A Unified Causal Framework for Nonlinear Electrodynamics Black Hole from Courant-Hilbert Approach: Thermodynamics and Singularity (2511.17407v1)

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 and show that the nature of the central singularity is determined by the matter fields sourcing the spacetime. Analysis of the Kretschmann scalar demonstrates how gravity and electromagnetism jointly control curvature blow-up in charged and ModMax black holes, with gravity providing the leading divergence. In contrast, (GBI) and logarithmic (NED) models imprint distinct singularity profiles governed by the specific functional form of their Lagrangians. Examining the near-origin behavior of the metric allows us to identify the parameter ranges that support an event horizon and to determine when the solution instead becomes a naked singularity. Overall, the comparison across models reveals that different causal NED theories produce identifiable signatures in both the strength of the singularity and the conditions for horizon formation.