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The JP black hole with a non-closed horizon shows an extended inner shadow absent in Kerr images.

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2026-08-01 14:28 UTC pith:GHZUUC5E

load-bearing objection Solid numerical imaging catalog, but the headline extended inner shadow may be an artifact of an untested disk inner-edge choice. the 2 major comments →

arxiv 2607.18751 v1 pith:GHZUUC5E submitted 2026-07-21 gr-qc

Novel extended inner shadow in images of Johannsen-Psaltis black holes with thin accretion disks

Xinyu Wang , Xiaobao Wang , Xin-Yang Wang , Minyong Guo This is my paper
classification gr-qc MSC 83C5783C10
keywords Johannsen-Psaltis metricblack hole imageinner shadowthin accretion diskbackward ray-tracingdeviation parameterKerr hypothesisno-hair theorem
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper uses numerical backward ray-tracing to image Johannsen-Psaltis (JP) black holes with thin accretion disks. It shows that when the JP deviation parameter ε3 makes the event horizon non-closed, the image contains an extended inner shadow beyond the ordinary inner shadow, a feature with no Kerr analogue. For closed horizons, increasing |ε3| enlarges the inner shadow faster than the photon ring, narrowing the direct-image band, and the growth is anisotropic, strongest along the direction probing the equatorial plane. These features are presented as observable signatures to test the Kerr paradigm and the no-hair theorem.

Core claim

The paper finds a new image feature: an extended inner shadow, distinct from the original inner shadow, in images of JP black holes whose event horizon is non-closed (dumbbell-shaped with an exposed singularity). Using backward ray-tracing and an approximate analytical framework, it shows the feature arises from the ε3 deviation. For closed horizons, the inner-shadow growth outpaces the photon-ring growth, and the evolution is dominated by the equatorial-plane direction where h(r,θ) is largest.

What carries the argument

The single deviation parameter ε3 enters the JP metric through h(r,θ)=ε3 M^3 r/Σ^2. Its angular dependence (largest at the equator, smallest at the poles) plus frame-dragging drives the anisotropic image deformation. Because JP spacetime is Petrov type I, photon motion is non-separable, so images are computed by numerical backward ray-tracing with a ZAMO frame and simplified radiative transfer.

Load-bearing premise

The extended inner shadow rests on assuming the thin disk stops at the Kerr horizon radius r_h^Kerr for non-closed horizons; there is no sensitivity test of that choice.

What would settle it

For a non-closed JP horizon, move the disk inner edge inward from r_h^Kerr (e.g., to half or to the singularity region) and re-run the ray-tracing; if the extended inner shadow disappears, it is an artifact of the disk inner-edge prescription.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Non-closed horizons produce an extended inner shadow, a direct image marker absent in Kerr.
  • Closed horizons: increasing |ε3| expands both inner shadow and photon ring, with the inner shadow catching up.
  • The growth anisotropy is tied to the equatorial plane, giving a geometric signature of ε3.
  • Maximum-blueshift position responds to ε3 in prograde but not retrograde flows.
  • These relations can be used to constrain deviations from Kerr with VLBI images.

Where Pith is reading between the lines

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

  • The extended inner shadow's existence depends on the disk inner edge being at the Kerr horizon radius; other choices might alter or erase it.
  • The directional anisotropy predicts that shadow ellipticity relative to the spin axis encodes ε3, offering a testable extension.
  • Further work could turn the approximate analytical framework into a closed-form relation between shadow size and ε3 for fast parameter estimation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper constructs images of Johannsen-Psaltris black holes surrounded by thin, optically thin accretion disks using numerical backward ray-tracing. For closed-horizon JP black holes it systematically studies how the inner shadow, photon ring, intensity profiles, directional scales, ellipticities, and redshift/blueshift maps depend on the deviation parameter ϵ3 for selected spins and observer inclinations. For non-closed horizon configurations it reports a new 'extended inner shadow' and proposes an approximate analytical formation mechanism in Sec. 3.3.

Significance. The claimed extended inner shadow, if robust, would be a qualitative image feature absent in Kerr and traceable to the JP deviation parameter, giving a new observable avenue for no-hair tests. The closed-horizon analysis has real strengths: the image quantities are computed by geodesic integration rather than fitted, the photon-ring peak is honestly excluded from quantitative claims because of resolution sensitivity, and the model inherits ϵ2=0 from LLR/PPN constraints. The non-closed-horizon claim, however, is not yet established because the imaging result depends on an ad hoc choice of the disk inner edge.

major comments (2)
  1. [Sec. 3.3, inner-disk-edge prescription r_in = r_h^Kerr] The headline 'extended inner shadow' is computed with the thin disk truncated at the Kerr horizon radius r_h^Kerr. For a non-closed JP horizon the spacetime interior to r_h^Kerr is regular up to the naked singularity, so there is no geometric reason why emitting matter must end there; no ISCO, photon-surface, or causality argument is used to select this radius. A dark region is produced whenever the backward-traced geodesic intersects the equatorial plane at r < r_in simply because no emitter exists there. The feature may therefore be a disk-truncation artifact rather than a new spacetime shadow. The paper should test the sensitivity of the extended-shadow size and intensity to r_in (e.g., r_in = 0.5, 0.8, 1.2 r_h^Kerr) and report how the claimed mechanism in Sec. 3.3 depends on this choice. Without such a test the central claim is not load-bearing.
  2. [Sec. 2, paragraph after Fig. 1, and Sec. 3.3] The paper uses the statement that photons from the vicinity of the naked singularity require infinite coordinate time to reach infinity as justification for neglecting the singular region in the non-closed case. This result is imported from the authors' previous work and not re-derived; it should be stated precisely or re-derived in the present framework. More importantly, even if the singularity itself is unobservable, this does not exclude emitting matter in the regular region r < r_h^Kerr. The actual assumption behind the extended inner shadow is the absence of emission interior to r_h^Kerr, and that assumption is neither physically derived nor numerically varied.
minor comments (5)
  1. [Introduction, Sec. 2] Typo: 'analytic deviation of photon trajectories' should be 'analytic derivation of photon trajectories.'
  2. [Fig. 3 and text] The caption/text color coding is inconsistent: the text describes yellow, green, cyan, and red regions, while the caption mentions black, yellow, green, blue, and red. Please align the labels.
  3. [Eqs. (3.14)-(3.23)] The notation I0 is used both for the observed intensity and for the Lorentz-invariant intensity at the observer; please distinguish these, for example with a different symbol or explicit subscript.
  4. [Sec. 3.2, discussion of average radii] The text refers to Fig. 5 when discussing the evolution of average radii and relative deviations; the relevant figure appears to be Fig. 4. Please correct the cross-reference.
  5. [Eq. (3.23) and fudge factor f_n] Setting f_n = 1 is an arbitrary normalization. The paper states that f_n mainly affects photon-ring brightness, and it excludes photon-ring peaks from quantitative claims, which mitigates the concern; however, this should be stated more explicitly so that the absolute intensity scale is not over-interpreted.

Circularity Check

0 steps flagged

No significant circularity; the non-closed-horizon disk inner-edge choice is a model limitation, not a circular reduction.

full rationale

The derivation chain is self-contained. Starting from the external JP metric (Johannsen & Psaltis, Ref. [46]) with epsilon_2=0 fixed by LLR/PPN constraints (Eqs. 2.7-2.9) and retaining only epsilon_3, the paper computes images by numerical backward ray-tracing plus the radiative-transfer expression I0 = sum_n f_n g_n^3 J_n (Eq. 3.23), with the emissivity J=exp(-(1/2)z^2-2z), z=log(r/r_H) (Eq. 3.24) taken from an externally published EHT-motivated model (Ref. [48]). No parameter is fitted to reproduce the extended inner shadow or any other target feature; shadow radii, intensity profiles, Doppler maps, and relative deviations are all forward-computed from the metric and ray tracing. The non-closed-horizon section does introduce an input choice: 'it is reasonable to assume that the inner radius of the accretion disk coincides with the event horizon radius of the corresponding Kerr black hole obtained in the limit where the deviation parameter vanishes, namely r_in = r_h^Kerr' (Sec. 3.3). This choice can strongly affect whether and how far a dark 'extended inner shadow' appears, and the absence of a sensitivity test is a real modeling limitation. But the feature is still obtained through lensing and geodesic transport rather than by definitional identification with the input radius, so it is not a circular reduction. The paper also cites the authors' earlier JP-shadow study for the approximate Carter constant and for the statement that photons from near the naked singularity take infinite coordinate time to reach infinity; those are reproducible numerical claims and are not the load-bearing derivation of the image results here. Overall, the central derivation is an independent forward calculation with no circular step, and the only caveats are model-choice and minor self-citation issues, which do not warrant a score above 2.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The computation rests on: (i) the JP metric as an effective spacetime with ϵ3 as the single scanned deviation parameter and all other ϵ_k zeroed; (ii) the geodesic/radiative-transfer prescription of Sec. 3.1; (iii) imported or hand-set inputs — the Gralla-Lupsasca emissivity fitted to EHT data in Ref. [48], f_n = 1, and the disk inner/outer radii; and (iv) for the non-closed case, the ad hoc inner edge r_in = r_h^Kerr and the imported infinite-coordinate-time property of singularity-bound photons. No new particles, forces, or conserved quantities are invented. The extended inner shadow is a derived image feature, not a postulated entity.

free parameters (4)
  • ϵ3 = scanned; |ϵ3| ∈ [0,8]
    The only retained JP deviation parameter. ϵ0, ϵ1, ϵ2 are set to zero (PPN/LLR constraints, Eqs. 2.7-2.9) and k>3 terms are truncated (Sec. 2). The paper scans ϵ3 rather than fitting it; the full image catalog depends on it.
  • Emissivity profile coefficients = J = exp(-z²/2 - 2z), z = log(r/rH)
    Adopted from Ref. [48], which fitted M87*/Sgr A* 230 GHz images; transported as an input to JP black hole images (Eq. 3.24). The coefficients 1/2 and 2 are fitting outputs of prior work, not derived here.
  • Fudge factor f_n = 1
    Normalized to unity 'for computational convenience' (Sec. 3.1); the paper states this affects only the brightness of narrow photon rings, not the overall image structure.
  • Disk inner edge for non-closed horizons = r_h^Kerr
    For non-closed-horizon JP black holes the accretion disk inner radius is set to the Kerr horizon radius (Sec. 3.3) because a JP event horizon is absent. This ad hoc choice directly conditions the image structure in which the extended inner shadow appears.
axioms (4)
  • domain assumption The JP metric (with only ϵ3 retained) is a valid effective spacetime for astrophysical black hole modeling, and photon motion is governed by null geodesics in it.
    The paper states the JP metric is not a solution of Einstein's equations but an effective Kerr-like framework (Sec. 1); all imaging analysis rests on computing null geodesics and radiative transfer in this metric.
  • domain assumption Photons originating near the naked singularity of a non-closed-horizon JP black hole take infinite coordinate time to reach infinity, so the exposed singularity does not visibly affect the image.
    Invoked in Sec. 2 (cited to the authors' prior Ref. [47]) to justify studying non-closed horizons despite naked-singularity formation; load-bearing for the non-closed-horizon images and the extended inner shadow interpretation.
  • ad hoc to paper The disk is optically thin with emissivity J = exp(-z²/2 - 2z); absorption, refraction, and the fudge factor are neglected (κ_n ≈ 0, f_n = 1).
    Eqs. 3.23-3.24 and the discussion in Sec. 3.1; the emissivity form is fitted to EHT wavelengths in prior work, and the fudge factor is normalized to unity. These choices set the intensity maps but not the geometric shadow structure.
  • domain assumption Accreting plasma follows timelike geodesics: circular orbits outside the ISCO and critical plunging orbits inside, with energy and angular momentum frozen at their ISCO values.
    Standard thin-disk prescription, Eqs. 3.1-3.4 and 3.25-3.30; used to compute the comoving-frame frequencies and redshift factors g_n that determine the Doppler maps and intensity distributions.

pith-pipeline@v1.3.0-alltime-deepseek · 46045 in / 13754 out tokens · 155935 ms · 2026-08-01T14:28:17.590340+00:00 · methodology

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read the original abstract

The Johannsen-Psaltis (JP) metric is constructed by introducing deviation parameters into the Kerr metric. The presence of these deviation parameters provides additional degrees of freedom for testing the extent to which astrophysical black holes are consistent with the Kerr paradigm and for examining the validity of the no-hair theorem. Since black hole imaging provides one of the most direct means of probing the properties of astrophysical black holes, a detailed investigation of the imaging characteristics of JP black holes constitutes a natural first step toward employing the JP metric to explore potential deviations from the Kerr geometry in astrophysical black hole systems. Accordingly, numerical backward ray-tracing simulations are employed to conduct a comprehensive investigation of the image structures, intensity distributions, and redshift-blueshift signatures of JP black holes with both closed and non-closed event horizons. The dependence of these imaging characteristics on the deviation parameter is also investigated systematically. Furthermore, the emergence of an extended inner shadow, distinct from the original inner shadow of the black hole, in images of JP black holes with non-closed event horizons motivated a detailed investigation of its physical formation mechanism using both numerical backward ray-tracing simulations and an approximate analytical framework.

Figures

Figures reproduced from arXiv: 2607.18751 by Minyong Guo, Xiaobao Wang, Xin-Yang Wang, Xinyu Wang.

Figure 1
Figure 1. Figure 1: (a) Parameter space of the JP metric in the ( [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Three classes of photon trajectories emitted from the accretion disk in the vicinity of the [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Images of JP black holes with closed event horizons. Different colors denote photon trajec [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The evolution of the relative deviations of the average radii of the inner shadow and critical [PITH_FULL_IMAGE:figures/full_fig_p035_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The evolution of the intensity distributions along the [PITH_FULL_IMAGE:figures/full_fig_p038_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The evolution of the intensity distributions along the [PITH_FULL_IMAGE:figures/full_fig_p042_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Definition of the characteristic scales of the inner shadow and the photon ring on the [PITH_FULL_IMAGE:figures/full_fig_p045_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The evolution of the relative deviations of the characteristic scales of the inner shadow [PITH_FULL_IMAGE:figures/full_fig_p046_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The evolution of the relative deviations of the characteristic scales of the inner shadow [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The evolution of the ellipticities of the inner shadow and photon ring in JP black hole images [PITH_FULL_IMAGE:figures/full_fig_p051_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The evolution of the ellipticities of the inner shadow and photon ring in JP black hole [PITH_FULL_IMAGE:figures/full_fig_p053_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The distributions of redshift and blueshift corresponding to the direct image of JP black [PITH_FULL_IMAGE:figures/full_fig_p055_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The distributions of redshift and blueshift corresponding to the lensed image of JP black [PITH_FULL_IMAGE:figures/full_fig_p057_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The evolution of the relative radial position of the maximum blueshift, [PITH_FULL_IMAGE:figures/full_fig_p059_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The distributions of redshift and blueshift corresponding to the direct image of JP black [PITH_FULL_IMAGE:figures/full_fig_p061_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The distributions of redshift and blueshift corresponding to the lensed image of JP black [PITH_FULL_IMAGE:figures/full_fig_p063_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The evolution of the relative radial position of the maximum blueshift, [PITH_FULL_IMAGE:figures/full_fig_p065_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Images of JP black holes with non-closed event horizons observed at an inclination angle of [PITH_FULL_IMAGE:figures/full_fig_p069_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Images of JP black holes with non-closed event horizons observed at an inclination angle [PITH_FULL_IMAGE:figures/full_fig_p072_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The evolution of the intensity distributions along the [PITH_FULL_IMAGE:figures/full_fig_p075_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The distributions of redshift and blueshift associated with the direct image of JP black [PITH_FULL_IMAGE:figures/full_fig_p080_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The distributions of redshift and blueshift associated with the lensed image of JP black [PITH_FULL_IMAGE:figures/full_fig_p085_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Identification of the distinct image structures in the image of a JP black hole with a non [PITH_FULL_IMAGE:figures/full_fig_p092_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Photon trajectories in the JP spacetime, displayed in the [PITH_FULL_IMAGE:figures/full_fig_p093_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Parameter space of the photon impact parameters ( [PITH_FULL_IMAGE:figures/full_fig_p102_25.png] view at source ↗

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