Scalarized Bardeen Spacetime

• Scalarized Bardeen Spacetime (SBS) is a class of solutions to Einstein's equations where a nonminimally coupled scalar field augments a regular, singularity-free Bardeen black hole geometry.
• It exhibits spontaneous and frozen scalarization regimes, determined by critical coupling thresholds and magnetic charge, which alter horizon structure and geodesic dynamics.
• The modified photon effective potential and altered thermodynamic phase transitions in SBS offer observable signatures in gravitational lensing, black hole shadows, and stability analyses.

Scalarized Bardeen Spacetime (SBS) denotes a class of solutions to the Einstein equations in which the regular (singularity-free) Bardeen black hole spacetime is supplemented by a nonminimally coupled scalar field, leading to the emergence of “scalar hair.” The scalarization process is governed by the interplay of nonlinear electrodynamics—specifically the Bardeen magnetic monopole—and scalar-curvature or scalar-matter couplings. SBS encompasses both the spontaneous onset of scalar fields (driven by tachyonic instabilities above a critical coupling parameter) and the rich phenomenology associated with horizonless solutions, frozen configurations, and modified photon trajectories.

1. Mechanism and Thresholds of Scalarization

Scalarization in the Bardeen spacetime proceeds via a nonminimal coupling between a scalar field and the electromagnetic Lagrangian density underlying the Bardeen model. A prototypical coupling adopts an exponential form: $f(\Phi) = e^{-\alpha \Phi^2}$ where $\Phi$ is the scalar field and $\alpha$ (often denoted $a$ in the SBS literature) is the coupling parameter. For small $\alpha$, the background remains scalar-free; but when $\alpha$ exceeds a threshold $\alpha_t$, a tachyonic instability in the scalar sector is triggered. The scalar field equation of motion admits a schematic form: $$\nabla^2 \Phi = \frac{1}{4} \frac{\partial f(\Phi)}{\partial \Phi} \mathcal{L}(\mathcal{F})$$ where $\mathcal{L}(\mathcal{F})$ is the nonlinear electromagnetic Lagrangian function. The threshold $\alpha_t$ sets the boundary for scalarization: for $\alpha > \alpha_t$, nontrivial scalar field solutions emerge; for $\alpha \leq \alpha_t$, the scalar field vanishes identically and the SBS reduces to the pure Bardeen spacetime (Huang et al., 10 Sep 2025).

The magnetic charge $q$ from the Bardeen sector plays a pivotal role: below a model-dependent critical charge $q_c$, scalarization is “spontaneous”—the scalar field interpolates smoothly to zero as $\alpha \to \alpha_t^+$; for $q \geq q_c$, the scalar remains nonvanishing and localizes around a critical radius even as $\alpha \to \alpha_t^+$ (“frozen” regime) (Huang et al., 10 Sep 2025).

2. Spontaneous vs. Frozen Scalarization Regimes

The SBS exhibits two qualitatively distinct scalarization patterns:

• Spontaneous Scalarization ($q < q_c$):
  • Scalar hair arises continuously above $\alpha_t$ and vanishes as $\alpha \to \alpha_t^+$.
  • The geometry transitions smoothly into the original Bardeen solution; the scalar charge $Q_s$ vanishes at threshold.
  • Dynamics are governed by traditional bifurcation arguments, similar to those found in spontaneous scalarization of neutron stars or Gauss–Bonnet black holes.
  • The solution is regular everywhere, and the external spacetime is indistinguishable from the original Bardeen model in the $\alpha \to \alpha_t^+$ limit (Huang et al., 10 Sep 2025, Zhang et al., 18 Sep 2024).
• Frozen Scalarization ($q \geq q_c$):
  • As $\alpha \to \alpha_t^+$, the scalar field persists and aggregates near a “critical horizon” $r_{cH}$, marked by a near-zero $-g_{tt}$ and vanishing $g^{rr}$ at $r_{cH}$.
  • The region $r < r_{cH}$ becomes dynamically “frozen,” with physical processes effectively halted due to extreme redshifting.
  • The geometry develops a quasi-horizon structure without an actual event horizon, reminiscent of “frozen star” or “critical surface” solutions found in SBS with charged scalar or Proca fields (Huang et al., 10 Sep 2025, Huang et al., 16 Jul 2024, Zhang et al., 20 Mar 2025).
  • Although the scalar charge measured at infinity vanishes, the scalar field is nontrivial in the interior, and the metric function $-g_{tt}$ can reach values as low as $10^{-16}$ (Huang et al., 10 Sep 2025).

3. Geometrical and Orbital Structure

The SBS spacetime maintains the Bardeen’s lack of curvature singularity, but the presence of a scalar field modifies the effective potential for both massive and massless (null) test particles. The general metric form is: $$ds^2 = -n(r)\sigma^2(r) dt^2 + \frac{dr^2}{n(r)} + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2)$$ where the metric functions $n(r)$ and $\sigma(r)$ encode scalar field backreaction.

Photon and massive particle geodesics are governed by modified radial equations: $$\dot{r}^2 + V_\text{eff}(r) = 0,\quad V_\text{eff}(r) = \frac{E^2 + g_{tt}(L^2/r^2)}{g_{tt} g_{rr}}$$ with energy $E$ and angular momentum $L$. The existence and location of photon spheres (light rings) and circular timelike orbits depend sensitively on both the magnetic charge and the scalar field profile.

Key features include:

• Appearance of paired light rings straddling the critical horizon for frozen SBS (one inner, stable and one outer, unstable), with their separation growing as the solution approaches the frozen regime (Zhang et al., 20 Mar 2025).
• Orbits with “many-world” and “two-world” connectivity in the geodesic structure for massive and massless particles, as inherited from the Bardeen background (Zhou et al., 2011).
• Direction reversal and velocity disparity in the precession of inner versus outer bounded orbits, a property expected to be modulated by scalarization (Zhou et al., 2011).

These modifications are of observational significance, impacting gravitational lensing, shadow size, and potentially developing signatures in high-precision pulsar or black hole X-ray timing analyses.

4. Thermodynamics, Stability, and Phase Structure

The scalarization process is closely tied to thermodynamic instabilities of the regular Bardeen black hole. In the absence of the scalar, Bardeen spacetimes support second-order phase transitions marked by divergences in the specific heat. The critical parameters for stability (such as magnetic charge) shift the locus of phase transitions in the $(S,T)$ or $(M,S)$ planes (Saleh et al., 2017, Wu et al., 29 Jul 2024, Rizwan et al., 2020).

Scalarized Bardeen spacetimes inherit and extend this phenomenology:

• The presence of additional scalar degrees of freedom can change the pattern and location of phase boundaries, potentially leading to new first- or second-order transitions, including Hawking–Page–like transitions between vacuum and black hole phases in AdS backgrounds (Wu et al., 29 Jul 2024).
• Stability of the fundamental ($n=0$) scalarized branch against radial perturbations is explicitly demonstrated for several couplings (quadratic or exponential), indicating astrophysical viability (Zhang et al., 18 Sep 2024).
• Microstructure analyses (Ruppeiner geometry) reveal that the underlying interactions in the scalarized regime can acquire repulsive character in certain phases (e.g., small black holes), providing a thermodynamic probe of scalarization (Rizwan et al., 2020).

5. Solitonic, Horizonless, and "Frozen" Solutions

Scalarization can also suppress horizon formation altogether, resulting in globally regular, horizonless compact objects that serve as black hole mimickers:

• Frozen Bardeen-Boson/Proca/Dirac Stars: When nonlinear matter fields (complex scalars, Proca fields, or Dirac fields—charged or neutral) are included, SBS solutions without a horizon but with frozen interiors arise as $\omega \rightarrow 0$ or effective frequencies vanish (Huang et al., 16 Jul 2024, Sun et al., 20 Nov 2024, Zhang et al., 20 Mar 2025, Zhang et al., 22 Sep 2024).
• The metric function $-g_{tt}$ becomes extremely small at a “critical horizon” $r_{cH}$, and matter fields (scalar, Dirac, or Proca) are tightly localized within; the ADM mass, Noether charge, and energy density distributions are modified compared to standard black holes.
• The transition between solitonic and standard SBS solutions can be parameterized via the scalar (or matter field) charge and coupling strength, as well as the magnetic charge of the underlying Bardeen background.
• These objects are distinguished by unique photon orbit structures, such as paired or smeared light rings and, for AdS spacetimes, modified maximum frequencies and mass versus frequency curves (Zhang et al., 22 Sep 2024).

6. Observational and Theoretical Implications

SBS models provide a spectrum of potential observational signatures:

• The elaborated photon effective potentials and location of light rings directly influence the black hole shadow and gravitational lensing, especially in the frozen regime where time dilation effects near $r_cH$ become extreme (Huang et al., 10 Sep 2025).
• Gravitational wave signals in mergers involving SBSs are expected to deviate from the standard Kerr paradigm, with differences manifest in the quasinormal mode spectrum—most notably as an “outburst” of overtones sensitive to scalar hair and quantum corrections (Konoplya et al., 2023, Ovchinnikov, 2023).
• The evaporation and dynamical evolution of regular black holes, including SBS, are influenced by energy extraction mechanisms (e.g., charged Penrose process), which can depopulate the magnetic charge required for regularity and potentially induce or end a scalarization phase (Vertogradov et al., 20 Aug 2025).
• Scalarized Bardeen stars and compact objects offer an alternative to neutron stars and traditional boson stars, with enhanced masses and modified M–R diagrams, directly impacting astrophysical mass limits and the modeling of pulsar-like objects (Shamir, 2020).

The theoretical landscape of regular black holes and scalarization is significantly enriched by SBS: the fine structure of scalar, electromagnetic, and matter field couplings to Bardeen’s nonlinear core underpins a diversity of stable, horizonless, and potentially observable gravitational configurations.

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Table: Scalarization Regimes Versus Magnetic Charge

Magnetic Charge $q$	Scalarization Behavior	Scalar Field Profile as $a\to a_t^+$
$q < q_c$	Spontaneous	$\Phi\to 0$ everywhere (vanishes smoothly)
$q \geq q_c$	Frozen	$\Phi$ nonzero near $r_{cH}$ (“frozen” zone)

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7. Mathematical Summary

Let $f(\Phi)$ be the scalar coupling (e.g., $e^{-\alpha\Phi^2}$); $\mathcal{L}(\mathcal{F})$ the Bardeen electromagnetic Lagrangian; $n(r),\ \sigma(r)$ metric functions; $\omega$ the matter field frequency.

• Scalarization Onset:

$$\nabla^2 \Phi = \frac{1}{4} \frac{\partial f(\Phi)}{\partial \Phi} \mathcal{L}(\mathcal{F}) \qquad a \geq a_t \implies \Phi\neq 0$$

• Metric:

$$ds^2 = -n(r)\sigma^2(r) dt^2 + \frac{dr^2}{n(r)} + r^2(d\theta^2 + \sin^2\theta\, d\phi^2)$$

• Photon Effective Potential:

$$V_\text{eff}(r) = \frac{E^2 + g_{tt}(L^2/r^2)}{g_{tt}g_{rr}}$$

• Light Ring Condition:

$$\frac{dV_\text{eff}}{dr} = 0\ ,\quad V_\text{eff} = 1/b^2$$

These equations, together with parameters $(q, a, g, \omega)$ and matter field type, determine the scalarized Bardeen spacetime’s structure and phenomenology.

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• Scalarization mechanism and threshold analysis: (Huang et al., 10 Sep 2025, Zhang et al., 18 Sep 2024).
• Geodesics, precession, and potential diagnostics for scalarization: (Zhou et al., 2011, Li et al., 2022).
• Thermodynamic structure, stability, microstructure: (Saleh et al., 2017, Wu et al., 29 Jul 2024, Rizwan et al., 2020, Amir et al., 2020).
• Horizonless and frozen solutions: (Huang et al., 16 Jul 2024, Sun et al., 20 Nov 2024, Zhang et al., 20 Mar 2025, Zhang et al., 22 Sep 2024).
• Quasinormal modes and observational signatures: (Konoplya et al., 2023, Ovchinnikov, 2023).
• Charge evolution and energy extraction: (Vertogradov et al., 20 Aug 2025).
• Compact stars and astrophysical objects: (Shamir, 2020).

Scalarized Bardeen Spacetime, through its capacity for both regular and horizonless scalarized solutions, offers a prominent framework in the paper of modified gravity, black hole mimicry, and the interplay between nonlinear electrodynamics and scalar-tensor interactions.