Gauss-Bonnet entropy and thermal dynamics of RN-AdS black holes

TL;DR

The study addresses how Gauss-Bonnet corrections affect the thermodynamics and phase structure of RN-AdS black holes by deriving a GB-modified entropy $S = \pi r_+^2 + 4 \alpha \pi$, the equation of state $P = \frac{T}{2 r_+} - \frac{1}{8 \pi r_+^2} + \frac{Q^2 \eta}{8 \pi r_+^4}$, and the Gibbs free energy $G = M - T S$ within an extended gravity framework. Using these relations, it identifies critical points with $\nu_c = Q \sqrt{\frac{24 \eta}{5}}$, $T_c = \frac{2 \sqrt{5}}{3 \pi Q \sqrt{24 \eta}}$, and $P_c = -\frac{5}{32 \pi \eta Q^2}$, demonstrating how the Gauss-Bonnet coupling and horizon radius modify stability and phase transitions. The thermal stability analysis via heat capacity $C$ and Hessian criteria $\tau \ge 0$ shows that the sign of the cosmological constant (as pressure) crucially affects stability regions, yielding a richer phase structure than standard GR. Overall, the work highlights observable consequences of higher-curvature corrections and motivates further GB-corrected holographic and higher-dimensional studies.

Abstract

We explore the thermodynamics of a novel solution for the Reissner-Nordström-Anti-de Sitter (AdS) black hole, uniquely incorporating the Gauss-Bonnet term. Unlike previous studies that primarily focused on standard General Relativity or other modifications, this inclusion allows for a modified entropy formulation, facilitating the computation of key thermodynamic quantities such as Gibbs free energy, the first law of thermodynamics, the equation of state, and Hawking temperature. We identify critical points and graphically represent the relationship between temperature and Gibbs free energy as a function of the horizon radius. Ultimately, we assess the thermal stability of the Reissner-Nordström-AdS black hole within the framework of Gauss-Bonnet gravity, emphasizing the influence of the Gauss-Bonnet term unlike previous studies that primarily focused on standard General Relativity or other modifications. As a result, it is found that the Gauss-Bonnet coupling significantly alters the thermodynamic behavior and stability structure of the black hole, revealing richer phase transition phenomena.

Figures (6)

• Figure 1: The behavior of $\textsl{T}$ versus $\textsl{r}_{+}$ for a fixed value of $\eta = -1$, different values of $\mathcal{Q}=1, 2, 3$ and P. Each line in the plot corresponds to a specific value of the thermodynamic pressure P= -0.02 (pink dashed line), P= -0.01 (green solid line), and P= 0.003 (blue dotted line).
• Figure 2: $\mathbb{P}$ versus $\textsl{r}_{+}$ for $\mathcal{Q}=1, 2, 3$ and $\eta = -1$ of RN-AdS BH.
• Figure 3: Function of $G$ versus $\textsl{r}_{+}$ for parameter values $\alpha=0.1$, $\mathcal{Q}=1$ and $\eta = -1$ of RN-AdS BH.
• Figure 4: Function of $\textbf{P}$ versus $\nu$ for $\mathcal{Q}=1, 2, 3$ and $\eta = 0.1$ of RN-AdS BH.
• Figure 5: $\textit{C} as a function of$ versus $\textsl{r}_{+}$ for parameter values $\mathcal{Q}=1$ and $\eta = 0.1$ of RN-AdS BH.