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arxiv: 2507.09563 · v3 · pith:FN2O533Cnew · submitted 2025-07-13 · 🌀 gr-qc

Light Rings, Accretion Disks and Shadows of Hayward Boson Stars

Zhen-Hua Zhao , Yi-Ning Gu , Shu-Cong Liu , Zi-Qian Liu , Yong-Qiang Wang This is my paper

Pith reviewed 2026-05-21 23:36 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Hayward boson starslight ringsaccretion disksshadowsquasi-horizonphoton ringscomplex scalar fieldEinstein-Hayward gravity
0
0 comments X

The pith

Hayward boson stars in frozen states produce shadows like Schwarzschild black holes without extra photon rings

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper numerically builds solutions for boson stars in Einstein-Hayward gravity coupled to a complex scalar field without self-interaction. It shows that in the frozen state the quasi-horizon radius and light ring radii both grow as the magnetic monopole charge rises. Ray-tracing calculations of accretion disks reveal that non-frozen states generate multiple photon rings inside the shadow. Frozen states instead produce images that match those of Schwarzschild black holes and lack any additional rings. These distinctions matter because they offer a possible way to tell such exotic objects from ordinary black holes through future light observations.

Core claim

We construct a class of Hayward boson star solutions in Einstein-Hayward gravity coupled to a complex scalar field without self-interaction. In the frozen state, both the quasi-horizon radius and the light ring radii increase with the magnetic monopole charge. Using the ray-tracing method, non-frozen states show multiple photon rings within the shadow region of the accretion disks, while frozen states produce images resembling those of Schwarzschild black holes with no additional photon rings.

What carries the argument

The frozen versus non-frozen distinction in the boson star solutions, which controls whether the quasi-horizon produces standard black-hole-like shadows or allows extra photon rings to appear inside them.

If this is right

  • Larger magnetic monopole charge produces bigger quasi-horizons and light ring radii in frozen states.
  • Non-frozen states create multiple photon rings visible inside the accretion disk shadow.
  • Frozen states yield accretion disk images that match Schwarzschild black holes with only the usual light rings.
  • The presence or absence of extra photon rings depends directly on whether the solution is frozen or non-frozen.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-resolution shadow images lacking extra rings could point to frozen boson stars rather than true black holes.
  • The number of observed photon rings might serve as a test for these boson star models in future telescope data.
  • Applying the same ray-tracing approach to other modified-gravity scalar solutions could uncover additional shadow varieties.
  • Confirming long-term stability of the numerical solutions would strengthen their relevance for real astrophysical objects.

Load-bearing premise

The numerically obtained solutions remain stable and the chosen scalar field boundary conditions correctly represent realizable physical configurations.

What would settle it

Detection of multiple photon rings inside the shadow of a compact object whose other measured properties match a frozen Hayward boson star would show that the frozen-state claim does not hold.

Figures

Figures reproduced from arXiv: 2507.09563 by Shu-Cong Liu, Yi-Ning Gu, Yong-Qiang Wang, Zhen-hua Zhao, Zi-Qian Liu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Solutions for ˜q [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plots of metric components [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. ADM mass and critical horizon radius ˜r [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Effective potentials for photon circular orbits with (a) ˜q [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The existence region (shaded) of LR solutions on the [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The existence region (shaded) of LR solutions on the [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. ˜r [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. ˜r [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Light rings with the parameter ˜ω [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. ˜r [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. ˜r [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Schematic diagram of the coordinate system configuration. The X-Z plane is shaded [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Shadows and spectral distribution on the accretion disk with [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. (a) Photon trajectory cross-section (˜q [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Magnified view of the central region in Fig. 18(b), with (a) fov = 5 [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The incident angles of light rays 1, 2, and 3 are 0 [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Shadows and spectral distribution on the accretion disk with [PITH_FULL_IMAGE:figures/full_fig_p025_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Magnified view of Fig. 21c [PITH_FULL_IMAGE:figures/full_fig_p026_22.png] view at source ↗
read the original abstract

In this paper, we investigate the Einstein-Hayward gravity coupled to a complex scalar field without self-interaction. Using numerical methods, we construct a class of Hayward boson star solutions and examine their fundamental properties as well as the optical appearance of the accretion disk. Our results show that in the frozen state, both the quasi-horizon radius and the light ring radii increase with the magnetic monopole charge. Furthermore, using ray-tracing method, we find that for non-frozen states, the absence of an quasi-horizon results in the appearance of multiple photon rings within the shadow region of the accretion disks. In contrast, for frozen states, the presence of a quasi-horizon causes their images to resemble those of Schwarzschild black holes, with no additional photon rings appearing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper numerically constructs a class of Hayward boson star solutions in Einstein-Hayward gravity coupled to a complex scalar field without self-interaction. It examines their fundamental properties and uses ray-tracing to investigate the optical appearance of accretion disks. Key results include the increase of quasi-horizon radius and light ring radii with magnetic monopole charge in the frozen state, and the presence of multiple photon rings in non-frozen states versus Schwarzschild-like images without additional rings in frozen states.

Significance. If validated, these findings could help differentiate boson stars from black holes observationally through their accretion disk shadows and photon ring patterns in the context of modified gravity theories, potentially informing strong-field tests of general relativity.

major comments (2)
  1. [Numerical construction of solutions] The manuscript does not report any convergence tests, error bars, or sensitivity analyses for the numerical solutions under changes in discretization, grid resolution, or boundary conditions. This is critical as the central claims about the dependence on magnetic monopole charge and the distinct photon ring structures in frozen versus non-frozen states depend on the accuracy and robustness of these solutions.
  2. [Ray-tracing and optical appearance] Without explicit checks on the stability of the solutions under perturbations or verification that the light ring radii are insensitive to numerical parameters, the reported differences in photon rings (multiple in non-frozen, none additional in frozen) risk being numerical artifacts rather than physical features.
minor comments (2)
  1. Clarify the definitions of 'frozen' and 'non-frozen' states early in the paper, perhaps with a reference to the relevant section or equation.
  2. [Abstract] The abstract could specify the range of the magnetic monopole charge and scalar field parameters explored in the study.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address each major comment point by point below, providing clarifications and indicating the revisions we will make to improve the numerical validation of our results.

read point-by-point responses

  1. Referee: [Numerical construction of solutions] The manuscript does not report any convergence tests, error bars, or sensitivity analyses for the numerical solutions under changes in discretization, grid resolution, or boundary conditions. This is critical as the central claims about the dependence on magnetic monopole charge and the distinct photon ring structures in frozen versus non-frozen states depend on the accuracy and robustness of these solutions.

    Authors: We agree with the referee that explicit documentation of numerical convergence and error analysis is essential for validating the results. In our work, the solutions were obtained using a shooting method to solve the coupled ODE system, with the error in satisfying the Einstein equations monitored to be less than 10^{-7}. To fully address this comment, we will revise the manuscript to include a detailed description of the numerical procedure, along with convergence tests varying the grid resolution and step size, as well as sensitivity checks to the asymptotic boundary conditions. These additions will confirm that the increase in quasi-horizon and light ring radii with the magnetic monopole charge, and the differences between frozen and non-frozen states, are not affected by numerical parameters. revision: yes

  2. Referee: [Ray-tracing and optical appearance] Without explicit checks on the stability of the solutions under perturbations or verification that the light ring radii are insensitive to numerical parameters, the reported differences in photon rings (multiple in non-frozen, none additional in frozen) risk being numerical artifacts rather than physical features.

    Authors: We appreciate this concern regarding potential numerical artifacts in the ray-tracing results. The light ring positions are determined analytically from the metric functions by finding the extrema of the effective potential for equatorial null geodesics, and we have ensured consistency by using high-precision numerical integration. We will add verification that the light ring radii remain stable under small variations in the metric functions consistent with the numerical accuracy. However, performing a full stability analysis of the boson star solutions under perturbations would require solving the time-dependent perturbation equations or conducting numerical relativity simulations, which is not within the scope of the present paper. The observed distinction in photon ring structures is tied to the presence of the quasi-horizon in frozen states, which alters the light propagation in a manner analogous to black holes. revision: partial

standing simulated objections not resolved
  • Full stability analysis of the solutions under perturbations

Circularity Check

0 steps flagged

No circularity; results follow from numerical integration of field equations with input parameters

full rationale

The paper constructs Hayward boson star solutions by numerically solving the Einstein-Hayward equations coupled to a complex scalar field. The magnetic monopole charge is an explicit input parameter, the scalar field ansatz (harmonic time dependence with radial profile) is stated as a modeling choice, and metric functions are obtained via direct integration. Light-ring locations, quasi-horizon radii, and ray-traced accretion-disk images are then computed from these metric functions. No step renames a fitted quantity as a prediction, no self-citation supplies a uniqueness theorem that forces the result, and no output is defined in terms of itself. The derivation chain is therefore self-contained against the field equations and the chosen numerical scheme.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The construction relies on a numerical ansatz for the metric and scalar field, the Hayward regularization term, and the absence of self-interaction in the scalar potential; no independent evidence is given for the stability of the solutions.

free parameters (2)
  • magnetic monopole charge
    Input parameter that controls the size of the quasi-horizon and light rings; its value is chosen rather than derived.
  • scalar field amplitude and frequency
    Parameters of the complex scalar field ansatz that are tuned to obtain stationary solutions.
axioms (2)
  • domain assumption The Hayward term regularizes the spacetime without introducing new degrees of freedom beyond the metric and scalar field.
    Invoked to justify the modified Einstein equations used in the numerical integration.
  • domain assumption Stationary, spherically symmetric ansatz is sufficient to capture the essential optical properties.
    Standard assumption in boson-star literature but not proven for the Hayward case.

pith-pipeline@v0.9.0 · 5672 in / 1442 out tokens · 26566 ms · 2026-05-21T23:36:52.460606+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We employ numerical methods to investigate the fundamental properties of Hayward boson stars and the optical appearance of their accretion disks in asymptotically anti-de Sitter (AdS) spacetime... for non-frozen states, the absence of an quasi-horizon results in the appearance of multiple photon rings... for frozen states, the presence of a quasi-horizon causes their images to resemble those of Schwarzschild black holes

  • IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The conditions for stable circular motion of photons are: V(r̃)=0, Vpho_eff′(r̃)=0 and Vpho_eff′′(r̃)>0

What do these tags mean?
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. $\ell$-Boson stars in anti-de Sitter spacetime

    gr-qc 2025-12 unverdicted novelty 5.0

    ℓ-boson stars are constructed and their properties studied in asymptotically anti-de Sitter spacetime.

  2. Observational appearance and photon rings of non-singular black holes from anisotropic fluids

    gr-qc 2025-11 unverdicted novelty 5.0

    Simulations of photon rings around non-singular black holes from anisotropic fluids show they are difficult to distinguish from Schwarzschild black holes due to entangled theoretical, numerical, and observational unce...

Reference graph

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