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Horizonless star based on regular black hole with finite radius and its observational signatures

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Horizonless star emits echo trains and chaotic photon rings

desk verdict A coherent new gravastar template with useful photon-ring and echo predictions, but the imaginary sound speed in the crust and missing radial stability analysis undercut the claim that the object is viable. read the letter →

arxiv 2508.18072 v1 pith:YBJXBJO7 submitted 2025-08-25 gr-qc

M. F. Fauzi , B. N. Jayawiguna , H. S. Ramadhan , A. Sulaksono This is my paper · ORCID
classification gr-qc MSC 83C5783C3583C55 PACS 04.70.-s04.30.-w
keywords regularblackholeshorizonlessultracompactobjectanisotropicgravastarphotonringsgravitational-waveechoesRegge-Wheelerequationraytracingofstate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a regular black hole with a finite surface radius, completed with an anisotropic gravastar equation of state, is a viable horizonless ultracompact star with two distinctive observational signatures. The first is optical: for configurations compact enough to possess photon spheres, images of the object surrounded by a thin accretion disk contain chaotic minor photon rings between the first two major photon rings, a feature not seen in thin-shell gravastars. The second is gravitational: for the same compact configurations, the time-dependent Regge-Wheeler evolution produces trains of gravitational-wave echoes, while configurations without a photon sphere do not. These signatures matter because they give concrete ways to distinguish a horizonless star from a black hole in future high-resolution images and gravitational-wave ringdown observations.

What carries the argument

The central object is the finite-radius Hayward metric, whose energy density is the Hayward profile multiplied by a Tolman-like cutoff $1-(r/R)^n$, so the mass function is continuous up to the surface and matches Schwarzschild outside. The carrying mechanism is the equation-of-state ansatz $\bar p(y) = -\bar\epsilon(y)[1-F(y)\Theta(x-1)]$ with $F(y) = T(y)[1+a(\bar\epsilon/\bar\epsilon_0)^{\gamma-1}]$ and a tanh activation $T(y)$, which deforms the de Sitter core into a gravastar-like pressure profile while preserving regularity at the center and a smooth surface. The dimensionless ratio $x=\alpha/\alpha_c$ controls the size of the negative-pressure core; it also sets the threshold $x_m$ at which a marginally stable photon sphere appears, and hence which combination of the two observational signatures, chaotic photon rings or echo trains, is present.

What would settle it

Vary $\omega$, $\sigma_t$, $a$, and $\gamma$ within the conditions of Table II, or use a different smooth function $F(y)$ satisfying those conditions; if chaotic minor photon rings or gravitational echo trains disappear for some $x>x_m$, those signatures are artifacts of the specific ansatz rather than generic properties. Observationally, a next-generation very-long-baseline image of an ultracompact candidate that resolves the photon-ring region and shows no minor rings between the first two major rings would contradict the claimed image signature, as would a post-merger ringdown with no echo train for a configuration with $x>x_m$.

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Extended reading notes

Core claim

The paper's claim is that the horizonless branch of a modified Hayward regular black hole, once completed with a phenomenologically chosen anisotropic equation of state, forms a class of ultracompact star whose exterior is Schwarzschild, whose interior has a de Sitter core plus a positive-pressure crust and atmosphere, and whose observables differ from both black holes and thin-shell gravastars. Concretely, for $\bar R = 1.1$ and $1.5$ and the compactness parameter $x=\alpha/\alpha_c$, the photon-sphere threshold is $x_m \approx 0.654$–$0.656$; for $x > x_m$ the ray-traced images show chaotic minor photon rings between the first two major rings, and numerically evolved axial perturbations produce echo trains. At $x=x_m$ the potential well is too shallow to trap modes and no echoes appear. The paper also claims that approaching horizon formation forces a violation of the dominant energy condition in the transverse pressure, and that at the extremal configuration the time metric component freezes below $y_c$, mimicking a frozen star.

Load-bearing premise

The entire pressure profile, the dominant-energy-condition violation, the photon-sphere threshold $x_m$, and the echo behaviour follow from the hand-picked function $F(y)$; if a different allowed $F(y)$ changes or removes the signatures, the paper's predictions rest on that un-derived choice.

Editorial extensions

If this is right

  • For objects with $x > x_m$ and a transparent interior, the optical appearance is a set of photon rings rather than a shadow, with chaotic minor rings between the first and second major rings.
  • Gravitational-wave echoes appear only when the effective potential has a sufficiently deep well, namely $x > x_m$; at the marginally stable photon-sphere threshold no echo trains exist.
  • An anisotropic gravastar approaching horizon formation must violate the dominant energy condition, giving a concrete finite-radius realization of the earlier polarisation argument.
  • Reproducing the 72 Hz echo frequency reported from GW170817 requires an $\ell$ of roughly $10^5$ m and an object mass of about $115.7\,M_\odot$, a requirement the paper treats as disfavouring the model.
  • A limiting choice of the same parameters recovers the thin-shell gravastar model, and in that limit the transverse pressure violates the weak energy condition at the surface.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A parameter scan over the free constants in $F(y)$ would show whether the chaotic rings and echo trains are generic properties of this gravastar completion or specific to the tanh choice.
  • Applying the same cutoff-plus-ansatz construction to other regular black hole densities, such as a Bardeen-like profile, would test whether the threshold $x_m$ and its two signatures survive changes in the core profile.
  • Because the predicted echo time is set by the integral of $\sqrt{-g_{rr}/g_{tt}}$ from the center to the photon sphere, future broadband gravitational-wave searches are effectively measuring this time-delay integral and would fix the combination of $x$ and $\ell$.
  • The imaging prediction assumes light passes through the interior without interacting; if the positive-pressure atmosphere radiates, the central brightness pattern could change and the chaotic rings might be washed out.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper constructs a horizonless star by modifying the Hayward regular black hole with a finite-radius cutoff and by proposing a phenomenological anisotropic equation-of-state ansatz. It then solves the TOV equations to obtain pressure profiles, metric functions, and energy conditions, and uses the resulting spacetime to compute photon geodesics, ray-traced images with GLM1/GLM2 accretion disks, axial Regge-Wheeler potentials, quasinormal modes, and time-domain echo solutions. The central claims are that (i) configurations with photon spheres (x > x_m) produce chaotic minor photon rings between the first two major rings, and (ii) gravitational echo trains exist for x > x_m. A match to the 72 Hz GW170817 echo candidate is reported but requires ℓ ~ 1.5 × 10^5 m and a stellar mass of about 116 solar masses.

Significance. If the model is dynamically stable and the signatures are robust, this would be a useful new example of a horizonless compact object whose optical and gravitational-wave signatures differ from both black holes and thin-shell gravastars. The paper's forward calculations are explicit and benchmarked against the Schwarzschild photon-sphere results and against thin-shell gravastar images, and the statement that near-horizon configurations violate the dominant energy condition is a concrete, checkable result. The main reservations are that the equation of state is an ad hoc phenomenological ansatz, that the crust has imaginary sound speed, and that no radial stability analysis is given; these issues currently leave the 'viable' part of the central claim unproven.

major comments (3)
  1. [Sec. IV, Figs. 5-6 and text after Eq. (40)] The model has a crust region where d pbar/d ebar < 0, i.e. an imaginary adiabatic sound speed; this is visible in the bottom row of Fig. 6 and acknowledged in the discussion following Eq. (40). Because the paper's central claim is that this is a viable horizonless star, the absence of a radial (l=0) stability analysis is load-bearing: the axial Regge-Wheeler evolution in Sec. VI involves only toroidal fluid displacements and cannot detect radial collapse modes. I request either a linear radial-stability analysis (polar l=0, or at minimum a Chandrasekhar-style variational argument) or a substantial restriction of the viability claim, since an unstable configuration would not provide a physically realizable background for the ray-traced images and echo waveforms.
  2. [Sec. IV C, Eq. (47)] The entire construction rests on the phenomenological ansatz F(y) = T(y)[1 + a(epsilon/epsilon0)^(gamma-1)], with omega, sigma_t, a, and gamma selected by hand to enforce gravastar boundary conditions via Eqs. (49)-(52). The resulting predictions - the threshold x_m, the chaotic minor photon rings, the potential-well depth, and the echo threshold - are all contingent on this particular functional choice, and the paper gives no sensitivity analysis or microphysical derivation. The earlier model in Ref. [29] already shows that a different F(y) changes the signatures; I ask the authors to either scan the allowed parameter region and quantify how x_m, ring structure, and echo existence vary, or explicitly frame all predictions as properties of this particular ansatz rather than of the class of horizonless Hayward stars.
  3. [Sec. VI C, Eqs. (97)-(101)] The '72 Hz can be achieved' statement is presented as an observational match, but it is a parameter inversion: Eqs. (100)-(101) simply solve for ℓ such that f_echo = 1/(2 tau_echo) equals the GW170817 candidate value, and the resulting ℓ ~ 1.5 × 10^5 m and M ~ 116 M_sun are consequences of that choice. Moreover, tau_echo in Eq. (97) is defined as a null travel time from r=0 to r=3M, which is not obviously the inter-echo interval extracted from the time-domain solutions in Figs. 21-22. The identification should either be justified, or the text should present this as a calibration of ℓ rather than a prediction.
minor comments (5)
  1. [Sec. VI opening] The text twice refers to the 'Reggae-Wheeler' equation; this should be Regge-Wheeler.
  2. [Fig. 5 caption] The caption says 'with Rbar = 1.1 and Rbar = 1.1'; the second radius should presumably be Rbar = 1.5.
  3. [Table III] Table III is difficult to read: the n and omega columns are duplicated and some rows mix l=2 and l=3 values; please reformat into separate blocks with clear columns.
  4. [Sec. V B and Figs. 13-14] The paper repeatedly states that light does not interact with the interior, but the GLM2 disk extends to the center and rays are traced through the interior; please clarify whether interior absorption/emission is neglected and how this is consistent with the stated assumption.
  5. [Sec. VII, first paragraph] The discussion says the deviation is 'proportional to a tangential function'; the function used in Sec. IV is a hyperbolic tangent, not a tangent, so the wording should be corrected.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor fitted-input inversion: the 72 Hz echo 'match' is solved from the free scale ℓ; central photon-ring and echo-train predictions are forward computations, not circular.

  1. fitted input called prediction [Abstract; Sec. VI C, Eqs. (97)-(101)]
    "By comparing the echo time with the GW170817 observation, we find that a frequency of 72 Hz can be achieved, albeit at the cost of requiring a relatively high value of ℓ."

    Eq. (97) defines τecho = ∫ sqrt(-g_rr/g_tt) dr = ℓ τ̃, and the text sets f_echo = 1/(2τecho). Inserting the target '72 Hz' and solving for ℓ yields ℓ_{R̄=1.1}=1.51×10^5 m and ℓ_{R̄=1.5}=1.33×10^5 m, which is exactly the inversion f_echo = 1/(2ℓ τ̃). The frequency is therefore an input fitted to GW170817, not an output of the model; '72 Hz can be achieved' is equivalent to choosing ℓ = 1/(2×72 Hz×τ̃). The paper frames this transparently as requiring a large ℓ, so the circularity is mild and non-load-bearing for the central imaging and echo-train results.

full rationale

No significant circularity in the main derivation chain. The modified Hayward metric (Sec. III) is fixed by the chosen energy-density profile and the EoS ansatz F(y) (Eq. 47); the photon-sphere threshold x_m, the chaotic minor photon rings, and the echo trains are then computed forward from this metric via the effective potential (Eq. 60) and the time-dependent Regge-Wheeler equation (Eq. 84), with external checks against the Schwarzschild photon sphere and the thin-shell gravastar images of Ref. [35]. The straight-photon-path feature is re-derived analytically (Eqs. 62-64) rather than imported. The DEC-violation statement is broad (the paper itself notes F ∝ 1-2m/y would avoid it, Sec. IV A), but that is a model-dependence and correctness caveat, not an input-output inversion. Self-citations to [29] and [76] provide background and prior motivation; the claimed features are recomputed here, so they are not load-bearing. The only reduction-by-construction found is the 72 Hz echo comparison: f_echo = 1/(2ℓ τ̃) is inverted to solve for ℓ from the target frequency, so '72 Hz can be achieved' is a fitted constraint rather than a prediction; the paper presents it transparently as a requirement of large ℓ. Accordingly, the circularity score is 2.

Assumptions & free parameters 9 free parameters · 7 assumptions · 0 invented entities

The model rests on one new length scale ℓ and a set of EoS parameters (n, σs, ω, σt, a, γ) that are not fixed by first principles. The compactness x and surface radius R̄ are scanned as free inputs. The 72 Hz echo match is an inversion: ℓ is solved from the observed frequency. No new particles, fields, or geometric entities are postulated.

free parameters (9)
  • n = 3 (fixed)
    Exponent in the finite-radius cut-off (1-(r/R)^n); fixed to 3 for all numerical results in the paper; not derived.
  • σs = 1e-3 (fixed)
    Smoothness parameter of the Θ(x-1) cutoff in Eq. (31); set for simplicity in Sec. IV.C.1.
  • ω = 0.7 R̄ (chosen)
    Location of the pressure transition in F(y); chosen from Fig. 4 to balance regularity and surface conditions.
  • σt = 0.15 (chosen)
    Smoothness of the tanh transition in F(y); chosen from Fig. 4 while maintaining regularity at ~1e-4 order.
  • a = Determined by Eq. (52)
    Coefficient in F(y) set so that F'(R̄)|_{x→1}=0, effectively fixing the surface pressure profile; depends on R̄, n, σt.
  • γ = >1 (not specified)
    Polytropic-like exponent in F(y); required to be >1 so F(R̄)≈1; exact value not given.
  • x = varied {0.3, 0.5, 0.9, 1.0, 1.1}
    Dimensionless compactness α/αc; scanned to show horizonless (x<1) and horizonful (x>1) configurations.
  • R̄ = 1.1 and 1.5 (chosen)
    Surface radius in units of ℓ; chosen to be comparable to ℓ so the finite-radius modification is significant.
  • ℓ = 1.51e5 m (R̄=1.1) or 1.33e5 m (R̄=1.5) for 72 Hz
    New physics scale; set to ℓ=1 for imaging, and solved for the GW170817 echo frequency match in Sec. VI.C.
assumptions (7)
  • standard math General relativity with anisotropic perfect fluid source
    Einstein field equations and TOV equation used throughout Secs. II-IV.
  • domain assumption Hayward-type regular black hole energy density template
    Adopted from Ref. [9] as the starting point before adding the finite-radius cut-off (Sec. II.B).
  • ad hoc to paper The EoS ansatz p̄(y) = -ε̄(y)[1 - F(y)Θ(x-1)] with the tanh form of F(y)
    Central phenomenological assumption; parameters ω, σt, a, γ are chosen by hand, not derived from microphysics (Sec. IV.A and IV.C).
  • domain assumption Smooth matching to Schwarzschild exterior with Φ(R)=0
    Exterior spacetime is assumed exactly Schwarzschild; internal time metric is matched by integrating Φ backward with Φ(R)=0 (Sec. IV.B).
  • domain assumption Photons do not interact with the star's interior and the accretion disk is thin
    Explicitly assumed in Sec. V; stated as an idealized model.
  • standard math WKB/Bohr-Sommerfeld approximation for quasinormal modes
    Used in Sec. VI.B following Ref. [94] to estimate trapped-mode frequencies.
  • domain assumption Axial perturbations reduce to a single Regge-Wheeler master equation with potential Eq. (85)
    Chandrasekhar's formalism is applied to the anisotropic fluid interior; the potential uses local anisotropic pressure and density.

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Pith. "Pith review of Horizonless star based on regular black hole with finite radius and its observational signatures." pith.science (2026). https://pith.science/paper/YBJXBJO7

@misc{pith2026250818072,
  author       = {Pith},
  title        = {Pith review of: Horizonless star based on regular black hole with finite radius and its observational signatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBJXBJO7}},
  note         = {Machine review of arXiv:2508.18072}
}
abstract

The horizonless configuration of regular black holes has recently attracted attention as a model for ultracompact stars. In this paper, we propose a new class of regular black hole models sourced by a de Sitter vacuum with a finite radius. We focus on studying its horizonless configuration, which is modified into an anisotropic gravastar by proposing an ansatz of equation of states. We confirm that an anisotropic gravastar approaching horizon formation must violate the dominant energy condition. We also found that the proposed object has an effectively similar structure as a frozen star on the time geometry at the extremal configuration. From the proposed model, we investigate the photon geodesics inside the object and predict the optical appearance of the object surrounded by a thin accretion disk. Our imaging results indicate that, assuming light does not interact with the object's interior, its optical appearance differs from that of a thin-shell gravastar. ``Chaotic" photon ring merges for $x>x_{m}$, where $x_{m}$ represents the minimum value required for the photon sphere to exist. In addition to its optical appearance, we investigate the axial gravitational perturbations emitted by this horizonless star. Notably, echo trains are found to exist for $x>x_{m}$, as determined by numerically solving the time-dependent Regge-Wheeler equation. By comparing the echo time with the GW170817 observation, we find that a frequency of 72 Hz can be achieved, albeit at the cost of requiring a relatively high value of $\ell$.

Figures

Figures reproduced from arXiv: 2508.18072 by the authors.

Figure 1
Figure 1. FIG. 1. A close up image of our star model surrounded by [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Values of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Qualitative sketch of anisotropic gravastar’s pressure [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Regularity ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Radial (left) and transverse pressure (right) profile [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Energy conditions and speed of sound representation for several configurations of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (First and second row) Shift metric component [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Effective photon potential for several values of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Photon trajectory around the object for several values of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Number of photon orbits [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Two-dimensional illustration of axial and inclined [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Optical appearence of thin shell gravastar based on [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Optical appearence of our anisotropic gravastar model for axial observation with GLM1 (ISCO) accretion disk. The [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Optical appearence of our anisotropic gravastar model for axial observation with GLM2 (center) accretion disk. The [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Optical appearence of our anisotropic gravastar [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Axial GW effective potential profile of the [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Effective potential profile vs tortoise coordinate [PITH_FULL_IMAGE:figures/full_fig_p024_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20. QNM profile for our anisotropic gravastar model [PITH_FULL_IMAGE:figures/full_fig_p025_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Gravitational echo profiles of the anisotropic gravas [PITH_FULL_IMAGE:figures/full_fig_p026_21.png]

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    GLM1; characterized by ξ = −2, µ = RISCO , and σj =M/4, with RISCO is the radius of the innermost stable circular orbit (ISCO) for a mas- sive particle. This type of accretion disk assumes that the emission peaks and stops near the ISCO radius. This model is considered because any mas- sive particle will eventually fall towards the center if it passes wit...

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