commutative (NC) gauge theory of gravity. According to our previous work, we modify the first law of the black hole (BH) thermodynamics by the physical quantity (NC potential) $\mathcal{A}$ which is the conjugate to the NC parameter $\Theta$, which leads to this expression $d\hat{M}=\hat{T}d\hat{S}+\mathcal{A}d\Theta$. Our result shows that the NC SBH has a phase transition, and the non-commutativity affected this transition. And the NC potential $\mathcal{A}$ is effective only in the final stage of the BH evaporation, where it increases the Gibbs free energy at this stage, and the NC parameter in this study can represent the tension of the spacetime. Then the study of the pressure of the SBH in the modified first law of the BH thermodynamic shows a second-order phase transition, and the critical value of the thermodynamical variables are related to each value of $\Theta$, and that leads to this parameter to play the same role as a thermodynamical variable.