Neutron Stars with Nonlinear Magnetic Monopoles

• Neutron stars with nonlinear magnetic monopoles are compact stellar objects whose structure is redefined by self-regularizing magnetic fields.
• Nonlinear electrodynamics modifies the TOV equations, revealing critical phenomena such as frozen state configurations and altered mass–radius relations.
• Observable signatures include shifts in ADM mass and distinctive metric profiles that signal extreme, horizonless compact stars.

Neutron stars with nonlinear magnetic monopoles are compact stellar objects whose structure, macroscopic properties, and observable signatures are fundamentally altered by the presence of nonlinear electromagnetic fields sourced by magnetic monopole charge. These systems are investigated within the Einstein–nonlinear electrodynamics (NED) framework, in which the electromagnetic sector departs from standard Maxwell theory to admit regularized, self-interacting field configurations. Critical phenomena such as the emergence of “frozen states” at threshold magnetic charge, modifications to the Tolman–Oppenheimer–Volkoff (TOV) equations, and consequences for the endpoint of gravitational collapse frame this research area at the intersection of relativistic astrophysics and nonlinear field theory (Tan et al., 11 Sep 2025).

1. Nonlinear Electromagnetic Framework and Magnetic Monopoles

Neutron stars with nonlinear magnetic monopoles are described by extending the total action to include a nonlinear electromagnetic Lagrangian, ℒ₍NED₎, typically formulated such that

$$S = \int \sqrt{-g} \, d^4x \left[ \frac{c^3}{16\pi G} R + \frac{1}{c} \mathcal{L}_m + \frac{1}{c} \mathcal{L}_{NED} \right].$$

Here, $\mathcal{L}_{NED}$ is a function of the electromagnetic field invariant $F = \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$. Specific models (e.g., Bardeen, Hayward) are employed to regularize the singular energy density associated with monopoles in linear Maxwell theory. Nonlinear Lagrangian choices typically depend on magnetic charge $q$, a coupling parameter $s$, and are selected to produce finite field invariants throughout the interior region. The matter sector (ordinary neutron star matter) is coupled to gravity and to the magnetic monopole field, providing additional contributions to both energy density and pressure.

2. Structure Equations: Modified Tolman–Oppenheimer–Volkoff Equations

Static, spherically symmetric neutron stars are modeled with metrics of the form

$$ds^2 = -e^{2\alpha(r)} c^2 dt^2 + e^{2\beta(r)} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2),$$

and perfect fluid energy–momentum tensors

$$T_{\mu\nu} = (\rho c^2 + p)U_{\mu}U_{\nu} + p g_{\mu\nu}.$$

Nonlinear magnetic monopoles contribute additional terms,

$T^{NED}_{\mu\nu},$

modifying the Einstein equations and leading to generalized TOV equations. The mass function satisfies

$m'(r) = 4\pi r^2 \rho(r),$

and combining energy–momentum conservation

$(\rho c^2 + p)\alpha'(r) + p'(r) = 0$

with the $G_{rr}$ equation yields the modified hydrostatic equilibrium equation, which contains functions of the monopole field components. In symbolic form,

$$r (\rho c^2 + p) \cdot \mathcal{D}[p(r), T^{NED}_{11}, \ldots] = 0,$$

where $\mathcal{D}$ is a derivative operator involving $e^{-2\beta(r)}$ and field-dependent quantities. As the magnetic charge $q\rightarrow 0$, the equations revert to the standard TOV system.

3. Formation and Characteristics of Frozen States

A defining phenomenon is the existence of “frozen states”, appearing when the magnetic charge $q$ reaches a critical value $q_c$. At this threshold:

• The metric component $-g_{tt} = e^{2\alpha(r)} \to 0$ across the stellar interior.
• The inverse radial metric $1/g_{rr} = e^{-2\beta(r)}$ develops a sharp minimum, essentially vanishing at the stellar surface $r=R$.

These features mimic an extremal horizon, similar to those of maximally charged black holes, with the notable distinction that no true event horizon is formed. The system is “frozen” in the sense that the exterior observer perceives the star as stationary at its gravitational radius—a phenomenon rooted in the nonlinear field dynamics rather than in conventional collapse. The value of $q_c$ is sensitive to the choice of NED model, equation of state (EoS), and central density, with softer EoS and higher core densities resulting in lower $q_c$.

4. Impact on Macroscopic and Observable Properties

Nonlinear magnetic monopoles modify several global properties of neutron stars:

• Gravitational Mass: In the frozen state regime, the ADM mass increasingly reflects contributions from the monopole field, with the baryonic mass constituting a diminishing fraction.
• Radius and Metric Structure: The mass–radius relation and interior metric profiles deviate from canonical neutron stars; the approach to $q=q_c$ is marked by near-vanishing $g_{tt}$ and $g_{rr}$ at the boundary.
• Equation of State: The pressure gradients governing hydrostatic balance are strongly altered by field contributions, affecting structural stability and compactness.
• Collapse Endpoint: Instead of forming a standard black hole with an event horizon, neutron stars may form horizonless extremely compact objects whose boundaries display metric “freezing”—an extension of formerly bosonic frozen state phenomenology to systems composed of baryonic matter.

5. Nonlinear Electrodynamics: Distinction from Linear Field Theory

Standard Maxwell theory cannot support regular solutions with magnetic monopoles due to energy density divergence. Nonlinear electrodynamics (with Bardeen or Hayward Lagrangians, for instance) enables the existence of regular monopole–sourced fields and self-consistently incorporates their gravitational effects. This approach not only ensures finite field invariants but also admits critical phenomena—such as the appearance of effective horizons in the metric—that are absent in linear theory. The significance lies in producing regular, horizonless compact objects whose gravitational and electromagnetic field structure is dictated by nonlinear source terms.

6. Extension to Ordinary Matter and Future Directions

Historically, frozen states were confined to configurations involving field-theoretical (bosonic) matter. The extension to neutron stars with ordinary baryonic matter plus nonlinear magnetic monopoles broadens the landscape of allowable compact objects. Observable consequences may include distinctive gravitational wave signatures and peculiar electromagnetic emission profiles resulting from the modified global structure and metric behavior around the critical horizon.

A plausible implication is that similar frozen-state configurations could arise in other compact astrophysical systems provided suitable nonlinear field contributions are present. This motivates further investigation into the end states of gravitational collapse and the taxonomy of horizonless compact stars, both from theoretical and observational vantage points.

7. Summary Table: Key Aspects of Frozen States in Nonlinear Magnetic Monopole Neutron Stars

Phenomenon	Mathematical Manifestation	Physical Consequence
Frozen state	$-g_{tt} \to 0$, $e^{-2\beta(r)} \to 0$ at $r=R$	Apparent freezing at gravitational radius
Critical charge	$q_c$ (model/EoS dependent)	Onset of frozen metric configuration
ADM mass shift	Increased monopole field contribution	Redefines stellar mass–radius relation
Nonlinear effect	$\mathcal{L}_{NED}$ regularizes monopole fields	Enables compact horizonless states

The presence of nonlinear magnetic monopoles in neutron stars fundamentally transforms compact stellar structure, mass–radius relations, and collapse outcomes, and provides pathways to horizonless, metric-frozen configurations in systems composed of both field-theoretic and ordinary matter. These insights extend the phenomenology of compact objects beyond conventional neutron stars and black holes (Tan et al., 11 Sep 2025).