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

Abstract: We construct a two-scale, static, spherically symmetric regular black hole in Einstein gravity sourced by magnetic nonlinear electrodynamics (NED). The zero-point length $\ell$ regularizes the mass and charge profiles, whereas $q$ is the asymptotic magnetic charge. The geometry approaches Reissner-Nordström at large radius, reduces to the neutral zero-point-length solution for $q=0$, and coincides geometrically with the Ayón-Beato-García solution for $\ell=|q|$. For $q\neq0$, inverse reconstruction gives a single-valued magnetic Lagrangian with Maxwell asymptotics and a finite strong-field limit. The center is regular and is de Sitter, locally Minkowski, or anti-de Sitter according to the sign of $2M\ell-q<sup>2$; the weak energy condition holds globally if and only if $3M\ell\geq2q<sup>2$. We derive the extremality curve, the exact heat capacity, and homogeneous horizon-variation and Smarr identities while retaining the Wald area entropy. We also prove that every charged black hole in this family has a nondegenerate extraordinary NED optical metric throughout the domain of outer communication. The associated capture shadow is selected by the global minimum of the optical impact-parameter function and generally differs from the background-geodesic shadow. Weak-field calculations yield the periapsis, bending, time-delay, and redshift corrections; in particular, $\ell$ first appears beyond the standard first-post-Newtonian parameters. Finally, for minimally coupled test radiation in a cold transparent plasma, we obtain exact parametric shadow relations for power-law density profiles and combine Hamiltonian ray tracing with a Novikov-Thorne disk model. A separate extraordinary NED-plasma continuation is displayed only as a phenomenological prescription because a material plasma breaks the conformal ambiguity of the vacuum characteristic metric.

• The paper constructs a two-scale, magnetically charged regular black hole supported by a reconstructed nonlinear-electrodynamics theory, with finite curvature, Reissner–Nordström asymptotics, and independent charge and zero-point length scales.
• It proves that global weak-energy-condition satisfaction requires the stronger bound 3Mℓ ≥ 2q² and establishes positive, nondegenerate extraordinary optical characteristics outside the event horizon for all physical charged solutions.
• The model produces observational signatures including a 10.3% difference between NED and background-geodesic shadow sizes, zero-point-length corrections to weak-field tests, and plasma-driven suppression of disk emission by approximately 50% in selected regimes.

Overview and construction

This paper constructs a static, spherically symmetric regular black hole in Einstein gravity sourced by magnetic nonlinear electrodynamics (NED), with two independent dimensionful scales: the asymptotic magnetic charge $q$ and a zero-point length $\ell$ motivated by string T-duality and Padmanabhan's path-integral duality. The metric function is

$$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$

with mass function $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$. The geometry approaches Reissner–Nordström (RN) at large radius, reduces to the neutral zero-point-length solution for $q=0$, and coincides geometrically with the Ayón-Beato–García (ABG) solution when $\ell=|q|$. The authors are careful to state that these are geometric limits only; the underlying electromagnetic theories need not be identical.

The central result of the matter sector is an inverse reconstruction: for $q\neq0$, the Einstein equations yield a single-valued local Lagrangian $L(F)$ with Maxwell weak-field asymptotics ($L_F\to1$) and finite strong-field limit $\lim_{F\to\infty}L(F)=6(2M\ell-q^2)/\ell^4$. The leading weak-field correction is proportional to $\ell$0, so the reconstructed theory is nonanalytic in integer powers of $\ell$1, and higher derivatives such as $\ell$2 need not remain finite in the strict vacuum limit. Crucially, $\ell$3 depends explicitly on $\ell$4; the paper concedes that the model is therefore a parameter-dependent effective NED representation rather than a family of states of one universal microscopic electromagnetic theory.

Regularity and energy conditions

All curvature invariants are finite at the origin for any nonzero $\ell$5: for example, $\ell$6. The core is de Sitter, locally Minkowski, or anti-de Sitter according to the sign of $\ell$7. The energy-condition analysis is exact rather than numerical. Using dimensionless variables with $\ell$8, monotonicity arguments show that $\ell$9 holds globally if and only if $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$0, while the full weak energy condition (WEC), including $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$1, holds globally if and only if

$$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$2

This condition is strictly stronger than positivity of the central density, so a de Sitter core alone does not guarantee global WEC satisfaction — a point the paper emphasizes against common practice in the regular-black-hole literature.

Horizons and thermodynamics

Extremality is governed by the polynomial $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$3, which admits a unique positive root for each charge ratio; the causal structure is RN-like. The neutral extremal remnant has $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$4. The Hawking temperature is obtained exactly,

$$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$5

and the Wald entropy retains the Bekenstein–Hawking area form because the gravitational action is pure Einstein–Hilbert.

Two structural features distinguish the thermodynamics from scale-free black holes. First, the asymptotic-force nonextremality parameter $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$6 does not vanish at extremality (for the neutral remnant, $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$7); the correct horizon non-extremality parameter is $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$8. Second, allowing $$f(r)=1-\frac{2Mr^2}{(r^2+\ell^2)^{3/2}}+\frac{q^2r^2}{(r^2+\ell^2)^2},$$9 to scale as a coupling renders the horizon mass homogeneous, yielding an exact state differential $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$0 and two equivalent Smarr-type identities, one involving the physical Hawking temperature with a factor $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$1. The authors stress that $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$2 is not an entropy correction but accounts for the parameter dependence of the reconstructed matter sector. The exact heat capacity identifies Davies-type transition points via $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$3, interpreted as transitions in effective horizon thermodynamics rather than phase transitions of a fixed microscopic theory. A notable mixed result: the Penrose entropy inequality $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$4 and both specific-heat bounds hold throughout the outer branch, whereas the stronger scale-free quantity $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$5 becomes negative for large horizons whenever $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$6 — a quantitative thermodynamic signature of the additional scale.

Optical admissibility theorem

A key analytic contribution is the proof that every charged black hole in this family possesses a nondegenerate extraordinary NED characteristic geometry throughout its domain of outer communication. Defining $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$7 and $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$8, the paper shows analytically that $$m(r)=Mr^3/(r^2+\ell^2)^{3/2}-q^2r^3/[2(r^2+\ell^2)^2]$$9 and $q=0$0 for all $q=0$1 when $q=0$2, using monotonicity of the auxiliary polynomials $q=0$3 and $q=0$4. On the WEC branch, $q=0$5 does acquire a negative region near the center, but its noncentral zero satisfies $q=0$6, i.e., it is always hidden behind the event horizon. Consequently optical positivity imposes no additional constraint beyond the physical domain $q=0$7, whose boundary switches from extremality to the WEC at $q=0$8.

The shadow is defined by the global minimum of the impact-parameter function $q=0$9, a prescription the authors argue supersedes selection of an unspecified "outermost" stationary point. For the benchmark $\ell=|q|$0, $\ell=|q|$1, the extraordinary shadow radius $\ell=|q|$2 differs from the background-geodesic value $\ell=|q|$3 by approximately 10.3%, demonstrating that the NED propagation effect is observationally separable from the geometric effect. A fixed-$\ell=|q|$4 perturbative expansion about RN gives the explicit separation between the two contributions to order $\ell=|q|$5.

Weak-field tests

The infrared expansion permits complete classical-test calculations. The periapsis advance contains the correction $\ell=|q|$6 beyond the standard GR and charge terms; light bending through fourth post-Minkowskian order yields the leading zero-point-length response $\ell=|q|$7; and the Shapiro delay acquires a $\ell=|q|$8 term. In isotropic coordinates, $\ell=|q|$9 and $q\neq0$0 first enters $q\neq0$1 at order $q\neq0$2, so the Cassini bound on $q\neq0$3 cannot directly constrain $q\neq0$4. The paper presents conditional one-parameter sensitivities (e.g., Mercury periapsis giving $q\neq0$5 km) but explicitly disclaims them as fitted bounds, since no magnetically charged solar interior model exists; they quantify observable reach only.

Plasma shadows and disk emission

For minimally coupled test radiation in a cold transparent plasma with power-law profile $q\neq0$6, the paper derives exact parametric circular-ray and shadow relations, including the exceptional $q\neq0$7 case where the shadow obeys $q\neq0$8 exactly. Below a cutoff frequency fixed by a plasma equilibrium radius, dispersive reflection removes the central capture shadow entirely, though direct and reflected disk rays remain visible.

Combining Hamiltonian backward ray tracing with a Novikov–Thorne disk and invariant radiative transport, the benchmark configuration gives ISCO at $q\neq0$9 (3.39% inward relative to Schwarzschild), efficiency $L(F)$0, and maximum flux increased by about 10.3%. Spectral results split into two regimes: for shallow profiles ($L(F)$1), the spectral peak shifts upward by roughly 21% and its amplitude drops by roughly 51% purely through dispersive reflection, with no absorption assumed; for steep profiles ($L(F)$2), the spectrum differs from vacuum by less than 1% near its peak. Image maps show the disappearance and reappearance of the silhouette across the cutoff frequency.

An important methodological caveat runs through this section: appending a dimensionful plasma-frequency term to the extraordinary NED characteristic Hamiltonian breaks the conformal ambiguity of the vacuum optical metric, so no unique NED-plus-plasma dispersion relation follows without a microscopic constitutive theory. The paper accordingly presents its extraordinary NED–plasma continuation explicitly as a phenomenological prescription, not a prediction.

Limitations and open questions

The paper is candid about several restrictions. The inverse magnetic reconstruction is defined only for $L(F)$3; the $L(F)$4 limit is smooth at the metric level but singular in the NED formulas, optical functions, and extraordinary-ray integrals, all of which contain inverse powers of $L(F)$5. The parameter-dependent nature of $L(F)$6 means Davies points and thermodynamic quantities describe effective horizon thermodynamics, not states of a fixed microscopic theory. The Solar-System sensitivities presuppose an unconstructed charged solar source. Open questions left by the work include the electric (Hamiltonian $L(F)$7) formulation of the same geometry, perturbative stability of the coupled gravitational–electromagnetic system, calibrated radiative-transfer fits to actual EHT data requiring an electron-density normalization, and rotating generalizations that do not assume Newman–Janis constructions preserve spacetime or optical regularity.

Conclusion

The paper delivers a self-contained two-scale regular black hole whose charge and ultraviolet length remain independent parameters, supported for $L(F)$8 by an exactly reconstructible magnetic NED sector. Its strongest analytic results are the necessary-and-sufficient global WEC condition $L(F)$9, the exterior optical-admissibility theorem guaranteeing nondegenerate extraordinary characteristics for every charged member of the family, and the generalized first-law and Smarr relations incorporating $L_F\to1$0 as a scaling coupling. The quantitative separation of the extraordinary NED shadow from the background-geodesic shadow (~10% at the benchmark) and the demonstration that plasma dispersion alone can suppress disk spectra by ~50% identify concrete observational channels, while the conformal obstruction to combining material plasmas with NED characteristics remains the principal unresolved theoretical issue.