A Unified Causal Framework for Nonlinear Electrodynamics Black Hole from Courant-Hilbert Approach: Thermodynamics and Singularity

TL;DR

The paper presents a unified, causal framework for Einstein gravity coupled to Generalized Nonlinear Electrodynamics in AdS, built from a root-$T\bar T$ (via the Courant–Hilbert approach) that preserves duality and causality. It systematically constructs GNED models (ModMax, GBI, logarithmic, and extensions) and derives exact or perturbative AdS black hole solutions, revealing van der Waals–type phase transitions and swallowtail structures in free energy. The study links thermodynamics to interior geometry through the Kretschmann scalar, showing how mass, charge, and nonlinear electromagnetic couplings shape horizon formation and the emergence of naked singularities. The framework offers a versatile platform for holographic explorations of transport, RG flows, and universal black hole thermodynamics in strongly coupled systems.

Abstract

We develop a unified framework for analyzing black hole thermodynamics and spacetime structure in Einstein gravity coupled to causal nonlinear electrodynamics (NED) in asymptotically anti-de Sitter backgrounds. The electromagnetic sector is governed by a Generalized Nonlinear Electrodynamics (GNED) Lagrangian obtained from a root-$T\bar T$ deformation constructed via the Courant-Hilbert approach, ensuring both duality invariance and causal propagation. This theory contains ModMax, Generalized Born-Infeld (GBI), and self-dual logarithmic electrodynamics as continuous limits. Within this framework we obtain exact charged AdS black hole solutions and perform a detailed study of their thermodynamic properties, including mass, temperature, entropy, and free energy. The resulting phase structure exhibits van der~Waals-type transitions between small and large black holes and features a characteristic swallowtail in the free energy at the critical point. we further investigate the internal geometry, showing that the nature of the central singularity is determined by the matter fields that source the spacetime. An analysis of the Kretschmann scalar reveals how mass and electric charge jointly govern curvature divergence in ModMax black holes. We derive an explicit charge-to-mass bound within a causal logarithmic electrodynamic theory. This extends the finite self-energy property of the point charge beyond the standard Born-Infeld model. This bound cleanly distinguishes black holes from naked singularities, showing that naked singularities occur precisely when the mass parameter is smaller than the electromagnetic self-energy of the point charge. This provides a clear energetic criterion for horizon formation in the absence of a Cauchy horizon.

Figures (8)

• Figure 1: Plots of the metric function $f(r)$ versus $r$ for Born-Infeld gravity with $k=0$, $L=1$, and $r_h = 1$ are shown. In Fig. \ref{['fig:frBIQ']}, with $\lambda = 0.1$, the black hole transitions from a single-horizon to a two-horizon state as the electric charge $Q$ and $\gamma$ are increased beyond a threshold of $e^{-\gamma/2}Q\approx 0.414$. Conversely, in Fig. \ref{['fig:frBIlam']}, with a fixed charge and $\gamma$ at $e^{-\gamma/2}Q = 0.410$, the same transition to a two-horizon state occurs as the Born-Infeld parameter $\lambda$ is decreased below a value of $\lambda \approx 0.1$.
• Figure 2: \ref{['fig:TSBIQ']}: The temperature vs entropy for (GBI) black hole for spherical ($k=1 ,L=1$) case by fixed $\lambda=0.02$ for different variable effective electric charge $\tilde{Q}=e^{-\gamma/2}Q$, and \ref{['fig:FTBIQ']}: The Helmholtz free energy vs temperature for a (GBI) black hole for the spherical ($k=1 ,L=1$) case, fixed in $\lambda=0.02$ for different variable effective electric charge $\tilde{Q}=e^{-\gamma/2}Q$.
• Figure 3: Plots of the metric function $f(r)$ versus $r$ for Log-$AdS$ gravity with $k=0$, $L=1$, and $r_h = 1$ are shown. In Fig. \ref{['fig:frLogQ']}, with $\lambda = 0.1$, the black hole transitions from a single-horizon to a two-horizon state as the electric charge $Q$ and $\gamma$ is increased beyond a threshold of $e^{-\gamma/2}Q\approx 0.377$. Conversely, in Fig. \ref{['fig:frLoglam']}, with a fixed charge and $\gamma$ at $e^{-\gamma/2}Q = 0.377$, the same transition to a two-horizon state occurs as the coupling $\lambda$ is decreased below a value of $\lambda \approx 0.1$.
• Figure 4: \ref{['fig:TSlogQ']}: The temperature vs entropy for Log-$AdS$ black hole for spherical ($k=1 ,L=1$) case by fixed $\lambda=0.02$ for different values of the effective electric charge $\tilde{Q}=e^{-\gamma/2}Q$, and \ref{['fig:FTlogQ']}: The Helmholtz free energy vs temperature for a Log-$AdS$ black hole for the spherical ($k=1 ,L=1$) case, fixed in $\lambda=0.02$ for different values of the effective electric charge $\tilde{Q}=e^{-\gamma/2}Q$.
• Figure 5: The inverse Hawking temperature vs entropy for different theories for spherical ($k=1 ,L=1$) case by fixed $\lambda=0.02$ and $e^{-\gamma/2} \,Q=0.09$. Here we considered $q=3/5$ for the $q$-deformed theory.