The starting point is a construction in which the electric potential of a point charge is regularized, taking the form phi = Q divided by the square root of r0 squared plus r squared, where Q is the electric charge and r0 is a nonlinear scale parameter. The corresponding electric field is regular everywhere, and the resulting nonlinear electrodynamics Lagrangian recovers the Maxwellian limit at large distances while encoding nonlinear corrections that become significant near the origin. When coupled to Einstein gravity, the resulting static, spherically symmetric spacetime is characterized by three independent parameters: the ADM mass M, the electric charge Q, and the nonlinear scale r0. This extra degree of freedom, absent in the linear electrovacuum case, is what generates the model’s unusually rich structure.

One striking feature of the model concerns the mathematical structure of the nonlinear electrodynamics source itself. The invariant F of the theory is not monotonic: it decreases from zero at the origin to a minimum at r = r0 divided by the square root of 2, and then increases back toward zero at infinity. As a consequence, eliminating the radial coordinate between the field variables does not define a globally single-valued constitutive relation. Instead, the model consists of two local branches, and the author emphasizes that these branches are not physically equivalent. The outer branch possesses a hyperbolic effective optical geometry and connects continuously to Maxwell electrodynamics at spatial infinity, whereas on the inner branch the standard hyperbolicity condition for transverse electromagnetic disturbances is lost. The analysis therefore treats the physically admissible, Maxwell-connected outer branch as the relevant sector for propagating electromagnetic modes.

The causal structure of the solutions is classified using the dimensionless parameters m = M/r0 and q = Q/r0, which split the solution space into four distinct configurations. Depending on the values of these parameters, the spacetime admits black holes with single non-degenerate horizons, double-degenerate extremal horizons, and even triple-degenerate ultra-extremal horizons, alongside naked singular geometries with no horizon at all. The nature of the central singularity is controlled by the combination m minus two-thirds q squared: when this quantity is non-negative the singularity at the origin is spacelike, and when it is negative the singularity is timelike. Penrose diagrams constructed for each case clarify the global structure, showing that black hole solutions hide a spacelike singularity behind an event horizon while the naked solutions expose a timelike singularity.

The classification is remarkably detailed. In the third configuration, where m lies between two critical values, six qualitatively different cases appear as the charge increases: a singular black hole with a single simple event horizon, a black hole with a simple event horizon and a double inner horizon, a black hole with three simple horizons, a black hole with a double event horizon and a simple inner horizon, and finally naked singular solutions. In the fourth configuration, with even larger mass, the sequence includes an extremal black hole with a double event horizon and a timelike singularity at the center. This diversity of horizon structures goes well beyond the standard Reissner-Nordström case and highlights the nontrivial role played by nonlinear electromagnetic effects in shaping spacetime geometry.

Thermodynamic stability forms the second pillar of the analysis. The author derives the Hawking temperature and the heat capacity of the solutions, noting that the entropy obeys the standard Bekenstein-Hawking area law because the nonlinear electrodynamics enters only through the matter sector and involves no curvature couplings. The heat capacity analysis reveals two critical values of the charge parameter that govern the thermal behavior. Below the first critical value, q squared equal to about 1.052647, the Hawking temperature decreases monotonically with horizon radius and the heat capacity remains negative, meaning the black hole is thermally unstable. Above this critical charge, the temperature develops a local minimum followed by a local maximum, and the heat capacity diverges and changes sign at these Davies points, signaling phase transitions. Between the two Davies points the heat capacity is positive, producing a finite thermally stable region.

A second critical charge, q squared equal to three-quarters plus the square root of two over two, marks the formation of a degenerate, extremal horizon with vanishing Hawking temperature. For charges exceeding this value, the minimum of the temperature becomes negative, and the physically admissible stable region is bounded by the larger zero of the Hawking temperature and the larger Davies point. In other words, stable black holes exist only within a finite interval of horizon radii, either between the two Davies points or between the extremal radius and the larger transition point. The study also examines the relation between mass and charge for the existence of black hole solutions in general, finding that the horizon radius never exceeds twice the mass and decreases as the charge-to-mass ratio increases, with a critical charge beyond which no black hole solution exists for each fixed value of the nonlinear scale.

The near-horizon geometry of the most exotic configuration, the triple-degenerate or ultra-extremal black hole, receives special attention. At such a horizon the metric function and its first two derivatives vanish, so the curvature invariants take particularly simple forms determined entirely by the horizon radius. The metric function vanishes cubically near the horizon, and the resulting two-dimensional time-radius sector is not maximally symmetric and does not reduce to AdS2, unlike the extremal Reissner-Nordström case. The proper radial distance to the horizon diverges, and the redshift effect is more pronounced due to the higher-order vanishing. The vanishing first derivative implies a vanishing surface gravity and hence zero Hawking temperature, confirming the ultra-extremal character. Notably, this configuration does not admit the local mapping to colliding plane wave geometries that the extremal Reissner-Nordström near-horizon geometry enjoys.

The final part of the study addresses electromagnetic photon propagation and the black hole shadow. In nonlinear electrodynamics, electromagnetic disturbances generally do not follow the null geodesics of the background spacetime; instead, their characteristics are governed by an effective optical metric determined by the nonlinear Lagrangian. The resulting photon-sphere condition differs from the one obtained from null geodesics of the background metric, and the critical impact parameter, which separates captured rays from escaping rays and thus determines the apparent shadow, must be computed on the physically admissible Maxwell-connected branch. The calculations show that the nonlinear optical corrections generally increase the critical impact parameter relative to the Reissner-Nordström case, with the strongest deviations occurring for smaller values of the dimensionless mass, particularly in the strong-field regime.

An intriguing subtlety emerges in the neutral limit. For exactly zero charge the spacetime reduces to the Schwarzschild geometry, with the familiar critical impact parameter of three times the square root of three times the mass. However, the limit taken along the charged branch of the nonlinear theory is nonuniform: although the electromagnetic invariant vanishes with the charge squared, a derivative of the Lagrangian contains an inverse power of the charge squared, so the combination entering the effective optical geometry remains finite. The effective optical result for nonzero charge therefore cannot be continuously extrapolated to the exactly neutral solution. Taken together, the results demonstrate that nonlinear electrodynamics significantly enriches the causal structure, thermodynamic phase space and electromagnetic propagation properties of charged black holes, offering characteristic deviations from the Reissner-Nordström paradigm that may provide potential observational signatures of nonlinear electromagnetic effects in strong-field gravitational systems.

Subject of Research: Theoretical analysis of charged black hole solutions in Einstein gravity coupled to a nonlinear electrodynamics model based on a regularized electric potential

Article Title: Physical properties of charged black holes from the nonlinear electrodynamics model based on electric potential regularization

Article References: Mazharimousavi, S. H. (2026). Physical properties of charged black holes from the nonlinear electrodynamics model based on electric potential regularization. The European Physical Journal C, 86(10), Article 1240. https://doi.org/10.1140/epjc/s10052-026-16505-2

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16505-2

Keywords: black holes, nonlinear electrodynamics, general relativity, horizon structure, thermal stability, Hawking temperature, heat capacity, photon sphere, black hole shadow, Reissner-Nordström, theoretical physics, Penrose diagrams