Phase Transitions of Electromagnetically Charged Black Holes in Lovelock Gravity with Nonconstant Curvature Horizons

TL;DR

This work analyzes phase transitions of electromagnetically charged black holes in third-order Lovelock gravity with nonconstant-curvature horizons in even dimensions. It derives the most general charged solutions, formulates the field equations including the Maxwell sector, and reduces the problem to a cubic master equation for $\psi(r)$ that determines the metric; in addition, two horizon-geometry–induced parameters $\hat{\eta}_2$ and $\hat{\eta}_3$ arise, with $\hat{\eta}_2$ acting as an effective magnetic charge. The authors compute the Hawking temperature $T$, Wald entropy $S$, and mass density $M$, and perform a thorough stability analysis in both the grand canonical and canonical ensembles, identifying first- and second-order transitions including van der Waals–like behavior. The results show that horizon topology and higher-curvature corrections strongly influence stability and phase structure, revealing a rich landscape of evaporation stabilization mechanisms and thermodynamic behavior in higher-dimensional gravity.

Abstract

We present the most general class of charged black hole solutions in third-order Lovelock gravity within even-dimensional spacetimes in the presence of an electromagnetic field. These solutions feature nonconstant-curvature horizons that affect geometry when n>=8. The near-origin behavior of the metric reveals a timelike singularity for electrically charged cases, in contrast to the spacelike singularity found in the uncharged case. We investigate thermodynamic stability in both the grand canonical and canonical ensembles. In the grand canonical ensemble, stability is determined by the positivity of both the Hessian determinant and the temperature. In the canonical ensemble, the sign of the heat capacity governs stability. We identify both first- and second-order phase transitions, including a van der Waals-like behavior characterized by instability at intermediate black hole sizes. Our results reveal a rich phase structure influenced by Lovelock corrections and electromagnetic fields, and demonstrate how conserved charges affect black hole evaporation and stabilization.

Figures (5)

• Figure 1: Behavior of $Det(H)$ with respect to $r_h$ for $k=0$, $n=10$, $\hat{{\alpha_2}}=1$, $\hat{{\alpha_{0}}}=0.3$, $\hat{{\alpha_{3}}}=0.8$, $\hat{\eta}_2=2$ and $\hat{\eta}_3=-0.03$
• Figure 2: Behavior of $Det(H)$ and $T$ with respect to $r_h$ for $n=8$, $k=-1$, $\hat{{\alpha_2}}=0.2$, $\hat{{\alpha_{0}}}=0.2$, $\hat{{\alpha_{3}}}=4$, $\hat{\eta}_2=4$, $\hat{\eta}_3=-0.05$, $q_{_E}=10$ and $q_{_M}=0.1$
• Figure 3: Behavior of $C_Q$ and $T$ with respect to $r_h$ for $n=8$, $\hat{{\alpha_2}}=0.2$, $\hat{{\alpha_{0}}}=0.2$, $\hat{{\alpha_{3}}}=4$, $\hat{\eta}_2=4$, $\hat{\eta}_3=-0.05$, $q_{_E}=10$ and $q_{_M}=0.1$
• Figure 4: Behavior of $C_Q$ and $T$ with respect to $r_h$ for $n=8$, $\hat{{\alpha_2}}=0.8$, $\hat{{\alpha_{0}}}=-3$, $\hat{{\alpha_{3}}}=1.05$, $\hat{\eta}_2=1$, $\hat{\eta}_3=-0.005$, $q_{_E}=0.002$ and $q_{_M}=10$
• Figure 5: Behavior of $C_Q$ and $T$ with respect to $r_h$ for $k=0$, $n=8$, $\hat{{\alpha_2}}=2$, $\hat{{\alpha_{0}}}=0.4$, $\hat{{\alpha_{3}}}=0.3$, $\hat{\eta}_2=4$, $\hat{\eta}_3=-3$, $q_{_E}=0.0013$ and $q_{_M}=0.01$