Two-scale magnetically charged regular black holes from nonlinear electrodynamics and a T-duality-inspired zero-point length

Technical Summary: Two-scale magnetically charged regular black holes from nonlinear electrodynamics and a T-duality-inspired zero-point length

Problem and Motivation
Classical black-hole solutions in Einstein gravity typically exhibit curvature singularities, signaling a breakdown of the Einstein–Maxwell description in the strong-field ultraviolet regime. While regular black hole (RBH) geometries have been proposed to resolve these singularities, many rely on phenomenological metrics or specific nonlinear electrodynamics (NED) sources where the matter Lagrangian is not uniquely determined or depends explicitly on integration constants. Furthermore, the interplay between ultraviolet (UV) regularization scales and electromagnetic charges remains an open question: can the charge and the UV length be treated as independent parameters within a local magnetic NED representation? Additionally, observational motivations necessitate distinguishing between the background spacetime metric and the effective optical geometry governing high-frequency electromagnetic perturbations in NED, a distinction that becomes critical when modeling black-hole shadows and accretion disks in the presence of plasma.

Methodology
The authors construct a static, spherically symmetric, two-scale regular black hole solution in Einstein gravity sourced by magnetic NED. The metric is defined by a mass function incorporating a zero-point length $\ell$ (inspired by path-integral duality and T-duality) and an asymptotic magnetic charge $q$.

1. Inverse Reconstruction: For $q \neq 0$, the authors employ an inverse-reconstruction method to derive the magnetic NED Lagrangian $\mathcal{L}(F)$ that supports the prescribed metric. This involves solving the Einstein-NED field equations to express the Lagrangian and its derivative in terms of the metric function and its derivatives.
2. Thermodynamic Analysis: The thermodynamics is analyzed using the Wald entropy formula. Crucially, because the reconstructed NED Lagrangian depends explicitly on the mass $M$, charge $q$, and length $\ell$, the authors treat these as independent thermodynamic variables. They derive generalized first-law and Smarr relations by applying Euler's theorem to the homogeneous scaling of the horizon mass.
3. Optical and Ray Tracing Analysis: The study distinguishes between background null geodesics and the "extraordinary" NED characteristic rays, which follow an effective optical metric determined by the NED constitutive relations. The authors derive the effective optical metric, identify the photon sphere, and calculate the shadow radius defined by the global minimum of the impact-parameter function.
4. Plasma and Accretion Modeling: The system is embedded in a cold, transparent, non-magnetized plasma with a power-law density profile. Using a Hamiltonian ray-tracing approach combined with the Novikov–Thorne thin-disk model, the authors compute frequency-dependent shadow images and emission spectra. They distinguish between test radiation propagating on the background metric and a phenomenological continuation of the extraordinary NED cone in plasma.

Key Contributions and Results

• Metric and Regularity: The constructed metric approaches the Reissner–Nordström (RN) solution at large radii, reduces to a neutral zero-point-length solution for $q=0$, and coincides with the Ayón-Beato–García solution when $\ell = |q|$. The central curvature invariants are finite for any finite $\ell > 0$. The nature of the core (de Sitter, Minkowski, or anti-de Sitter) is determined by the sign of $2M\ell - q^2$.
• Energy Conditions: The weak energy condition (WEC) is satisfied globally if and only if $3M\ell \geq 2q^2$. This condition is stricter than the requirement for a de Sitter center ($2M\ell > q^2$).
• NED Lagrangian: The reconstructed magnetic Lagrangian is single-valued, possesses Maxwell asymptotics, and remains finite in the strong-field limit. However, the Lagrangian depends explicitly on the parameters $M, q, \ell$, implying that the family of solutions represents a parameter-dependent effective NED representation rather than a continuous state space of a single universal microscopic theory.
• Thermodynamics: The authors derive exact expressions for the Hawking temperature, heat capacity, and generalized Smarr relations. They demonstrate that the standard scale-free thermodynamic relations do not hold unchanged due to the independent dimensionful scale $\ell$. Specifically, a proposed scale-free entropy bound $Z$ is generally violated when $\ell \neq 0$, providing a thermodynamic signature of the T-duality-inspired scale.
• Optical Admissibility: A key analytic result is the proof that for every charged black hole in this family satisfying the WEC, the NED optical metric functions ($L_F$ and $\Phi$) remain positive throughout the domain of outer communication. This ensures the non-degeneracy of the extraordinary optical metric outside the event horizon.
• Shadows and Light Bending: The extraordinary NED shadow differs from the background-geodesic shadow. The authors derive the leading-order corrections to the shadow radius and light bending, showing that the zero-point length $\ell$ first appears beyond the standard first-post-Newtonian parameters in weak-field expansions.
• Plasma Effects: In the presence of plasma, the shadow becomes frequency-dependent. The authors derive exact parametric relations for the shadow radius and identify a cutoff frequency below which the central capture shadow disappears due to plasma reflection. Numerical results for a benchmark model show that the NED optical response reduces the vacuum shadow radius by approximately 10.3% compared to the background geodesic shadow.
• Accretion Disks: Using the Novikov–Thorne model, the authors calculate the energy flux and efficiency. The regular geometry with T-duality corrections shifts the innermost stable circular orbit (ISCO) inward and increases the radiative efficiency compared to Schwarzschild and RN black holes with the same mass and charge.

Significance and Claims
The paper claims to provide a consistent framework for studying two-scale regular black holes where the UV length scale and magnetic charge are independent parameters. The work highlights the necessity of distinguishing between the background metric and the effective optical metric in NED theories, particularly for interpreting observational data like black-hole shadows. The authors emphasize that their reconstructed NED model is an effective representation dependent on the solution parameters, rather than a derivation from a fixed microscopic Lagrangian. Consequently, the thermodynamic analysis requires an extended phase-space treatment where the mass and charge are varied as couplings. The study provides exact analytical and numerical tools to test these geometries against weak-field classical tests (periapsis advance, light bending, time delay) and strong-field observables (shadows, accretion disk spectra) in the presence of plasma, offering specific predictions for how T-duality-inspired regularization modifies these phenomena. The authors remain modest regarding Solar-System constraints, noting that applying these black-hole exterior solutions to the Sun requires assumptions about the solar interior's magnetic hair and regularization scale that have not been constructed.