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

TL;DR

This paper introduces a unified framework for analyzing black hole thermodynamics and singularities in Einstein gravity coupled with causal nonlinear electrodynamics, revealing new phase transitions and criteria for horizon formation.

Contribution

It develops a generalized nonlinear electrodynamics model from Courant-Hilbert deformation, providing exact black hole solutions and insights into their thermodynamics and singularity structure.

Findings

Black hole phase transitions resemble van der Waals behavior.

Derived charge-to-mass bounds for horizon formation.

Identified conditions distinguishing black holes from naked singularities.

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.