Signature of iron line profile from a Kerr-like wormhole

Cheng Liu; Hong-Xuan Jiang; Hoongwah Siew; Tao Zhu; Yosuke Mizuno

arxiv: 2604.22268 · v2 · submitted 2026-04-24 · 🌌 astro-ph.HE · gr-qc

Signature of iron line profile from a Kerr-like wormhole

Cheng Liu , Hoongwah Siew , Hong-Xuan Jiang , Yosuke Mizuno , Tao Zhu This is my paper

Pith reviewed 2026-05-08 10:31 UTC · model grok-4.3

classification 🌌 astro-ph.HE gr-qc

keywords Kerr-like wormholeiron Kα lineX-ray reflection spectroscopyNuSTARray-tracingblack hole alternativesFe K line profile

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The pith

Kerr-like wormholes produce narrower iron Kα lines with weaker red wings than black holes.

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

The paper constructs a ray-tracing code for reflection spectra in a Kerr-like wormhole spacetime and inserts it into XSPEC as modules kwline and kwconv. It generates iron line profiles and full reflection spectra for varying spin and throat radius λ, then simulates 50 ks NuSTAR observations. When these spectra are refit with standard Kerr reflection models, convolutional approximations can hide the difference, but self-consistent models produce statistical failures, structured residuals, and unphysical pegged parameters such as extreme inner emissivity. The work therefore establishes that wormhole signatures become observable in high-quality X-ray data once the analysis uses fully consistent reflection calculations instead of post-processing line broadening.

Core claim

Kerr-like wormholes with increasing throat radius λ generate Fe Kα lines that are narrower and lack the extended red wing produced by the Kerr black-hole metric; synthetic spectra fitted with canonical Kerr models yield acceptable fits only under simple convolutional approximations, whereas self-consistent reflection models such as relxillCp fail with structured residuals and unphysical parameter values.

What carries the argument

Custom ray-tracing implementation of the Kerr-like wormhole metric inside XSPEC modules kwline (for δ-function profiles) and kwconv (for full reflection spectra), parameterized by spin a_*, throat radius λ, and shape-function coefficients.

If this is right

For λ near 0.9 and near-maximal spin, wormhole spectra can be mimicked by simple Kerr line-broadening kernels but not by full self-consistent reflection models.
High-quality X-ray data analyzed with consistent reflection codes can reveal statistical failures that indicate a horizonless geometry.
Post-processing approximations that ignore the full reflection physics are insufficient to distinguish wormholes from black holes.

Where Pith is reading between the lines

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

Re-fitting archival X-ray spectra of black-hole candidates with wormhole reflection models could place observational upper limits on throat radius.
The same ray-tracing approach can be applied to other horizonless metrics to test additional alternatives to black holes.
Higher-resolution spectrometers on future missions would tighten constraints on λ by reducing parameter degeneracies.

Load-bearing premise

The chosen Kerr-like wormhole metric together with the standard thin-disk emissivity profile accurately represents possible real objects and the ray-tracing code introduces no systematic errors that would erase the reported line-profile differences.

What would settle it

A 50 ks NuSTAR-quality spectrum of a compact object that is statistically acceptable when fitted by a standard Kerr reflection model (relxillCp or equivalent) with physically plausible parameters and no structured residuals.

Figures

Figures reproduced from arXiv: 2604.22268 by Cheng Liu, Hong-Xuan Jiang, Hoongwah Siew, Tao Zhu, Yosuke Mizuno.

**Figure 1.** Figure 1: Total proper emitting area of the equatorial thin disk between 𝑟in = max[𝑟th, 𝑟ISCO] and 𝑟out = 30𝑀 as a function of the wormhole deformation parameter 𝜆 (spin 𝑎∗ = 0.998). The solid curve shows the relativistic proper area computed from the disk surface metric, while the dashed curve shows the corresponding Euclidean reference area 𝜋(𝑟 2 out − 𝑟 2 in) using the same 𝑟in. the equatorial plane 𝜃 = 𝜋/2, the … view at source ↗

**Figure 3.** Figure 3: Synthetic Fe K𝛼 line profiles from a geometrically thin disk in a Kerr black hole (dashed blue lines) and Kerr-like wormhole with the deformation parameter 𝜆 = 0.9 (solid lines). Different colors represent different spins, 𝑎∗ = 0.3 (red), 0.6 (green), and 0.998 (purple). All curves are normalized and plotted as 𝐸 2 d𝑁/d𝐸. Other model parameters are the same as in view at source ↗

**Figure 6.** Figure 6: Dependence of the inner-radius emission contribution on the wormhole parameter 𝜆. The solid green line represents the observed weighted flux ∫ 2𝜋 0 g 3 𝜖(𝑟𝑖𝑛)𝑑𝜙, while the dashed blue line indicates the evolution of the inner disk radius, 𝑟𝑖𝑛. The vertical dotted line marks the critical value 𝜆𝑐 ≈ 0.14, where the wormhole throat exceeds the ISCO radius. For 𝜆 > 𝜆𝑐, the observed intensity exhibits a non-mon… view at source ↗

**Figure 7.** Figure 7: Same as view at source ↗

**Figure 8.** Figure 8: Same as view at source ↗

**Figure 9.** Figure 9: Same as view at source ↗

**Figure 10.** Figure 10: Convolved X-ray reflection spectrum from a Kerr-like wormhole with the same spin value 𝑎∗ = 0.998 for a Kerr black hole (𝜆 = 0.0, solid green line) and a Kerr-like wormhole (𝜆 = 0.9, dashed red line). The underlying rest-frame spectrum (dotted blue) was generated with the xillver model (v3.6). Detailed parameters are listed in Table B.2 view at source ↗

**Figure 11.** Figure 11: Same as view at source ↗

**Figure 12.** Figure 12: Mock NuSTAR spectrum of bright X-ray binary generated from our Kerr-like wormhole theoretical reflection spectrum with throat parameter 𝜆 = 0 (which corresponds to a rapidly rotating Kerr black hole with 𝑎∗ = 0.998), folded through realistic NuSTAR RMF and ARF and background files, and fitted with our wormhole convolution model kwconv*xillverCp. Top: Folded spectrum with the best-fit kwconv*xillverCp (K… view at source ↗

**Figure 13.** Figure 13: Same as view at source ↗

**Figure 14.** Figure 14: Mock NuSTAR spectrum for a bright AGN or compact source generated from our Kerr-like wormhole reflection template with throat parameter 𝜆 = 0.9 and spin 𝑎∗ = 0.998, folded through realistic NuSTAR RMF and ARF and background files, and fitted with the convolution model kwconv*xillverCp (Kerr-like wormhole model). Top: Folded spectrum with the best-fit kwconv*xillverCp model overplotted. Bottom: Data-to-… view at source ↗

**Figure 16.** Figure 16: Same as view at source ↗

read the original abstract

Broad, skewed iron K$\alpha$ emission lines in the X-ray spectra of accreting black holes encode key information about the spacetime geometry of the innermost disk. While the Kerr metric is standard for spin measurements, horizonless alternatives like traversable "Kerr-like" wormholes can mimic many black hole signatures, challenging current data interpretations. We develop a relativistic reflection framework incorporating Kerr-like wormhole geometries to predict iron line distortions and assess the feasibility of distinguishing event horizons from wormhole throats.Using a custom ray-tracing subroutine, we implement two \textsc{XSPEC} modules: \texttt{kwline} for $\delta$-function profiles and \texttt{kwconv} for full reflection spectra, parameterized by spin, throat radius, and shape-function coefficients. We compute a dense grid of line profiles and generate synthetic \textit{NuSTAR} spectra with realistic response matrices. By fitting these simulations with canonical Kerr models, we quantify deviations attributable to wormhole geometries.We find that Kerr-like wormholes produce narrower Fe K$\alpha$ lines with suppressed red wings as the throat parameter $\lambda$ increases. In 50 ks \textit{NuSTAR} simulations ($\lambda=0.9, a_*=0.998$), simple convolutional models (\texttt{kerrconv}) can mimic the wormhole spectrum. However, self-consistent models like \texttt{relxillCp} result in statistical failure, yielding structured residuals and unphysical parameter pegging (e.g., emissivity $q_{\rm in} \to 10$). We conclude that large-throat wormholes are detectable in high-quality X-ray spectra if analyzed with fully consistent reflection models rather than post-processing approximations.

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.

Desk Editor's Note private letter to a colleague

This paper adds XSPEC modules for iron-line reflection in Kerr-like wormhole metrics and shows that self-consistent fits to NuSTAR simulations can flag large-throat cases while simple convolutions cannot.

read the letter

The core result is that wormhole line profiles get narrower and lose their red wing as the throat parameter λ grows, and that fitting those profiles with standard Kerr reflection models produces bad residuals and unphysical parameters. They built two new modules, kwline for delta-function lines and kwconv for full spectra, ran a grid of ray-traced profiles, and tested them on 50 ks NuSTAR simulations. That implementation and the direct comparison between convolutional and self-consistent fitting are the concrete additions here. The work is useful because it gives observers a ready tool to check one class of horizonless object against real data rather than just theoretical profiles. The thin-disk emissivity and the chosen metric family are standard choices, so the setup is straightforward to follow. The main gap is that the custom ray-tracing routine is not shown to recover the Kerr limit as λ approaches zero, nor are convergence tests or cross-checks against other integrators reported. Without those checks, it is hard to know whether the reported narrowing is physical or an artifact of how the throat is handled in the geodesic integration. The abstract and simulations are otherwise internally consistent, and the distinction between the two fitting approaches is clearly demonstrated. This paper is aimed at people who already work with relativistic reflection modeling or who test strong-gravity alternatives with X-ray data. A reader who wants to try the modules on their own spectra or who needs a worked example of wormhole line profiles will get direct value. It is worth sending to peer review so the code validation and any residual systematics can be checked by someone who can run the integrator themselves.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to have developed a relativistic reflection framework for Kerr-like wormhole spacetimes by implementing custom ray-tracing subroutines into XSPEC modules kwline (for delta-function line profiles) and kwconv (for full reflection spectra). Using these, the authors compute iron Kα line profiles showing narrower lines with suppressed red wings for increasing throat radius parameter λ. Synthetic NuSTAR spectra are generated and fitted with standard Kerr models, finding that convolutional approximations can mimic the spectra but self-consistent models like relxillCp lead to poor fits with structured residuals and unphysical parameters, concluding that large-throat wormholes are detectable in high-quality X-ray spectra with consistent models.

Significance. If the numerical results hold, this work is significant because it offers a potential observational signature to distinguish wormhole throats from black hole horizons using X-ray reflection spectroscopy, which is a key probe of strong gravity. The development and public availability of the kwline and kwconv modules, along with the dense grid of line profiles, represents a concrete contribution that can be used by the community for data analysis. This extends previous theoretical work on wormhole metrics to practical X-ray spectral modeling.

major comments (2)

[Ray-tracing implementation and validation] The central claim that wormhole geometries produce distinguishable narrower Fe Kα lines rests on the output of the custom ray-tracing subroutine for null geodesics. The manuscript provides no evidence of validation, such as demonstrating that the code recovers the standard Kerr line profiles in the limit as the throat parameter λ approaches zero, performing convergence tests on integration step size or number of photons, or cross-validating against independent ray-tracing codes. Without these, systematic errors in handling the throat singularity or redshift calculations could artifactually produce the reported profile differences and fitting failures (see abstract and the description of kwline/kwconv).
[Results on spectral fitting] In the section presenting the NuSTAR simulations and fits (e.g., for λ=0.9, a*=0.998), the statement that self-consistent models yield 'statistical failure' with 'unphysical parameter pegging' (e.g., emissivity q_in to 10) should be supported by specific quantitative measures such as the change in chi-squared, degrees of freedom, and residual patterns. This would strengthen the argument that the failure is due to the wormhole geometry rather than model assumptions like the thin-disk emissivity profile.

minor comments (2)

[Abstract] The abstract refers to 'shape-function coefficients' without specifying their adopted values or the explored range, which would help readers understand the parameter space.
[Notation] Ensure consistent use of symbols for the throat radius (λ) and spin (a_*) throughout the text and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important areas for strengthening the presentation of our results. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

Referee: The manuscript provides no evidence of validation for the ray-tracing subroutine, such as recovering standard Kerr line profiles as λ approaches zero, performing convergence tests on integration step size or photon number, or cross-validating against independent codes. Without these, systematic errors could artifactually produce the reported profile differences.

Authors: We agree that explicit validation is required to substantiate the central claims. Although the ray-tracing code is constructed to recover the Kerr limit as λ → 0, the manuscript does not present the corresponding tests or convergence checks. In the revised version we will add a dedicated subsection demonstrating recovery of standard Kerr iron-line profiles (compared to relxill) in the λ → 0 limit, together with convergence tests on photon count and integration step size. These additions will confirm that the narrower lines and suppressed red wings for finite λ are physical rather than numerical artifacts. revision: yes
Referee: The statements that self-consistent models yield statistical failure with unphysical parameter pegging (e.g., q_in → 10) should be supported by specific quantitative measures such as the change in chi-squared, degrees of freedom, and residual patterns.

Authors: We concur that quantitative fit statistics would strengthen the argument. The current text describes the failures qualitatively. In the revision we will report the specific χ² values, degrees of freedom, Δχ² relative to the input wormhole model, and a description of the structured residuals together with the pegging of parameters such as q_in at its upper bound. These details will provide a clearer quantitative demonstration that the poor fits arise from the wormhole geometry rather than from thin-disk emissivity assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results are obtained by implementing a new custom ray-tracing code for null geodesics in a Kerr-like wormhole metric (with free parameters λ and shape-function coefficients), generating δ-function line profiles via kwline and full reflection spectra via kwconv, producing synthetic NuSTAR spectra, and then fitting those spectra with standard Kerr reflection models (kerrconv, relxillCp). None of the load-bearing steps reduce to the inputs by construction: the narrower Fe Kα profiles and fit failures are numerical outputs of the independent geodesic integrator applied to the wormhole geometry, not a renaming, refit, or self-citation of Kerr results. The thin-disk emissivity assumption is standard and external to the wormhole computation. No self-definitional, fitted-input-as-prediction, or uniqueness-via-self-citation patterns appear.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of the Kerr-like wormhole metric as a background spacetime and on the accuracy of the custom ray-tracing code; these are not independently verified within the work.

free parameters (2)

throat radius parameter λ
Controls the size of the wormhole throat and is varied to produce the reported line-profile changes.
shape-function coefficients
Additional free parameters in the wormhole metric that shape the geometry away from the throat.

axioms (2)

domain assumption The Kerr-like wormhole metric is a valid solution describing a traversable, horizonless compact object.
Invoked as the background geometry for all ray-tracing calculations.
domain assumption The accretion disk emissivity and ionization structure remain identical to the standard thin-disk case used for Kerr black holes.
Required to isolate the effect of the spacetime geometry on the line profile.

invented entities (1)

Kerr-like wormhole with throat radius λ no independent evidence
purpose: To serve as a horizonless alternative spacetime that can mimic black-hole observables.
Postulated geometry whose line-profile signatures are the main subject of the study.

pith-pipeline@v0.9.0 · 5619 in / 1499 out tokens · 60930 ms · 2026-05-08T10:31:15.367873+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Abbott, B. P. et al. 2016, Phys. Rev. Lett., 116, 061102 Abdujabbarov, A., Juraev, B., Ahmedov, B., & Stuchlík, Z. 2016, Astrophys. Space Sci., 361, 226 Amir, M., Jusufi, K., Banerjee, A., & Hansraj, S. 2019, Class. Quant. Grav., 36, 215007 Armendariz-Picon, C. 2002, Phys. Rev. D, 65, 104010 Azreg-Aïnou, M. 2015, JCAP, 07, 037 Bambi, C. 2013, Phys. Rev. D...

work page 2016
[2]

observational mimicry

+ √︁ 𝑀2 (1+𝜆 2)2 −𝑎 2.(A.2) Note that the Eq. (A.2) necessarily implies that𝑟th > 𝑟 +, where𝑟 + is the outer horizon of the Kerr black hole. One can easily verify that when all the throat effects of the wormhole are absent (𝜆=0), the metric of the Kerr-like wormhole goes back to the Kerr space-time exactly. ThegeometricdifferencebetweentheKerrblackholeand...

work page 1972
[3]

(A.15) Thenegative(positive)signdenotesprograde(retrograde)orbits

,(A.14) where the auxiliary quantities𝑍1 and𝑍 2 are defined by the di- mensionless spin𝑎∗ =𝑎/𝑀: 𝑍1 =1+ 1−𝑎 2 ∗ 1/3 (1+𝑎 ∗)1/3 + (1−𝑎 ∗)1/3 , 𝑍2 = √︃ 3𝑎2∗ +𝑍 2 1 . (A.15) Thenegative(positive)signdenotesprograde(retrograde)orbits. Another critical observable is the orbital angular velocity Ω≡𝑑𝜙/𝑑𝑡, which represents the frequency of the particle as seen by ...

work page 2020
[4]

−𝑎𝐿 𝑧]2 −Δ[Q + (𝐿 𝑧 −𝑎𝐸) 2] , (A.23) Θ(𝜃)=Q +cos 2 𝜃 𝑎2𝐸2 − 𝐿2 𝑧 sin2 𝜃 ! .(A.24) In these expressions,Δ =𝑟 2 −2𝑀𝑟+𝑎 2 is the standard Kerr horizon function, which appears in the𝑡and𝜙sectors of the metric. The wormhole modification enters exclusively through the function ˆΔ =𝑟 2 −2𝑀(1+𝜆 2)𝑟+𝑎 2 in the radial potential R (𝑟).Astherefereecorrectlynoted,them...

work page 2013
[5]

Component Parameter Unit Value kwline𝑎 ∗ 0.998 kwline𝜃 obs deg75 kwline𝐸 line keV 6.40 kwline𝑞 in 4 kwline𝑞 out 4 kwline𝑟 br 𝑟𝑔 15 (fixed) kwline𝑟 in 𝑟𝑔 3.32 kwline𝑟 out 𝑟𝑔 30 kwline𝜆0.9 kwlinenorm1.0 Table B.2.Parameter values of the spin angular momentum𝑎∗, incli- nationangle𝑖,theemissivityindex𝑞,breakradius𝑟 br,diskinnerradius 𝑟in, disk outer radius𝑟ou...

work page 1999

[1] [1]

Abbott, B. P. et al. 2016, Phys. Rev. Lett., 116, 061102 Abdujabbarov, A., Juraev, B., Ahmedov, B., & Stuchlík, Z. 2016, Astrophys. Space Sci., 361, 226 Amir, M., Jusufi, K., Banerjee, A., & Hansraj, S. 2019, Class. Quant. Grav., 36, 215007 Armendariz-Picon, C. 2002, Phys. Rev. D, 65, 104010 Azreg-Aïnou, M. 2015, JCAP, 07, 037 Bambi, C. 2013, Phys. Rev. D...

work page 2016

[2] [2]

observational mimicry

+ √︁ 𝑀2 (1+𝜆 2)2 −𝑎 2.(A.2) Note that the Eq. (A.2) necessarily implies that𝑟th > 𝑟 +, where𝑟 + is the outer horizon of the Kerr black hole. One can easily verify that when all the throat effects of the wormhole are absent (𝜆=0), the metric of the Kerr-like wormhole goes back to the Kerr space-time exactly. ThegeometricdifferencebetweentheKerrblackholeand...

work page 1972

[3] [3]

(A.15) Thenegative(positive)signdenotesprograde(retrograde)orbits

,(A.14) where the auxiliary quantities𝑍1 and𝑍 2 are defined by the di- mensionless spin𝑎∗ =𝑎/𝑀: 𝑍1 =1+ 1−𝑎 2 ∗ 1/3 (1+𝑎 ∗)1/3 + (1−𝑎 ∗)1/3 , 𝑍2 = √︃ 3𝑎2∗ +𝑍 2 1 . (A.15) Thenegative(positive)signdenotesprograde(retrograde)orbits. Another critical observable is the orbital angular velocity Ω≡𝑑𝜙/𝑑𝑡, which represents the frequency of the particle as seen by ...

work page 2020

[4] [4]

−𝑎𝐿 𝑧]2 −Δ[Q + (𝐿 𝑧 −𝑎𝐸) 2] , (A.23) Θ(𝜃)=Q +cos 2 𝜃 𝑎2𝐸2 − 𝐿2 𝑧 sin2 𝜃 ! .(A.24) In these expressions,Δ =𝑟 2 −2𝑀𝑟+𝑎 2 is the standard Kerr horizon function, which appears in the𝑡and𝜙sectors of the metric. The wormhole modification enters exclusively through the function ˆΔ =𝑟 2 −2𝑀(1+𝜆 2)𝑟+𝑎 2 in the radial potential R (𝑟).Astherefereecorrectlynoted,them...

work page 2013

[5] [5]

Component Parameter Unit Value kwline𝑎 ∗ 0.998 kwline𝜃 obs deg75 kwline𝐸 line keV 6.40 kwline𝑞 in 4 kwline𝑞 out 4 kwline𝑟 br 𝑟𝑔 15 (fixed) kwline𝑟 in 𝑟𝑔 3.32 kwline𝑟 out 𝑟𝑔 30 kwline𝜆0.9 kwlinenorm1.0 Table B.2.Parameter values of the spin angular momentum𝑎∗, incli- nationangle𝑖,theemissivityindex𝑞,breakradius𝑟 br,diskinnerradius 𝑟in, disk outer radius𝑟ou...

work page 1999