Regular Magnetically Charged BHs (NRCBHs)

• Regular magnetically charged black holes (NRCBHs) are spacetime solutions where nonlinear electrodynamics replaces central singularities with smooth cores or bounces.
• They encompass diverse models—such as Bronnikov, Fan–Wang, and Kruglov types—with varying energy conditions, horizon structures, and thermodynamic behaviors.
• Extensions into modified gravity frameworks (e.g., 4D EGB, ECG, f(R,T)) reveal distinct perturbative, optical, and stability features that impact theoretical and observational studies.

Searching arXiv for recent and foundational papers on regular magnetically charged black holes, nonlinear electrodynamics, and related perturbative/observational studies. Regular magnetically charged black holes (NRCBHs) are magnetically charged black-hole geometries sourced most often by nonlinear electrodynamics (NED) and constructed so that the Reissner–Nordström central singularity is replaced either by a regular core or, in black-bounce realizations, by a minimal-area bounce. In the standard spherically symmetric sector the metric is typically written as $ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2$, while the magnetic field is a monopole with $F_{\theta\phi}=Q_m\sin\theta$; depending on convention, the electromagnetic invariant is written either as $F=2Q_m^2/r^4$ or $\mathcal{F}=q^2/(2r^4)$. The literature includes minimally coupled GR constructions, Fan–Wang families, Bronnikov- and Kruglov-type models, rational and power-Maxwell theories, and extensions in 4D Einstein–Gauss–Bonnet (EGB), Einstein cubic gravity (ECG), and $f(R,T)$ gravity (Li et al., 2014, Meng et al., 2022, Ma, 2015, Jusufi, 2020, Tangphati et al., 2023, Liang et al., 16 Jun 2025).

1. Core definitions and source theories

A common starting point is Einstein gravity minimally coupled to NED,

$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$

or closely related sign-convention variants. In this framework, Bronnikov-type magnetic solutions use

$$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$

while Fan and Wang introduced a one-parameter family

$$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$

with $\mu=3$ and $\nu=1,2,3$ corresponding to Maxwellian, Bardeen-like, and Hayward-like subclasses, respectively. In these models the magnetic potential is $F_{\theta\phi}=Q_m\sin\theta$0, the field invariant is $F_{\theta\phi}=Q_m\sin\theta$1, and the magnetic parameter is related by $F_{\theta\phi}=Q_m\sin\theta$2 (Li et al., 2014, Meng et al., 2022).

Other minimally coupled NED models generate regular magnetic geometries with different ultraviolet behavior. Examples include Kruglov’s exponential NED,

$F_{\theta\phi}=Q_m\sin\theta$3

the rational model

$F_{\theta\phi}=Q_m\sin\theta$4

the cosh-type model

$F_{\theta\phi}=Q_m\sin\theta$5

and the deformed birefringent model

$F_{\theta\phi}=Q_m\sin\theta$6

Across these papers, the weak-field Maxwell limit is built in, while magnetic charge rather than electric charge is the preferred route to regularity in the minimally coupled setting (Jusufi, 2020, Kruglov, 2021, Kruglov, 2018, Ma, 2015).

The same theme extends beyond GR. Exact or perturbative NRCBHs have been constructed in 4D EGB gravity, in ECG coupled to NLE, and in $F_{\theta\phi}=Q_m\sin\theta$7 gravity coupled to NLED. These modified-gravity versions preserve the magnetic monopole sector but alter the branch structure, the field equations, and the thermodynamic and perturbative spectra (Jusufi, 2020, Lessa et al., 2023, Tangphati et al., 2023).

2. Regularity, core geometries, and energy conditions

The defining issue is whether the center is genuinely regular. Different NRCBH families realize this in different ways. The Bronnikov magnetic black hole has

$F_{\theta\phi}=Q_m\sin\theta$8

for which $F_{\theta\phi}=Q_m\sin\theta$9 as $F=2Q_m^2/r^4$0, the NED Lagrangian and energy density decay exponentially, and the curvature invariants satisfy

$F=2Q_m^2/r^4$1

yielding a Minkowski-like core. A related cosh-NED model likewise has $F=2Q_m^2/r^4$2, exponentially vanishing energy density, and curvature invariants that vanish at the center (Li et al., 2014, Kruglov, 2018).

By contrast, the Fan–Wang families with $F=2Q_m^2/r^4$3 approach

$F=2Q_m^2/r^4$4

so the core is de Sitter with

$F=2Q_m^2/r^4$5

and the curvature scalars remain finite: $F=2Q_m^2/r^4$6 The modified Hayward geometry sourced by rational NLED also has

$F=2Q_m^2/r^4$7

so its core is de Sitter with $F=2Q_m^2/r^4$8, and the Ricci and Kretschmann invariants are bounded; the paper explicitly states that the limiting curvature conjecture takes place. The $F=2Q_m^2/r^4$9 constructions likewise admit a regular center, with $\mathcal{F}=q^2/(2r^4)$0 for $\mathcal{F}=q^2/(2r^4)$1, and finite $\mathcal{F}=q^2/(2r^4)$2 and $\mathcal{F}=q^2/(2r^4)$3 for $\mathcal{F}=q^2/(2r^4)$4 (Meng et al., 2022, Kruglov, 2021, Tangphati et al., 2023).

Energy conditions are model dependent. The deformed birefringent NED model of Ma is built to satisfy the weak energy condition, and the paper verifies $\mathcal{F}=q^2/(2r^4)$5, $\mathcal{F}=q^2/(2r^4)$6, and $\mathcal{F}=q^2/(2r^4)$7 everywhere for the magnetic solution. In $\mathcal{F}=q^2/(2r^4)$8+NLED, the null, weak, and strong energy conditions can all be arranged to hold outside the outer event horizon by appropriate parameter choices. By contrast, some papers do not analyze the weak, null, or dominant energy conditions at all; this is explicit in the 4D EGB exponential-NED work and in the rational-NED modified Hayward model (Ma, 2015, Tangphati et al., 2023, Jusufi, 2020, Kruglov, 2021).

The label “regular” is therefore not uniform across the literature. The 4D EGB magnetic solution of Hegde, Karananas, and collaborators is discussed in the context of regular black holes, but the paper does not compute $\mathcal{F}=q^2/(2r^4)$9, $f(R,T)$0, or $f(R,T)$1, and does not provide a near-origin expansion proving regularity at $f(R,T)$2. In that case, the central regularity claim is contextual rather than invariant-based (Jusufi, 2020).

3. Horizons, extremality, and parameter space

In essentially all NRCBH models, horizons are defined by the zeros of $f(R,T)$3 or $f(R,T)$4, and the same qualitative trichotomy recurs: two horizons, one degenerate extremal horizon, or no horizon. The Bronnikov solution has subextremal, extremal, and horizonless sectors; for $f(R,T)$5 the paper gives

$f(R,T)$6

obtained from $f(R,T)$7. Fan–Wang solutions show analogous behavior, with allowed charge intervals

$f(R,T)$8

for black-hole horizons when $f(R,T)$9 (Li et al., 2014, Meng et al., 2022).

The horizon structure is particularly sensitive to modified-gravity couplings. In 4D EGB with exponential NED, the physically viable branch is the negative branch of the square-root solution, and the paper reports explicit critical values for horizon disappearance. For $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$0, $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$1, and $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$2, increasing $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$3 yields

$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$4

and for $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$5 one has $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$6 and thus no horizon. In ECG, sufficiently large cubic coupling can also eliminate horizons and produce naked singularities; the paper gives the neutral examples $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$7 as nakedly singular and $S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$8 as still black-hole-like (Jusufi, 2020, Lessa et al., 2023).

Some recent observationally oriented NRCBH metrics exhibit larger extremal magnetic charges than Reissner–Nordström. One 2025 model reports

$S=\frac{1}{16\pi}\int d^4x\sqrt{-g}\,[R-L(F)],$9

from $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$0, while also noting that its own table and figure captions contain inconsistent values around $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$1 for $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$2. That paper therefore provides a concrete example in which the existence of an extremal bound is clear, but its precise numerical value is internally unsettled (Aydiner et al., 7 Jul 2025).

4. Thermodynamics and phase structure

The thermodynamic analysis is usually built on the standard surface-gravity formula

$$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$3

with entropy given either by the area law $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$4 in GR and minimally coupled NED, or by Wald’s entropy in higher-curvature gravity. In many NRCBH families, $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$5 vanishes at extremality, develops a maximum at intermediate radius, and is accompanied by heat-capacity divergences interpreted as second-order phase transitions. This pattern appears in Bronnikov-type, cosh-NED, rational-NED, Fan–Wang-type, and ECG-corrected models (Li et al., 2014, Kruglov, 2018, Kruglov, 2021, Jusufi, 2020, Lessa et al., 2023).

The detailed phase structure depends strongly on the matter model. In the cosh-NED magnetic solution,

$$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$6

at the extremal point $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$7, $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$8 reaches a maximum at $$\mathcal{L}(F)=F\cosh^{-2}\!\left(a\left|\frac{F}{2}\right|^{1/4}\right),$$9, and the fixed-charge heat capacity diverges there, signaling a second-order phase transition; the outer branch is locally stable for $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$0 and unstable for $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$1. In Ma’s birefringent rational-NED model, the fixed-charge heat capacity similarly changes sign at the temperature maximum, and the paper states that the smaller black hole is more stable in the usual fixed-charge ensemble (Kruglov, 2018, Ma, 2015).

A separate line of work emphasizes remnants. In the modified Hayward geometry supported by rational NLED, the temperature and heat capacity vanish at a nonzero radius $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$2, which the paper interprets as a black-hole remnant, while a second-order phase transition occurs at a larger $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$3 where $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$4. In ECG with small charge and small cubic coupling, the Hawking temperature is regularized and leaves a thermodynamically stable remnant for small $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$5. This suggests that remnant formation is not tied to a single NED model, although the mechanism is model specific (Kruglov, 2021, Lessa et al., 2023).

Modified frameworks add further layers. In 4D EGB with exponential NED, negative Hawking temperature is used as a diagnostic of parameter regions without physical black holes. In one recent NRCBH model, the tunneling temperature is corrected by a generalized uncertainty principle to

$$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$6

and the same paper studies a Joule–Thomson coefficient $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$7, finding that cooling behavior is favored for larger charge and larger horizon radius (Jusufi, 2020, Aydiner et al., 7 Jul 2025).

5. Perturbations, quasinormal modes, and stability

Linear perturbations do not produce a single universal picture across NRCBHs. For Bronnikov’s regular magnetic black hole, electromagnetic perturbations in NED obey a Schrödinger-like equation with an effective potential containing $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$8, $$L(F)=\frac{4\mu}{\alpha}(\alpha F)^{(\nu+3)/4}\left[1+(\alpha F)^{\nu/4}\right]^{-(\mu+\nu)/\nu},$$9, and $\mu=3$0. In that model the potential nearly coincides with the Reissner–Nordström electromagnetic potential away from the central region, and the quasinormal spectra are correspondingly very close to RN. The paper finds positive-definite effective barriers outside the event horizon, $\mu=3$1 in the convention $\mu=3$2, and damped time-domain ringdown, concluding stability against NED electromagnetic perturbations (Li et al., 2014).

The coupled gravito-electromagnetic problem can behave quite differently. In the Fan–Wang families analyzed through Chandrasekhar’s procedure, odd-parity gravitational perturbations couple to even-parity electromagnetic perturbations, and the master system is matrix valued,

$\mu=3$3

For $\mu=3$4 and $\mu=3$5, all computed modes satisfy $\mu=3$6, so the spacetimes are linearly stable in the explored sector, but the spectra depend strongly on $\mu=3$7 and on the family label $\mu=3$8, and are reported to differ markedly from Reissner–Nordström (Meng et al., 2022).

Scalar perturbations in modified gravity preserve the same broad stability conclusion. For the regular magnetic solutions in $\mu=3$9+NLED, massive scalar quasinormal modes were computed with sixth-order WKB plus Padé averaging, all imaginary parts were found to be negative, and an eikonal analysis was also given. In that case, increasing the trace coupling $\nu=1,2,3$0 lowers both $\nu=1,2,3$1 and $\nu=1,2,3$2, while increasing the magnetic parameter raises $\nu=1,2,3$3 and tends to reduce $\nu=1,2,3$4 (Tangphati et al., 2023).

Taken together, these studies show that “NRCBH ringdown” is not a single template. Some regular magnetic geometries are almost exterior-isospectral to RN, while others acquire substantial NED-induced gravito-electromagnetic mixing and branch structure. A plausible implication is that observational distinguishability depends more on the specific NED or modified-gravity realization than on regularity alone.

6. Optical appearance, lensing, and orbital dynamics

Weak-field and strong-field observables are likewise model dependent. For the Bronnikov regular magnetic black hole, the Gauss–Bonnet and geodesic analyses agree that the weak deflection angle is smaller than in the singular RN case with the same mass and magnetic charge, and a cold non-magnetized plasma increases the bending angle. In the shadow problem, the effective geometry induced by NLED must be used: for both Euler–Heisenberg and Bronnikov models, the shadow was computed from the effective metric rather than from the background metric, and comparison with the EHT image of M87* produced upper limits on magnetic charge, including

$\nu=1,2,3$5

for the Bronnikov case (Fu et al., 2021, Allahyari et al., 2019).

A more recent magnetically charged regular black-hole metric of the form

$\nu=1,2,3$6

was studied in the thin-disk and EHT context. There the parameter $\nu=1,2,3$7 decreases the photon-sphere radius and shadow size, lowers $\nu=1,2,3$8 and $\nu=1,2,3$9, modestly increases the Novikov–Thorne flux and disk temperature, and changes the widths of direct, lensed, and photon-ring impact-parameter intervals. Using the EHT shadow radii of M87* and Sgr A*, the paper reports

$F_{\theta\phi}=Q_m\sin\theta$00

in units with $F_{\theta\phi}=Q_m\sin\theta$01 (Zare et al., 2024).

Not all regular metrics are observationally distinct from RN outside the horizon. For the Balart–Vagenas Fermi–Dirac-based regular solution, the paper finds that photon effective potentials, photon circular orbits, shadow radii, stable circular orbits, and even the regular-versus-chaotic dynamics of charged particles in external magnetic fields are almost indistinguishable from RN outside the event horizon. The paper attributes this to an extremely small exterior difference between the two metric functions. This directly cautions against treating regularity as an automatically observable property (Zhang et al., 2022).

One 2025 NRCBH study extends the observational program to GUP-corrected thermodynamics, Gauss–Bonnet weak lensing, plasma refraction, orbital frequencies, quasi-periodic-oscillation proxies, and Joule–Thomson expansion. It finds that sufficiently large magnetic charge can drive the weak-deflection series negative, that plasma modifies the deflection through

$F_{\theta\phi}=Q_m\sin\theta$02

and that the Keplerian frequency $F_{\theta\phi}=Q_m\sin\theta$03 develops charge-sensitive maxima. The same paper explicitly warns that its extremal-charge numerics are internally inconsistent, so its observational trends are more robust than its quoted extremal threshold (Aydiner et al., 7 Jul 2025).

7. Extensions, caveats, and no-go results

The NRCBH literature now includes rotating and nonstandard global extensions. Toshmatov, Stuchlík, and Ahmedov constructed rotating regular black holes in GR coupled to NED using a Kerr-like metric with a radius-dependent mass function $F_{\theta\phi}=Q_m\sin\theta$04. Their Maxwellian class approaches Maxwell electrodynamics in the weak-field limit, is regular for $F_{\theta\phi}=Q_m\sin\theta$05 and $F_{\theta\phi}=Q_m\sin\theta$06, and yields epicyclic frequencies smaller than in Kerr at the same spin. The same work shows that the weak and strong energy conditions are violated inside the Cauchy horizon of the rotating geometries (Toshmatov et al., 2017).

A different extension is the black-bounce realization of magnetically charged regular black holes in Einstein–NLED coupled to a self-interacting phantom scalar. In those solutions the physical spacetime has a minimum areal radius $F_{\theta\phi}=Q_m\sin\theta$07, the non-Lorentzian region $F_{\theta\phi}=Q_m\sin\theta$08 is excised, and the causal structure interpolates among black bounces, one-way wormholes, and traversable wormholes according to the relation between the horizon scale and the bounce scale. These geometries are regular on the physical domain but rely on phantom matter and explicit weak-energy-condition violation in the total stress tensor (Cañate, 2022).

Two major caveats have become central. First, in 4D EGB gravity the original dimensional-regularization proposal is controversial; the magnetic exponential-NED paper explicitly notes that its solution should remain valid only for spherically symmetric spacetimes in the regularized theories, and not beyond spherical symmetry. Second, regularity of the background metric does not guarantee regularity of effective light propagation. The 2025 power-Maxwell analysis shows that the effective photon metric can become singular in electric sectors, and that in particular magnetic configurations nonradial photon trajectories can even become spacelike in the background spacetime. This suggests that causal and optical regularity are logically distinct notions (Jusufi, 2020, Liang et al., 16 Jun 2025).

The strongest recent limitation is a no-go result for nonminimally coupled electromagnetic fields. In theories containing $F_{\theta\phi}=Q_m\sin\theta$09, $F_{\theta\phi}=Q_m\sin\theta$10, or $F_{\theta\phi}=Q_m\sin\theta$11, static spherically symmetric magnetically charged regular black holes are generically excluded under mild smoothness assumptions. The obstruction arises from $F_{\theta\phi}=Q_m\sin\theta$12 terms in the trace equation near the center. A combined coupling

$F_{\theta\phi}=Q_m\sin\theta$13

admits a necessary fine-tuning condition

$F_{\theta\phi}=Q_m\sin\theta$14

to cancel the leading magnetic divergence, but this is not shown to be sufficient for a regular solution. A plausible implication is that minimally coupled NED remains the comparatively robust arena for NRCBH model building, whereas nonminimal electromagnetic-curvature couplings sharply constrain the existence of physically realistic regular magnetic cores (Bokulić et al., 11 Jun 2026).