Thermodynamics of Regular Black Holes in Anti-de Sitter Space

TL;DR

The work addresses the longstanding issue of singularities in general relativity by constructing regular black holes with AdS asymptotics in theories that resum an infinite tower of higher-curvature corrections, encapsulated by quasi-topological densities $\mathcal{Z}_n$ and the master function $h(\psi)$. It develops general regularity criteria for static, minimally coupled matter and demonstrates fully regular gravitational and electromagnetic fields for Maxwell, Born–Infeld, and RegMax electrodynamics, with the core geometry (de Sitter vs AdS) determined by the near-origin Misner–Sharp energy or the nonlinear self-energy balance, respectively. The thermodynamics reveals a van der Waals–like equation of state with a finite molecular volume, universal behavior for black branes, and a law of corresponding states across models; critical ratios $P_c v_c/T_c$ cluster within a narrow range, with Hayward-AdS uniquely matching the canonical $3/8$ value in any dimension. Collectively, these results imply robust interior regularity and rich phase structure for AdS black holes in a broad class of higher-curvature theories, with implications for holography and quantum gravity phenomenology.

Abstract

We construct regular black holes with anti-de Sitter asymptotics in theories incorporating infinite towers of higher-order curvature corrections in any dimension $D \ge 5$. We find that regular black branes are generically inner-extremal, potentially evading instabilities typically associated with inner horizons. Considering minimally coupled matter, we establish general criteria for the existence of singularity-free solutions. We analyze solutions coupled to Maxwell and nonlinear (Born--Infeld and RegMax) electrodynamics, demonstrating in the latter case the first examples of fully regular gravitational and electromagnetic fields for all parameter values. Here, we find that the ratio of the gravitational mass to the electrostatic self-energy determines whether the regular core is de Sitter or anti-de Sitter. We perform a detailed analysis of the black hole thermodynamics and show that the equation of state exhibits features akin to those of fluids with a finite molecular volume induced by the regularization parameter.

Figures (17)

• Figure 1: Metric functions: AdS black holes. Plots of the $k=1$ metric function for models I-VI for different values of the thermodynamic mass $M$. In each plot we have set $L_{\rm eff} = 1$, $G_{\rm eff}=1$, $a=0.1$, and $D = 5$. The top row represents models I–III from left to right, while the bottom row corresponds to models IV–VI in the same order. The gray dashed line represents the critical thermodynamic mass value associated with an extremal regular black hole. The values of the thermodynamic mass $M$ in the color order [blue, gray, red] are: $[0.3, 0.706, 1]$ (top left), $[0.2,0.362,0.55]$ (top center), $[1.2,1.849,2.8]$ (top right), $[0.2,0.326,0.5]$ (bottom left), $[0.15,0.253,0.4]$ (bottom center), and $[0.15,0.217,0.35]$ (bottom right). All models I-VI yield metric functions that have exactly the same qualitative structure.
• Figure 2: Metric functions: Maxwell charged black holes. Plots of the $k=+1$ metric function for model IV for different values of the coupling and charge. In each plot, we have set $L = 1$, $m=1$, and $D = 5$. We have also included the corresponding metric function with $q=0$, shown in each panel as the dashed gray curve. The parameter values are: $\alpha = 1/100, q = 1/20$ (left), $\alpha = 1/100, q = 2$ (center) and $\alpha = 1/10, q = 1/20$ (right). Models V, VI and VII with ${\rm N}$ even yield metric functions that have exactly the same qualitative structure.
• Figure 3: Metric functions: black branes. Plots of the $k=0$ metric function for model IV for different values of the coupling and charge. In each plot, we have set $L = 1$, $m=1$, and $D = 5$. We have also included the corresponding metric function with $q=0$, shown in each panel as the dashed gray curve. The parameter values are: $\alpha = 1/100, q = 1/20$ (left), $\alpha = 1/100, q = 2$ (center) and $\alpha = 1/10, q = 1/20$ (right). Models V, VI and VII with ${\rm N}$ even yield metric functions that have exactly the same qualitative structure.
• Figure 4: Metric functions: charged black holes in nonlinear electrodynamics. Plots of the $k=1$ metric function for model IV, where we have set $m=1$, $L=1$, $\alpha=0.01$, $q=0.1$, and $D=5$ for two different NLE Lagrangians illustrated by the blue solid line. The Maxwell case is shown as a gray dashed line. Top: Born---Infeld case for $m>m_{\star}$ (left), $m=m_\star$ (middle), and $m<m_\star$ (right), achieved by choosing $b=1$, $b\simeq 20.07$, and $b=40$, respectively; Bottom: RegMax case for $m>m_{\star}$ (left), $m=m_\star$ (middle), and $m<m_\star$ (right), achieved by choosing $\beta=1$, $\beta\simeq 4.46$, and $\beta=15$, respectively. Both Born--Infeld and RegMax metric functions have exactly the same qualitative structure.
• Figure 5: Wald entropy: Hayward-AdS model. The Wald entropy is plotted as a function of the horizon radius for the Hayward model in $D = 5$. The blue curves show the entropy for different values of $a=\alpha/L^2$, ranging from $a= 0$ to the maximum allowed value of $a= 1$ from the lightest to the darkest hue. The black curve shows the entropy at extremality, with all blue curves originating along this line. Plots in higher dimensions are qualitatively similar.