Magnetically Charged NED Black Hole

• Magnetically Charged NED Black Hole is a regular solution to Einstein's equations that eliminates the central singularity using nonlinear electrodynamics with a purely magnetic source.
• It features an effective optical metric that alters photon propagation, leading to measurable shifts in shadow radius and polarization signatures compared to standard Reissner–Nordström black holes.
• The model exhibits varied horizon structures and thermodynamic phase transitions influenced by magnetic charge and mass, with implications observable via Event Horizon Telescope constraints.

A magnetically charged nonlinear-electrodynamics (NED) black hole is a static, spherically symmetric solution to Einstein's equations coupled to a nonlinear generalization of Maxwellian electrodynamics, sourced purely by magnetic charge. Such solutions, exemplified by the Bronnikov model, exhibit a globally regular geometry with the central singularity resolved by nonlinearities in the electromagnetic sector. The propagation of photons is governed by an effective optical metric distinct from the spacetime geometry, leading to observational signatures that differentiate these regular NED black holes from standard Reissner–Nordström solutions.

1. Nonlinear Electrodynamics: Action and Field Content

The total action is

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

where $R$ is the Ricci scalar and $\mathcal{L}(F)$ is the NED Lagrangian with $F = F_{\mu\nu}F^{\mu\nu}$. The canonical Bronnikov model adopts

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

where $q$ is the magnetic charge and $a$ is set by the requirement $M = |q|^{3/2}/(2a)$, with $M$ the ADM mass (Tlemissov et al., 11 Mar 2025, Kruglov, 2018).

Variation yields the Einstein equations and generalized Maxwell equations,

$$G_{\mu\nu}=8\pi T_{\mu\nu},\qquad \nabla_\mu(\mathcal{L}_F F^{\mu\nu})=0,$$

with

$R$0

A purely magnetic monopole field is employed: $R$1.

2. Spherically Symmetric Magnetically Charged Black Hole Solution

The metric ansatz is

$R$2

with the Bronnikov metric function in terms of the mass function $R$3,

$R$4

Properties:

• Asymptotically ($R$5): $R$6 (Reissner–Nordström-like).
• At the origin ($R$7): $R$8, $R$9: the geometry is regular, and curvature invariants are finite—there is no central singularity (Tlemissov et al., 11 Mar 2025, Kruglov, 2018).
• The total magnetic mass $\mathcal{L}(F)$0.

Horizon structure:

The number and type of horizons depends on the dimensionless parameter $\mathcal{L}(F)$1:

• For $\mathcal{L}(F)$2: two horizons.
• For $\mathcal{L}(F)$3: extremal (degenerate) horizon.
• For $\mathcal{L}(F)$4: horizonless, globally regular soliton (Kruglov, 2018).

3. Nonlinear Electrodynamics and the Effective Photon Geometry

Photon propagation in NED is not governed by the null cone of $\mathcal{L}(F)$5, but by an effective metric: $\mathcal{L}(F)$6 The effective line element is

$\mathcal{L}(F)$7

where $\mathcal{L}(F)$8. All light propagation, including lensing and shadow formation, is computed with this optical metric (Tlemissov et al., 11 Mar 2025, Allahyari et al., 2019).

Photon sphere and shadow:

The circular photon orbit satisfies

$\mathcal{L}(F)$9

yielding the photon sphere radius $F = F_{\mu\nu}F^{\mu\nu}$0. The critical impact parameter is

$F = F_{\mu\nu}F^{\mu\nu}$1

The shadow radius and photon sphere are significantly shifted for large $F = F_{\mu\nu}F^{\mu\nu}$2 by up to $F = F_{\mu\nu}F^{\mu\nu}$3 compared to Reissner–Nordström (Tlemissov et al., 11 Mar 2025, Allahyari et al., 2019).

4. Thermodynamics and Phase Structure

The Hawking temperature is

$F = F_{\mu\nu}F^{\mu\nu}$4

where $F = F_{\mu\nu}F^{\mu\nu}$5 is found from $F = F_{\mu\nu}F^{\mu\nu}$6. For the Bronnikov solution, in the parameterization $F = F_{\mu\nu}F^{\mu\nu}$7, $F = F_{\mu\nu}F^{\mu\nu}$8: $F = F_{\mu\nu}F^{\mu\nu}$9

Stability and phase transitions:

The heat capacity at fixed $$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$0 is

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

There is a divergence (second-order phase transition) at a critical $$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$2, beyond which the black hole is locally unstable ($$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$3); for $$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$4, stability holds ($$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$5) (Kruglov, 2018, Tlemissov et al., 11 Mar 2025). In the extremal limit ($$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$6), $$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$7.

5. Observational Signatures: Polarization, Shadows, and Lensing

Synchrotron emission polarization:

A thin, magnetized ring orbiting at radius $$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$8 produces polarized synchrotron radiation. The observed electric vector polarization angle (EVPA) is

$$\mathcal{L}(F) = F\,\cosh^{-2}\!\left(a\,|F/2|^{1/4}\right), \qquad F = \frac{2q^2}{r^4},$$9

with Stokes parameters $q$0 computed from projected electric field components. The normalized intensity difference with respect to RN is

$q$1

Differences in $q$2 and EVPA can reach tens of percent for near-extremal $q$3 and high inclination ($q$4) due to modifications in the photon sphere and lensing magnification (Tlemissov et al., 11 Mar 2025).

Shadow and lensing constraints:

• The shadow radius decreases as magnetic charge increases. For the regular Bronnikov black hole, Event Horizon Telescope observations of M87* set $q$5 (1σ) and $q$6 (2σ) (Allahyari et al., 2019).
• In the weak deflection regime, the bending angle is

$q$7

showing reduced lensing compared to RN (Fu et al., 2021).

6. Distinction from Reissner–Nordström Black Holes

In Maxwell electrodynamics ($q$8), photons propagate on the background spacetime, and the solution is the singular Reissner–Nordström black hole. In contrast, the NED case with the Bronnikov Lagrangian:

• Eliminates the curvature singularity at the core (for all $q$9).
• Permits higher extremal charges ($a$0 in some models).
• Alters the photon-sphere structure, shadow size, and polarization observables.
• Predicts phase transitions in specific heat and distinct stability boundaries.

Quantitative models demonstrate, for example, that at $a$1 the intensity peak of an accreting ring can exceed the RN value by up to a factor of $a$2, and EVPA swings by $a$3 (Tlemissov et al., 11 Mar 2025).

7. Summary Table: Key Features of the Bronnikov Magnetically Charged NED Black Hole

Property	Bronnikov Model [NED]	Reissner–Nordström [Maxwell]
Central singularity	Absent (regular core, $a$4)	Present ($a$5 curvature blow-up)
Photon propagation	Governed by effective optical metric $a$6	By background spacetime
Shadow radius	Decreases with $a$7, up to $a$8 shift	Standard GR formula
Maximal magnetic charge	$a$9 (model-dependent)	$M = |q|^{3/2}/(2a)$0
Heat capacity	Diverges (phase transition), stable/unstable regions	Usual RN instability/stability
Polarization features	Large $M = |q|^{3/2}/(2a)$1, EVPA swings at high $M = |q|^{3/2}/(2a)$2	Modest variation
Observational constraints	$M = |q|^{3/2}/(2a)$3 (M87*, 1σ)	N/A

Values and behavior are model-dependent; see (Tlemissov et al., 11 Mar 2025, Kruglov, 2018, Allahyari et al., 2019, Fu et al., 2021) for detailed expressions.

• (Tlemissov et al., 11 Mar 2025) – Effect of nonlinear electrodynamics on polarization distribution around black hole
• (Kruglov, 2018) – On a model of magnetically charged black hole with nonlinear electrodynamics
• (Allahyari et al., 2019) – Magnetically charged black holes from non-linear electrodynamics and the Event Horizon Telescope
• (Fu et al., 2021) – Weak deflection angle by electrically and magnetically charged black holes from nonlinear electrodynamics

These results establish magnetically charged NED black holes, particularly the Bronnikov solution, as theoretically distinct and potentially observationally distinguishable from Maxwellian charged solutions. Their phenomenology in polarimetric images and lensing, especially at high magnetic charge and large inclination, motivates further astrophysical and theoretical study.