---
title: Magnetically Charged NED Black Hole
url: https://www.emergentmind.com/topics/magnetically-charged-nonlinear-electrodynamics-black-hole
type: topic
---

# Magnetically Charged NED Black Hole

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 [2503.08294], [1805.07595].

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
\[
\mathcal{L}_F = \frac{d\mathcal{L}}{dF},\qquad T_{\mu\nu}=2\left( \mathcal{L}_F F_{\mu\alpha}F_\nu{}^\alpha -\tfrac{1}{4}g_{\mu\nu}\mathcal{L} \right).
\]
A purely magnetic monopole field is employed: \(F_{\theta\phi} = q\sin\theta\).

## 2. Spherically Symmetric Magnetically Charged Black Hole Solution

The metric ansatz is
\[
ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega^2,
\]
with the Bronnikov metric function in terms of the mass function \(m(r)\),
\[
f(r) = 1 - \frac{2\,m(r)}{r}, \qquad m(r) = M\Bigl[1-\tanh\Bigl(\frac{q^2}{2Mr}\Bigr)\Bigr].
\]

**Properties:**
- Asymptotically (\(r\gg q\)): \(f(r)\approx 1 - 2M/r + q^2/r^2 + \mathcal{O}(r^{-4})\) (Reissner–Nordström-like).
- At the origin (\(r\rightarrow 0\)): \(m(0)\to0\), \(f(0)\to1\): the geometry is regular, and curvature invariants are finite—there is no central singularity [2503.08294], [1805.07595].
- The total magnetic mass \(m_M = |q|^{3/2}/(2a)\).

**Horizon structure:**
The number and type of horizons depends on the dimensionless parameter \(b = \sqrt{\beta}/(\sqrt{2}Gq)\):
- For \(b < b_{\rm crit} \approx 0.27846\): two horizons.
- For \(b = b_{\rm crit}\): extremal (degenerate) horizon.
- For \(b > b_{\rm crit}\): horizonless, globally regular soliton [1805.07595].

## 3. Nonlinear Electrodynamics and the Effective Photon Geometry

Photon propagation in NED is not governed by the null cone of \(g_{\mu\nu}\), but by an effective metric:
\[
g_{\rm eff}^{\mu\nu} = \mathcal{L}_F g^{\mu\nu} - 4\mathcal{L}_{FF}F^\mu{}_\alpha F^{\alpha\nu},\qquad \mathcal{L}_{FF} \equiv \frac{d^2\mathcal{L}}{dF^2}.
\]
The effective line element is
\[
ds_{\rm eff}^2 = -\frac{f(r)}{\mathcal{L}_F}dt^2 + \frac{dr^2}{f(r)\mathcal{L}_F} + \frac{r^2}{\Phi}d\Omega^2,
\]
where \(\Phi = \mathcal{L}_F + 2F\mathcal{L}_{FF}\). All light propagation, including lensing and shadow formation, is computed with this optical metric [2503.08294], [1912.08231].

**Photon sphere and shadow:**
The circular photon orbit satisfies
\[
\frac{d}{dr}\Bigl[\ln\Bigl(\frac{r^2}{\Phi}\Bigr)\Bigr] = 2 \frac{f'}{f},
\]
yielding the photon sphere radius \(r_{\rm ph}\). The critical impact parameter is
\[
b_c = \frac{r_{\rm ph}}{\sqrt{f(r_{\rm ph})\,\Phi(r_{\rm ph})/\mathcal{L}_F(r_{\rm ph})}}.
\]
The shadow radius and photon sphere are significantly shifted for large \(q/M\) by up to \(\sim 10\%\) compared to Reissner–Nordström [2503.08294], [1912.08231].

## 4. Thermodynamics and Phase Structure

The Hawking temperature is
\[
T_H = \frac{f'(r_h)}{4\pi},
\]
where \(r_h\) is found from \(f(r_h) = 0\). For the Bronnikov solution, in the parameterization \(x = 2^{1/4}r/(\beta^{1/4}\sqrt{q})\), \(b=\sqrt{\beta}/(\sqrt{2}Gq)\):
\[
T_H = \frac{1 + b}{2^{7/4}\pi\beta^{1/4}\sqrt{q}\,x_h}\left(1-\frac{2}{(1+b)x_h}\right).
\]

**Stability and phase transitions:**
The heat capacity at fixed \(q\) is
\[
C_q = \frac{2\pi\,r_h\,T_H}{G\,\partial T_H/\partial r_h}.
\]
There is a divergence (second-order phase transition) at a critical \(x_h^*\), beyond which the black hole is locally unstable (\(C_q<0\)); for \(x_h<x_h^*\), stability holds (\(C_q>0\)) [1805.07595], [2503.08294]. In the extremal limit (\(b \to b_{\rm crit}\)), \(T_H \to 0\).

## 5. Observational Signatures: Polarization, Shadows, and Lensing

**Synchrotron emission polarization:**
A thin, magnetized ring orbiting at radius \(r_s\) produces polarized synchrotron radiation. The observed electric vector polarization angle (EVPA) is
\[
\mathrm{EVPA} = \frac{1}{2}\arctan\left(\frac{U}{Q}\right),
\]
with Stokes parameters \(Q, U\) computed from projected electric field components. The normalized intensity difference with respect to RN is
\[
\Delta I/I = \frac{I_{\rm NED} - I_{\rm RN}}{I_{\rm RN}}.
\]
Differences in \(\Delta I/I\) and EVPA can reach tens of percent for near-extremal \(q/M\) and high inclination (\(\theta_0 \gtrsim 60^\circ\)) due to modifications in the photon sphere and lensing magnification [2503.08294].

**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 \(Q_m \lesssim 0.7M\) (1σ) and \(Q_m \lesssim 1.0M\) (2σ) [1912.08231].
- In the weak deflection regime, the bending angle is
  \[
  \hat{\alpha} = \frac{4M}{b} - \frac{3\pi Q_m^2}{4b^2} + \frac{16MQ_m^3}{3b^3} + \mathcal{O}(b^{-4}),
  \]
  showing reduced lensing compared to RN [2101.08409].

## 6. Distinction from Reissner–Nordström Black Holes

In Maxwell electrodynamics (\(\mathcal{L}(F) = F\)), 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 \(q\)).
- Permits higher extremal charges (\(q_{\text{ext}} > M\) 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 \(q/M \to 0.99\) the intensity peak of an accreting ring can exceed the RN value by up to a factor of \(\sim 4\), and EVPA swings by \(\sim 20^\circ\) [2503.08294].

## 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, \(f(0)=1\))            | Present (\(r=0\) curvature blow-up) |
| Photon propagation      | Governed by effective optical metric \(g_{\rm eff}\) | By background spacetime      |
| Shadow radius           | Decreases with \(q\), up to \(10\%\) shift   | Standard GR formula           |
| Maximal magnetic charge | \(q_{\rm ext} \gtrsim M\) (model-dependent)  | \(q_{\rm ext} = M\)           |
| Heat capacity           | Diverges (phase transition), stable/unstable regions | Usual RN instability/stability |
| Polarization features   | Large \(\Delta I/I\), EVPA swings at high \(q/M\) | Modest variation              |
| Observational constraints| \(Q_m \lesssim 0.7M\) (M87*, 1σ)             | N/A                            |

Values and behavior are model-dependent; see [2503.08294], [1805.07595], [1912.08231], [2101.08409] for detailed expressions.

## References

- [2503.08294] – Effect of nonlinear electrodynamics on polarization distribution around black hole
- [1805.07595] – On a model of magnetically charged black hole with nonlinear electrodynamics
- [1912.08231] – Magnetically charged black holes from non-linear electrodynamics and the Event Horizon Telescope
- [2101.08409] – 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.

Source: https://www.emergentmind.com/topics/magnetically-charged-nonlinear-electrodynamics-black-hole