Frozen solitonic Hayward-boson stars in Anti-de Sitter Spacetime

We construct solitonic Hayward-boson stars (SHBSs) in Anti-de Sitter (AdS) spacetime, which consists of the Einstein-Hayward model and a complex scalar field with a soliton potential. Our results reveal a critical magnetic charge $q_c$. For $q\geq q_c$ in the limit of $ω\rightarrow 0$, the matter field is primarily distributed within the critical radius $r_c$, beyond which it decays rapidly, while the metric components $-g_{tt}$ and $1/g_{rr}$ become very small at $r_c$. These solutions are termed ``frozen solitonic Hayward-boson stars" (FSHBSs). Continuously decreasing $Λ$ disrupts the frozen state. However, we did not find a frozen solution when $q<q_c$. The value of $q_c$ depends both on the cosmological constant $Λ$ and the self-interaction coupling $η$. We also found that for high frequency solutions, increasing $η$ can yield a pure Hayward solution. However, for low frequency solutions, increasing $η$ reduces both $1/g_{rr}$ and $-g_{tt}$. Furthermore, we analyzed the effective potential of SHBSs and identified an extra pair of light rings in the second solution branch.

💡 Research Summary

This paper presents a comprehensive study on the construction and properties of “Solitonic Hayward-Boson Stars (SHBSs)” within Anti-de Sitter (AdS) spacetime. The model synthesizes the Einstein-Hayward regular black hole framework—sourced by a nonlinear electromagnetic field with magnetic charge q—with a complex scalar field endowed with a solitonic potential, characterized by a self-interaction coupling constant η. The primary goal is to explore static, spherically symmetric, horizonless solutions of this coupled system and uncover novel gravitational phenomena.

The core finding is the existence of a critical magnetic charge, q_c. When q ≥ q_c and the scalar field’s oscillation frequency ω approaches zero, a distinctive “frozen” state emerges. In this state, the scalar field profile ϕ(r) becomes highly concentrated within a critical radius r_c, decaying rapidly beyond it. Concurrently, the metric components -g_tt and 1/g_rr become exceedingly small at r_c, mimicking the appearance of a horizon to a distant observer. These configurations are termed “Frozen Solitonic Hayward-Boson Stars (FSHBSs).” The study reveals that this frozen state is maintained by the negative cosmological constant Λ; continuously decreasing Λ disrupts the freezing. Notably, no such frozen solutions are found for q < q_c. The value of the critical charge q_c is not universal but depends jointly on both Λ and η.

The influence of the self-interaction parameter η is bifurcated based on the frequency domain. For high-frequency solutions (larger ω), increasing η suppresses the contribution of the scalar field, causing the SHBS solutions to converge towards the pure Hayward regular black hole vacuum solution. In contrast, for low-frequency solutions (near the ω→0 limit), increasing η leads to a further reduction in both -g_tt and 1/g_rr, making the metric even “shallower” at the core.

For the case of q < q_c, where the frequency has a finite minimum, the paper details the ADM mass (M) and Noether charge (Q) as functions of ω for various parameters. In AdS spacetime (Λ < 0), the maximum frequency ω_max can exceed the scalar field mass (μ=1), a feature attributed to the cosmological constant’s influence. The M-ω and Q-ω curves can develop two local maxima for larger η² in asymptotically flat spacetime (Λ=0), a feature that is absent or altered in AdS.

A significant part of the analysis is devoted to the null geodesics and light rings of SHBSs. By studying the effective potential for photon orbits, the authors discover a rich structure of light rings whose number depends on the solution branch and parameters. For the first branch of solutions (starting from ω_max), the number of light rings can increase from zero to two (one stable inner and one unstable outer) as ω decreases. More remarkably, in the second branch of solutions (where ω increases from its minimum), the behavior is more complex: depending on q, Λ, and η, the number of light rings can decrease from two to zero and then increase back to two, or even yield up to four coexisting light rings within a specific frequency range for sufficiently large q. This indicates an exceptionally complex spacetime geometry influencing gravitational lensing signatures.

In summary, this work successfully constructs a new family of horizonless compact objects in AdS spacetime by merging regular black hole and boson star concepts. It uncovers a novel “frozen” phase controlled by a critical magnetic charge, demonstrates intricate parameter dependencies for critical phenomena, and reveals a potentially observable multi-light-ring structure. These findings enrich the theoretical landscape of ultra-compact objects and may provide new insights within the AdS/CFT correspondence, linking gravitational phase transitions to boundary field theory phases.