On Thermodynamics of Charged Black Holes via Extended Space-time Derivatives

TL;DR

This work introduces extended space-time derivatives built from a rank-2 antisymmetric tensor inspired by non-commutative string geometry and constructs charged black holes within a de Sitter gauge gravity framework. The resulting solutions feature a deformed metric $f_d(r)=f(r)+B g(r)$ with $f(r)$ containing the usual charged de Sitter terms and $g(r)$ providing a logarithmic deformation, controlled by the parameter $B$ and the charge $Q$. The authors perform a comprehensive thermodynamic analysis, obtaining expressions for $M$, $T$, $S$, $G$, and $C_p$, and demonstrate global and local stability as well as second-order phase transitions, including a $P$-$V$ criticality study that yields complex, parameter-dependent critical points. Through CUDA-based numerical computations, they map regions in $(B,Q)$ space where the black holes exhibit Van der Waals–like behavior and propose parameter-constraining strategies aligned with observational considerations from black hole shadows. The results offer a framework to explore string-inspired deformations in black hole thermodynamics and highlight GPU-accelerated methods for probing high-dimensional parameter spaces.

Abstract

Inspired by non-commutative geometry in string theory, we propose extended derivatives in black hole physics by incorporating a real antisymmetric tensor of rank 2 carrying similarities of certain stringy fields. Using gauge theory formulation of gravity via de Sitter group theory, we first find the associated black hole solutions by solving the Einstein field equations. Then, we study the thermodynamic properties by approaching the stability analysis, the criticality, and the phase transitions. Concretely, we investigate the P-V criticality behavior of the obtained solution. We compute and examine the Gibbs free energy revealing comparable attitudes with the Van der Waals phase transitions. Combining such results, we provide constraints on the deformed parameter B and the charge Q with the help of CUDA numerical methods exploited in machine learning computations. Precisely, we show that there are suitable ranges for such parameters where the obtained black holes behave like the Van der Waals fluid systems.

Figures (7)

• Figure 1: Metric function $f(r)$ in terms $r$ for different values of $Q$ with $B= -0,01$ and $\Lambda =0.1$.
• Figure 2: Hawking temperature in terms of $r_{h}$ by varying $Q$ for $B=-0,01$ and $\Lambda=1$.
• Figure 3: Gibbs free energy as a function of $r_h$ with $\Lambda=0.1$.
• Figure 4: Heat capacity in terms of $r_{h}$ for different values of $Q$.
• Figure 5: Pressure in terms of $v$ for different values of $T$ for $B=-3$ and $Q =0,11$.