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arxiv: 2605.29974 · v1 · pith:IQLM4XDQnew · submitted 2026-05-28 · 🌀 gr-qc · hep-th

Light rings and optical appearances of naked singularities, solitons, and black holes in beyond Horndeski gravity

Hyat Huang , Jutta Kunz , Rashmi Uniyal , Xiao Qian Wang This is my paper

Pith reviewed 2026-06-29 06:43 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords beyond Horndeski gravitynaked singularitieslight ringsblack hole shadowsthin disk imagesray tracingscalar hairoptical appearance
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The pith

In beyond Horndeski gravity, the number of horizons does not determine the optical image of compact objects.

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

The paper traces null geodesics and thin-disk images across a one-parameter family of static spacetimes that includes naked singularities, solitons, and black holes with varying horizon numbers. It shows that the image morphology is set by the shape of the photon potential and the location of the disk inner edge. Horizonless configurations can produce central depressions resembling shadows, while multi-horizon black holes can appear nearly identical to single-horizon ones when their outer light rings coincide. A sympathetic reader cares because this indicates that shadow observations alone may not distinguish these objects from standard black holes.

Core claim

For fixed theory parameter, varying the mass parameter produces a sequence of spacetimes from timelike naked singularities through regular solitons and black holes with one or more horizons to Schwarzschild-like black holes. Null geodesics reveal light rings whose critical impact parameters control the apparent size of central depressions. Thin-disk ray-tracing shows that horizonless objects can exhibit shadow-like features while multi-horizon black holes can closely match single-horizon images when exterior light-ring and disk structures are similar. The optical appearance is therefore governed mainly by the photon potential and disk inner edge, with deeper horizon structure leaving only an

What carries the argument

The photon potential together with thin-disk ray-tracing applied to the analytic family of metrics parameterized by theory and mass parameters.

If this is right

  • Images depend primarily on the critical impact parameter fixed by the outermost light ring.
  • Rescaled radial brightness profiles become similar across different horizon structures.
  • The morphological degeneracy holds when exterior light-ring and disk-edge properties coincide.
  • Deeper horizon features affect only higher-order image details.

Where Pith is reading between the lines

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

  • Shadow observations may fail to rule out horizonless objects in this class of theories.
  • Similar image degeneracies could appear in other scalar-tensor models with primary hair.
  • Thick-disk or polarized-light observations could break the degeneracy by probing closer to the would-be horizon.

Load-bearing premise

The given analytic solution accurately describes the spacetime metric of the theory and the thin-disk approximation captures the dominant light paths.

What would settle it

A ray-traced image of a horizonless object whose central depression size and brightness profile match those of a known single-horizon black hole only when their light-ring radii differ.

Figures

Figures reproduced from arXiv: 2605.29974 by Hyat Huang, Jutta Kunz, Rashmi Uniyal, Xiao Qian Wang.

Figure 1
Figure 1. Figure 1: The evolution of path A. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The evolution of path B. Path C: K > K2. For K > K2, the regular point lies inside the black-hole region, and therefore no regular soliton exists. For example, when K = 6 the solution starts as a timelike naked singularity at small µ. As the mass increases, two horizons form, corresponding to a RN-like black hole. At µ = √ 2 6 K, the singularity disappears and the solution becomes a regular black hole. For… view at source ↗
Figure 3
Figure 3. Figure 3: The evolution of path C. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The parameter space of the solution. 4 Geodesics and the effective potential The horizon and regularity classification discussed above provides only a first characterization of the different compact-object branches. To understand their optical appearance, one also has to analyze the geodesic structure outside the compact object. In particular, unstable light rings control the strong deflection of photons, … view at source ↗
Figure 5
Figure 5. Figure 5: The parameter space of light rings. The green dashed line represents K = 3.087, the blue dashed line represents K = 4.942, and the purple dashed line represents K = 5.511. For massless particles (ϵ = 0), the effective potential becomes Vgeo,0 = h(x) x 2 . (29) 10 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effective potentials for four different cases: K = 3 (for path A), K = 3.5 and K = 5 (for path B), and K = 6 (for path C). Insets highlight the fine structure near the potential maximum. For evolution along path C (i.e., K > K2), the solution always exhibits at least one light ring whether in the naked singularity phase or in the multi-horizon black hole phases. The distribution of light ring numbers in th… view at source ↗
Figure 7
Figure 7. Figure 7: Light ring radius xp as a function of the parameter µ for four representative cases with K = 3, 3.5, 5, 6. The curves illustrate how the number and location of light rings change with solution mass µ, corresponding to spacetimes with 0, 1, 2, or 3 horizons as labeled. 4.2 Massive particles, ISCO and static sphere For massive particles, we have ϵ = −1 and the effective potential becomes Vgeo,−1 = h(x) x 2 +… view at source ↗
Figure 8
Figure 8. Figure 8: The parameter space of ISCO and static sphere. The red dashed line represents K = 3.803, and the purple dashed line represents K = 5.511. Now we explain how to determine the boundaries. Firstly, an exterior static sphere requires h ′ (xs) = 0, h(xs) > 0. (34) The first condition gives rise to µ K = √ 2 6  1 + x 3 s (2 − x 2 s ) (1 + x 2 s ) 5/2  . (35) The stable and unstable static spheres are defined b… view at source ↗
Figure 9
Figure 9. Figure 9: Schematic picture of the ray-tracing setup. The compact object is surrounded by a thin accretion disk. Light rays emitted from the disk are deflected by the compact object and then reach the observer’s screen at infinity. We use the same ray-tracing method as in Refs. [24,57,58]. Since the spacetime and the disk configuration are both spherically symmetric and face-on, the final image is circularly 15 [PI… view at source ↗
Figure 10
Figure 10. Figure 10: Optical images along path A. From left to right, the panels give the total deflection angle ϕtot(b), the observed intensity Iobs(b) and the optical image. From top to bottom, the rows correspond to the naked singularity with µ = 0.2 and K = 2.5, the black hole with a static sphere xin = xs, µ = 0.6 and K = 2.5, and the black hole with an ISCO, µ = 2 and K = 2.5. 5.2 Optical images along path B For path B,… view at source ↗
Figure 11
Figure 11. Figure 11: Optical images for representative configurations in path B whose disks start from a static sphere. From left to right, the panels give the total deflection angle ϕtot(b), the observed intensity Iobs(b) and the optical image. From top to bottom, the rows correspond to the naked singularity with µ = 0.5 and K = 5, the black hole with a static sphere xin = xs and one light ring, µ = 1 and K = 4, and the blac… view at source ↗
Figure 12
Figure 12. Figure 12: Optical images for the black hole configurations with ISCO in path B. From left to right, the panels give the total deflection angle ϕtot(b), the observed intensity Iobs(b) and the optical image. From top to bottom, the rows correspond to the single-horizon black hole with an ISCO, µ = 1.3 and K = 4.5, and the three-horizon black hole with an ISCO, µ = 1.205 and K = 4.5. The three-horizon black hole has a… view at source ↗
Figure 13
Figure 13. Figure 13: Optical images along path C before and after the formation of horizons. From left to right, the panels give the total deflection angle ϕtot(b), the observed intensity Iobs(b) and the optical image. From top to bottom, the rows correspond to the naked singularity with µ = 1 and K = 6, and the two-horizon black hole with µ = 1.38 and K = 6. The naked singularity already exhibits strong bending, while the tw… view at source ↗
Figure 14
Figure 14. Figure 14: Optical images for the black-hole configurations with ISCO in path C. From left to right, the panels give the total deflection angle ϕtot(b), the observed intensity Iobs(b) and the optical image. From top to bottom, the rows correspond to the three-horizon black hole with an ISCO, µ = 1.5 and K = 6, and the single-horizon black hole with an ISCO, µ = 1.7 and K = 6. The images are mainly controlled by the … view at source ↗
read the original abstract

We investigate the geodesic structure and optical appearance of compact objects with primary scalar hair in shift- and parity-symmetric beyond Horndeski gravity. The analytic solution considered here depends on a theory parameter and a dimensionless mass parameter \cite{Bakopoulos:2023sdm}. For a fixed theory parameter, varying the mass traces a family of static spacetimes that can interpolate between timelike naked singularities, regular solitons, regular black holes, Reissner-Nordstr\"om-like black holes, multi-horizon black holes, and Schwarzschild-like black holes. We classify these branches by their horizon structure and analyze null and timelike geodesics, focusing on light rings, innermost stable circular orbits, and static spheres. We then compute thin-disk optical images by ray tracing. We find that the number of horizons is not directly encoded in the image: horizonless objects can show shadow-like central depressions, while multi-horizon black holes can closely resemble single-horizon black holes when their exterior light ring and disk structures are similar. Thus, the optical appearance is governed mainly by the photon potential and the disk inner edge, with the deeper horizon structure leaving only an indirect imprint. Quantitative radial-profile diagnostics confirm that the degeneracy is mainly morphological: the profiles differ at fixed impact parameter, but become much closer after rescaling by the critical impact parameter. These results provide a concrete example of how distinct compact object branches in beyond Horndeski gravity can share similar observational signatures.

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

0 major / 3 minor

Summary. The manuscript examines the geodesic structure and thin-disk optical appearances of a one-parameter family of static spacetimes in shift- and parity-symmetric beyond Horndeski gravity. For fixed theory parameter, varying the dimensionless mass traces branches from timelike naked singularities through regular solitons to single- and multi-horizon black holes. After classifying horizons and locating light rings, ISCOs and static spheres, the authors ray-trace thin accretion disks and conclude that horizon number is not directly encoded in the image; optical morphology is controlled by the photon potential and disk inner edge, producing degeneracies in which horizonless objects exhibit shadow-like depressions and multi-horizon black holes resemble single-horizon ones. Radial-profile comparisons before and after rescaling by critical impact parameter quantify the mainly morphological character of the degeneracy.

Significance. If the reported degeneracies survive more complete radiative-transfer calculations, the work supplies a concrete, analytic example in modified gravity of how distinct horizon structures can share similar strong-field signatures. The explicit branch classification, location of light rings and ISCOs, and quantitative before/after-rescaling profile diagnostics are clear strengths that advance the literature on observational degeneracies beyond GR.

minor comments (3)
  1. [optical-appearance section] The thin-disk ray-tracing approximation is standard but central to the degeneracy claim; a brief discussion of its domain of validity (e.g., possible impact of non-geodesic effects or thick-disk emission on the central depression) would strengthen the manuscript without altering the reported results.
  2. Figure captions and axis labels should explicitly identify the theory-parameter value and the branch (naked singularity, soliton, RN-like, multi-horizon, Schwarzschild-like) for each curve.
  3. [introduction or metric section] The abstract states that the solution depends on a theory parameter and a dimensionless mass parameter; the main text should give the explicit ranges explored for both parameters.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the manuscript's focus on geodesic structure, light rings, ISCOs, and thin-disk images across the family of solutions in shift- and parity-symmetric beyond Horndeski gravity, as well as the conclusion that optical morphology is controlled primarily by the photon potential and disk inner edge rather than horizon number.

Circularity Check

0 steps flagged

No circularity; results from explicit geodesic and ray-tracing computation on given metric family

full rationale

The paper obtains its central claims about horizon-independent optical appearances by classifying the analytic metric branches (from the cited solution), integrating the null geodesic equations to locate light rings, and performing thin-disk ray tracing to generate images and radial profiles. These steps are direct numerical/analytic evaluations of the geodesic structure and do not reduce to any fitted parameter, self-definition, or self-citation chain; the morphological degeneracy is an output of the integration, not an input. The cited metric source supplies the spacetime but carries no load-bearing uniqueness theorem or ansatz that the present work merely renames. No self-citation is invoked to justify the main result.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and validity of the analytic solution from the cited reference and on standard general-relativistic assumptions for null geodesics and thin-disk emission.

free parameters (2)
  • theory parameter
    The solution depends on a theory parameter whose value is held fixed while mass varies.
  • dimensionless mass parameter
    Varying this parameter traces the family of spacetimes from naked singularities to black holes.
axioms (2)
  • domain assumption The analytic solution from Bakopoulos:2023sdm correctly describes the metric in shift- and parity-symmetric beyond Horndeski gravity.
    Invoked in the abstract as the basis for the family of spacetimes.
  • domain assumption Standard null-geodesic equations and thin-disk ray tracing apply without additional radiative-transfer corrections.
    Implicit in the computation of light rings and optical images.

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discussion (0)

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Reference graph

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