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REVIEW 2 major objections 5 minor 75 references

Shadows of generalised Hayward spacetimes : in vacuum and with plasma

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The same two-parameter generalized Hayward metric that yields black holes and wormholes predicts that multi-peak wormhole mimickers fit the observed Sgr A* shadow better than single-peak ones, while regular Hayward black holes survive…

desk verdict Useful unified shadow catalog for the generalized Hayward family, but the EHT-based preference for multi-peak wormholes rests on an unjustified choice of shadow boundary. read the letter →

arxiv 2411.11970 v2 pith:3NP7TMHQ submitted 2024-11-18 gr-qc astro-ph.GAastro-ph.HE

Suvikranth Gera , Saurabh Kumar , Poulami Dutta Roy , Sayan Chakrabarti This is my paper · ORCID
classification gr-qcastro-ph.GAastro-ph.HE PACS 04.70.-s04.20.-q
keywords blackholeshadowsgeneralizedHaywardmetricregularholestraversablewormholesphotonspheresanti-photonplasmarefractionSgrA*shadowconstraints
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 one two-parameter metric, the generalized Hayward metric with separate mass functions in the time-time and radial-radial components, can serve as a single laboratory for black holes and their mimickers, and that its shadow predictions are testable against the observed Sgr A* shadow. It computes shadow radii for the six spacetime classes that arise from the parameters $(\sigma,\kappa)$, in vacuum and with two plasma profiles. The central result is that the Hayward-Damour-Solodukhin wormhole class can have one, two, or three photon spheres, and when several photon spheres exist the shadow is taken from the largest one; this makes multi-peak wormholes more consistent with the observed shadow than single-peak ones. Regular Hayward black holes remain viable only in a narrow parameter window once plasma is included. If the comparison is right, the same geometries that fit the shadow would also be expected to produce late-time gravitational-wave echoes, which future detectors could search for.

What carries the argument

The load-bearing object is the generalized Hayward metric $ds^2=-f_1(x)dt^2+dx^2/f(x)+x^2d\Omega^2$ with $f_1(x)=1-2\sigma x^2/(x^3+2\sigma\kappa^2)$ and $f(x)=1-2x^2/(x^3+2\kappa^2)$, which puts different mass parameters in the two metric functions. Photon orbits are found from the effective potential $V_{\mathrm{eff}}(x)=f_1(x)/x^2$; the branches $f(x)=0$ and $-f_1'(x)+2f_1(x)/x=0$ produce the photon-sphere radii, and the instability test $\dddot{x}|_{x_{\mathrm{ph}}}>0$ separates photon spheres from anti-photon spheres. For an asymptotic observer the vacuum shadow radius is just the critical impact parameter $x_{\mathrm{sh}}=b$, while in plasma the impact parameter carries factors of the plasma frequency through $\Omega(x)=\omega_p^2/E^2$, with $\Omega=k_0$ or $\Omega=k_x/x$ for the two profiles studied.

What would settle it

A full ray-tracing calculation of the lensed image of a Hayward-Damour-Solodukhin wormhole in the triple-photon-sphere parameter region, including emission from the inner anti-photon sphere, would settle whether the image diameter is actually set by the largest photon sphere; if the numerically computed image diameter differs from the paper's $x_{\mathrm{sh}}$ by more than the observational uncertainty, the Sgr A* comparison changes.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the generalized Hayward metric gives a unified parameter space whose different regions are the Schwarzschild black hole, a Schwarzschild wormhole, Damour-Solodukhin wormholes, Hayward wormholes, the regular Hayward black hole, and Hayward-Damour-Solodukhin wormholes. Working out the null geodesics shows that only the Hayward-Damour-Solodukhin class exhibits multiple photon spheres, and these are separated by anti-photon spheres. The shadow radius is then matched to the observed Sgr A* angular diameter through $r_{\mathrm{sh}}/r_o=\tan(\Phi/2)$, yielding the dimensionless bound $4.35548\le x_{\mathrm{sh}}\le 5.81695$. Under that bound the Schwarzschild wormhole and the Hayward wormhole are ruled out, the regular Hayward black hole is allowed but with its homogeneous plasma parameter confined to small values, and the Damour-Solodukhin and Hayward-Damour-Solodukhin wormholes are allowed mainly when their effective potential has two or three peaks. The paper points out that this shadow-based preference for multi-peak potentials is the opposite of the single-barrier preference from quasinormal-mode studies, and reads the tension as a hint that such wormholes would emit late-time echoes.

Load-bearing premise

The argument assumes that whenever a wormhole has several photon spheres, the observed shadow edge is set by the largest one, and that the Sgr A* angular diameter can be converted to a static, non-spinning shadow radius through Eq. (5.1); if inner photon/anti-photon spheres or spin and accretion morphology shape the image instead, the preference for multi-peak wormholes would not follow.

Editorial extensions

If this is right

  • The regular Hayward black hole remains compatible with the Sgr A* shadow across its full $\kappa$ range in vacuum, so observations do not yet distinguish it from Schwarzschild; with plasma the allowed homogeneous plasma parameter shrinks to small values.
  • The Schwarzschild wormhole and the Hayward wormhole are excluded by the shadow bound in vacuum and in both plasma profiles, because their shadow radii lie below the lower limit.
  • For Damour-Solodukhin and Hayward-Damour-Solodukhin wormholes, the observationally allowed region is concentrated at $\sigma$ close to 1, i.e. double- or triple-peak effective potentials; single-peak wormholes are disfavoured in vacuum and excluded for the non-homogeneous plasma profile.
  • The multi-peak wormholes that fit shadow data are not the ones preferred by earlier quasinormal-mode analyses, so these two observables are selecting different geometries from the same family.
  • If the multi-peak wormhole interpretation is correct, gravitational-wave ringdowns from comparable mergers should show late-time echoes, providing a separate observational channel to test the model.

Reading between the lines

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

  • My extension: the largest-photon-sphere rule is an analytic shortcut, not a ray-tracing result; full image calculations could show that inner photon and anti-photon spheres imprint observable substructure, which would alter the fitted shadow diameter and could change the ranking of multi-peak versus single-peak wormholes.
  • My extension: the exclusion of single-peak wormholes under non-homogeneous plasma depends on the chosen profile $\Omega=k_x/x$; other radial density laws could reopen or close different parameter regions, so the plasma profile should be varied before treating the exclusion as robust.
  • My extension: the shadow/quasinormal-mode tension suggests the two observables weight different parts of the effective potential; fitting both to the same source would be a stronger test than either alone.
  • My extension: applying the same analysis to a rotating generalized Hayward metric would replace isolated photon spheres with photon shells and make shadows non-circular, so spin is likely to broaden the viable parameter regions rather than preserve the exact spherical bounds.
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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

2 major / 5 minor

Summary. The paper studies the shadow radii of the generalized Hayward metric, a two-parameter (σ, κ) family that interpolates between a regular Hayward black hole, a Schwarzschild black hole, and several wormhole spacetimes (Schwarzschild, Damour-Solodukhin, Hayward, and Hayward-Damour-Solodukhin). Using the Hamilton-Jacobi formalism for null geodesics in static spherically symmetric spacetimes, the authors derive photon-sphere conditions and shadow radii for all six spacetime classes, in vacuum and with homogeneous (Ω = k0) and non-homogeneous (Ω = kx/x) plasma profiles. They then convert the EHT angular diameter of Sgr A* into a dimensionless shadow-radius bound and compare the predicted shadow radii to identify observationally viable parameter regions. The headline conclusions are that regular Hayward black holes remain viable only in a narrow parameter range, while Hayward-Damour-Solodukhin wormholes with multi-peak effective potentials are more consistent with the EHT shadow constraints than single-peak wormholes; the paper contrasts this with quasinormal-mode studies and suggests that detectable late-time echoes may accompany these wormholes.

Significance. If the computed shadow radii are correct, the paper provides a unified treatment of black-hole-mimicker shadows in a single metric family and identifies a tension with quasinormal-mode results that could be tested by future gravitational-wave observations. The vacuum and plasma formalisms follow the standard Perlick-Tsupko framework, the algebra in the single-photon-sphere cases checks out, and the classification of the six spacetime families is clear. However, the central observational claim for the Hayward-Damour-Solodukhin wormhole rests on an unverified assumption about which photon sphere sets the shadow boundary, so the significance of the EHT comparison is currently uncertain.

major comments (2)
  1. [Section 4.6, Fig. 12, Eq. (3.10)] The shadow boundary in a static, spherically symmetric spacetime is set by the global maximum of the effective potential V_eff(x) = f1(x)/x^2 over the accessible domain, not by the photon sphere at the largest radius. In Section 4.6 the authors state that for the Hayward-Damour-Solodukhin wormhole they 'compute the shadow radius corresponding to the largest of the photon spheres in case there are more than one photon sphere,' but they provide no argument that this largest-radius sphere realizes the maximum of V_eff. For the double- and triple-peak potentials shown in Fig. 13a, an inner or throat peak can be taller than the outer peak, in which case the true critical impact parameter b_crit = 1/sqrt(V_max) is smaller than the value obtained from the largest-radius photon sphere. Because the EHT Sgr A* band in Eq. (5.2) is narrow, a systematic overestimate of x_sh for multi-peak models could artificially make those models appear more consistent with the data than single-peak ones, thereby undermining the paper's headline conclusion. The authors should compute the shadow boundary from the global maximum of V_eff (including the throat boundary) for the full (σ, κ) grid, and explicitly verify whether the largest-radius photon sphere is the relevant one for each parameter choice.
  2. [Section 3.1 and Section 4.6 (Figs. 14-15)] The same largest-photon-sphere selection rule is used in the plasma calculations, where the effective potential is modified by the plasma term as in Eqs. (3.20)-(3.24). In the presence of plasma, the shadow radius is determined by the maximum of the relevant effective potential, not necessarily by the largest-radius circular orbit satisfying Eqs. (3.23)-(3.24). The paper applies the largest-photon-sphere rule when plotting x_sh in Figs. 14 and 15 and when deriving the EHT constraints with plasma in Section 5.2, so the potential error propagates to the plasma results as well. A separate check using the maximum of the plasma-modified effective potential is needed for both the homogeneous and non-homogeneous profiles before the multi-peak preference can be claimed.
minor comments (5)
  1. [Section 3, Eqs. (3.8)-(3.11)] The notation '˙x = ... = 0' for the circular-orbit condition is misleading because the right-hand side is only the condition for the derivative to vanish, not the derivative itself. The text should say 'the condition ˙x = 0 reduces to ...' and likewise for ¨x = 0.
  2. [Section 3.1, after Eq. (3.28)] The plasma parameter is denoted inconsistently: the text reads '0 ≤ κ0 < 1' while the parameter was defined as k0. Please use a single symbol throughout.
  3. [Section 4.2] The text says 'b2 = x2ph = 2' for the Schwarzschild wormhole with x_ph = 2; this should be b^2 = x_ph^2 = 4. The resulting shadow radius x_sh = 2 is correct, but the intermediate equality is a typo.
  4. [Fig. 16c caption] The caption says 'shadow radius of Damour-Solodukin BH' but the text identifies this panel as the regular Hayward BH. Please correct the caption.
  5. [Section 4.6, first paragraph] The sentence 'For the triple peak potential, there are two photon spheres out of which one is located at the throat, and the two photon spheres are separated by an anti-photon sphere' is confusing: a triple-peak potential should have three extrema, and the count of photon spheres needs to be stated precisely. Please rephrase to describe the number and locations of the peaks and the intervening minima.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: shadow radii are computed from geodesic equations and compared to external EHT data; the generalized Hayward metric is a model input, not a derived claim.

full rationale

The paper's derivation chain is: take the generalized Hayward metric (Eq. 2.3) from prior work [57]; integrate null geodesics (Eqs. 3.8–3.12); identify photon spheres; compute shadow radii (Eqs. 3.13, 3.26–3.30); and compare with the EHT Sgr A* angular diameter via Eq. (5.1). No parameter is fitted to EHT data and then renamed a prediction: the metric parameters (sigma, kappa) and plasma parameters (k0, kx) are scanned, and the EHT constraints are external observational numbers. The only author-overlap input is the generalized Hayward metric of [57], which is a starting model rather than a result being re-derived; the paper does not invoke a uniqueness theorem and does not use [57] to justify its shadow conclusions. The selection in Sec. 4.6 of the largest photon sphere in multi-peak cases is an assumption about which critical orbit casts the observed shadow; it is a potential correctness concern, but it is not equivalent to the derived shadow radius by construction, nor does it turn the EHT comparison into a fit. The paper also explicitly acknowledges its simplifying limitations (Sec. 6), which further supports that the central claim is presented as a conditional model prediction rather than a circularly defined result.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The shadow results inherit the generalized Hayward metric from [57] and the cold-plasma photon Hamiltonian from the Perlick-Tsupko formalism. The metric parameters (sigma, kappa) and plasma amplitudes (k0, kx) are scanned and constrained, not derived. Two modeling choices do real work: the selection of the largest photon sphere as the shadow boundary and the conversion of the EHT angular diameter into a dimensionless static shadow radius. No new physical entities are introduced; 'anti-photon sphere' is the paper's name for a stable null circular orbit.

free parameters (4)
  • sigma (mass ratio M1/M) = 0 <= sigma <= 1 (scanned; EHT-allowed ranges derived)
    Controls the mismatch between the gtt and grr mass functions and determines whether the solution is a black hole or a wormhole. Chosen by hand, not derived.
  • kappa (ratio ell/M) = 0 <= kappa <= 4/(3 sqrt(3)) for Hayward-type cases (scanned)
    Regularization length scale of the Hayward metric; it is a parameter of the model, not derived from first principles.
  • k0 (homogeneous plasma amplitude, Omega = k0) = 0 <= k0 < 1 theoretically; EHT imposes a narrower range
    Dimensionless plasma density-to-energy ratio. The homogeneous profile is chosen for analytic tractability, not derived from a matter model.
  • kx (non-homogeneous plasma amplitude, Omega = kx/x) = bounded by photon-sphere existence and EHT constraints
    Amplitude of the 1/x plasma falloff. The profile is chosen ad hoc for analytic tractability.
assumptions (5)
  • domain assumption Generalized Hayward metric (2.3) is a valid spacetime with the stated matter content and classification.
    The paper takes the metric from [57] as given and does not re-derive its matter sources or energy conditions; all shadow results inherit this assumption.
  • domain assumption Null geodesics in a non-magnetized cold plasma are governed by the Hamiltonian with refractive index n^2 = 1 - omega_p^2 / omega^2.
    Section 3.1 adopts this plasma model; it neglects magnetic fields, accretion flows, and dispersion beyond the cold-plasma approximation.
  • ad hoc to paper For multi-photon-sphere geometries, the observed shadow is set by the largest photon sphere, and inner or anti-photon spheres do not alter the shadow boundary.
    Section 4.6 states that the shadow radius is computed from the largest photon sphere; this selection is not derived from radiative transfer or ray tracing.
  • domain assumption The EHT angular diameter of Sgr A* can be mapped to the dimensionless static shadow radius using Eq. (5.1) with VLTI/GRAVITY mass and distance priors.
    Section 5 assumes this conversion; it neglects spin, inclination, and accretion-flow morphology, and uses one set of priors for the quoted bound.
  • standard math Standard Hamilton-Jacobi separability for static spherically symmetric metrics.
    Section 3 relies on the Hamilton-Jacobi method for null geodesics; no original mathematical machinery is introduced.

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Pith. "Pith review of Shadows of generalised Hayward spacetimes : in vacuum and with plasma." pith.science (2026). https://pith.science/paper/3NP7TMHQ

@misc{pith2026241111970,
  author       = {Pith},
  title        = {Pith review of: Shadows of generalised Hayward spacetimes : in vacuum and with plasma},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NP7TMHQ}},
  note         = {Machine review of arXiv:2411.11970}
}
abstract

We investigate the shadow properties of a wide class of spacetimes arising from different parameter regimes of the generalized Hayward metric, characterized by two independent parameters $(\sigma, \kappa)$ (Phys. Rev. D 106, 044028). This metric extends the original Hayward regular black hole solution by introducing distinct mass functions in the $g_{tt}$ and $g_{rr}$ components, giving rise to four types of wormholes ( which include multi-peak effective potentials), a regular black hole, and a singular black hole solutions allowing for a unified treatment of black hole mimickers. We compute the shadow radii for all spacetimes in vacuum and in the presence of plasma, using both homogeneous and non-homogeneous plasma profiles. Our results show that certain wormhole solutions particularly the Hayward-Damour-Solodukhin class can exhibit multiple photon spheres, leading to shadow features that differ significantly from the Schwarzschild black hole. When these results are compared with Event Horizon Telescope observations of Sgr~$A^\star$, we find that regular black holes remain observationally viable but only within a narrow parameter space. In contrast, wormhole solutions with multi-peak effective potentials are more consistent with shadow constraints than those with single peaks. This contrasts with quasinormal mode studies, which favored single-barrier potentials, and may imply detectable late-time echoes in gravitational wave signals.

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