Profiled spectral lines of Keplerian rings orbiting in the regular Bardeen black hole spacetimes

Jan Schee; Zdenek Stuchlik

REVIEW 2 major objections 5 minor 86 references

Profiled spectral lines of Keplerian rings orbiting in the regular Bardeen black hole spacetimes

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Photons moving through the nonlinear-electrodynamics effective geometry of a Bardeen black hole would make disk line profiles deep-red and 15 to 50 times narrower than standard geometry predicts.

desk verdict A clean, internally consistent calculation of Bardeen-NED line profiles that only covers the extraordinary photon mode; as an unpolarized observable it likely collapses to the standard Bardeen line once the ordinary mode is included. read the letter →

arxiv 1908.07197 v1 pith:NZ6CMINH submitted 2019-08-20 gr-qc

Jan Schee , Zdenek Stuchlik This is my paper · ORCID

classification gr-qc MSC 83C5783C50 PACS 04.70.-s

keywords BardeenblackholesnonlinearelectrodynamicseffectivegeometryprofiledspectrallinesKeplerianringsaccretiondisksregular

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 asks whether the nonlinear electrodynamics that removes the singularity in the regular Bardeen black hole leaves a measurable imprint on the light the hole's accretion flow emits. The authors compute photon motion not in the spacetime metric alone but in the effective geometry that the NED Lagrangian imposes on surfaces of discontinuity of the electromagnetic field, and they ray-trace monochromatic emission from Keplerian rings and disks. They claim the resulting profiled spectral lines are pushed to the red edge of the spectrum by at least an order of magnitude and narrowed by a factor of roughly 15 to 50 compared with lines generated by the spacetime geometry alone, for every magnetic charge and observer inclination they study. For an extended inner disk the peak specific flux is additionally suppressed by more than six orders of magnitude. If this is right, ordinary X-ray line profiles from accreting compact objects become a direct observational test of whether the central object is a regular Bardeen black hole.

What carries the argument

The load-bearing object is the effective geometry for photons, the metric replacement $\tilde{g}_{\alpha\beta} = L_F g_{\alpha\beta} - L_{FF} F_{\alpha\lambda} F^{\lambda}{}_{\beta}$ obtained from the Hadamard discontinuity method (Eq. 13). In the Bardeen spacetime this becomes a diagonal metric with the lapse f(r) divided by L_F and angular parts scaled by $\Phi = L_F + 2F L_{FF}$ (Eqs. 14-16). Null geodesics of this effective metric are integrated by Hamiltonian ray tracing; their constants of motion separate in radial and latitudinal coordinates, and the line profile is assembled from the impact parameters, the frequency shift g, and the Jacobian of the map from detector coordinates to emitter radius and shift. A second, simpler ingredient carries the redshift: for static emitters and observers the effective-geometry shift is $g_{\mathrm{NED}} = \sqrt{f(r_e)/f(r_o)}\, L(o)/L(e)$, while the Maxwell case is just $\sqrt{f(r_e)/f(r_o)}$; their ratio $R = L(e)/L(o)$ is of order 10, which directly explains the deep-red displacement.

What would settle it

Take a Keplerian ring at r_e = 10 M around a Bardeen black hole with q_m = 0.5 M viewed at θ_o = 60°. The effective-geometry prediction is a line spanning g ≈ 0.0154-0.0338, while the spacetime-geometry prediction is g ≈ 0.643-1.202. A spectrally resolved observation of such a source showing the line peaked near g ≈ 1 with width Δg ≈ 0.56, instead of a deep-red narrow line, would falsify the claimed NED effect for that system.

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Extended reading notes

Core claim

The central claim is that the NED effect on photon propagation is not a small correction to the Bardeen spacetime geometry; it dominates the observed line. For radiating rings at the innermost stable circular orbit, at 10 M and at 20 M, the frequency shift g computed in the effective geometry is of order 0.005-0.07, versus 0.4-1.4 in the pure spacetime geometry, so the effective-geometry line sits deeper in the red by more than one order and its width shrinks by factors from 15 to 50 across the surveyed charges q_m=0.05M, 0.5M, 0.768M and inclinations 30°, 60°, 85°. The authors trace the deep redshift to the ratio $R = L(e)/L(o)$ of the NED Lagrangian evaluated at emitter and observer, which is itself of order 10 and only mildly dependent on the magnetic charge; Doppler shifting and gravitational lensing then reshape the line without changing its qualitative redshift. For extended Keplerian disks the same effect suppresses the peak flux by more than six orders and visibly changes the line shape. The paper therefore positions profiled spectral lines, not just the shadow, as the observable signature of the effective geometry.

Load-bearing premise

Everything in the ray tracing rests on the assumption that photons in nonlinear electrodynamics propagate along the effective metric of Eq. (13), obtained from the Hadamard discontinuity method, rather than along the background spacetime metric.

Editorial extensions

If this is right

Accreting Bardeen black holes should show iron-line profiles displaced to energies far below the rest energy, with line widths factors of 15 to 50 smaller than standard relativistic disk-line predictions.
The NED effect remains visible even for rings at 20 M, where the dependence on the magnetic charge has nearly disappeared, so a deep-red narrow line can serve as a charge-independent indicator of effective-geometry photon motion.
For emission integrated over the inner disk from the ISCO out to 20 M, the peak flux drops by more than six orders of magnitude and the line shape is substantially modified, not just shifted.
Comparison with Reissner-Nordstrom spacetimes of the same charge isolates the nonlinear-electrodynamics contribution, because the two agree in the Maxwellian weak-field limit.

Reading between the lines

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

A parameter-free test can be built from the static-emitter ratio R = L(e)/L(o): measure the centroid shift of a line from a large-radius, nearly static part of a disk and compare it with the value predicted by the same charge that fits the continuum; a mismatch would favor a different effective geometry, such as one with birefringence.
The same effective geometry also shifts photon circular orbits, which the paper connects to electromagnetic quasinormal modes; this suggests that ring line profiles, shadow measurements, and ringdown signals could jointly constrain the NED Lagrangian better than any one observable alone.
The deep-redshift prediction depends quantitatively on the specific NED Lagrangian chosen; applying the same ray-tracing procedure to other regular-black-hole Lagrangians would rank the candidate theories by how much line centroid and width they predict.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: the paper has a clean, internally consistent numerical result, but the headline claim is stated for the wrong photon mode. The effective geometry in Eq. (13) is the extraordinary birefringent mode of NED. The same field equations admit an ordinary mode with a conformally related metric whose null geodesics are exactly the Bardeen ones. The paper never says the emitting ring is single-polarization. For a polarization-independent source, the ordinary mode's g values are near standard (0.5–1.4) while the extraordinary mode's are ~0.005–0.03, and because flux scales as g^3 the ordinary mode dominates the summed profile. So the deep red shift, 15–50x narrowing, and six-order flux suppression in the abstract and conclusions do not describe the total unpolarized line. This is load-bearing, not a quibble.

What is genuinely new: this is the first computation of profiled spectral lines in the Bardeen effective geometry, and it extends the authors' own shadow/lensing program in a sensible way. The derivation of the effective geodesic equations in Sec. 3 is standard and clearly laid out. Table 1 gives a full grid of 27 configurations showing the factor 15–60 narrowing in the extraordinary mode, and Eq. (49) reduces the central effect to a simple algebraic ratio R = L(e)/L(o). That ratio is not fitted to any target, so there is no circularity or parameter-fitting worry. I agree with the reader's conditional verdict, but I would make the condition stronger than 'missing verification details.'

Other soft spots are lesser but real. The frequency-shift formula (36) is quoted from the authors' unpublished paper [59], which makes a key step hard to verify. The emissivity index p in Eq. (40) is never specified for the disk profiles. The text still contains unresolved citation placeholders [?] in Sections 2 and 4. The abstract's 'for all values' overstates the three sampled charges and three inclinations. No code or convergence information is given, though the algebraic ratio and the consistency of Table 1 suggest the numerics are not the main risk.

Who gets value from this: people working on NED regular black hole phenomenology and X-ray line profile modeling. It deserves a serious referee rather than a desk reject, but the referee should insist on an explicit polarization analysis. If the emission is genuinely single-polarization, the extraordinary-mode prediction stands as an interesting extreme case; if not, composite profiles including the ordinary mode should replace the headline numbers. I would not cite it in its current form.

Recommendation: send to peer review. Require the authors to state the polarization assumption, provide ordinary-mode or summed profiles, and either derive Eq. (36) or cite a published source. The central claim as written is broader than the physics supports.

Referee Report

2 major / 5 minor

Summary. The paper studies profiled spectral lines from Keplerian rings orbiting regular Bardeen black holes, under the assumption that photons follow null geodesics of the NED effective geometry g̃_{αβ}=L_F g_{αβ}-L_{FF}F_α^λ F_{λβ} given in Eq. (13). It constructs ray-traced line profiles for three magnetic charges, three ring radii, and three observer inclinations, compares them with profiles generated in the Bardeen (or Reissner-Nordström) spacetime geometry, and reports deep red shifts (more than one order of magnitude), line narrowing by factors of 15 to 50, and up to about 10^6 specific-flux suppression for extended disks. A closed-form static redshift ratio R=L(e)/L(o) is derived in Eq. (49), and Table 1 provides quantitative width data for 27 configurations.

Significance. If correct, the predicted NED signature would be an observationally distinctive way to test regular Bardeen black holes using iron-line profiles from SgrA* and AGN disks. The paper has clear strengths: the parameter coverage in Table 1 is systematic; the comparison quantity R in Eq. (49) is a closed-form, parameter-free ratio; and no model parameters are fitted to the target line profiles. However, two load-bearing physical issues affect the interpretation of every numerical result: the frequency-shift formula appears to mix the effective metric with the physical observer frame, and the polarization content of the emitted radiation is not specified even though NED predicts two photon modes. These issues prevent the paper's central claims from being accepted as they stand.

major comments (2)

[§3.1, Eq. (13); §5, Eqs. (39)-(40)] The paper uses a single effective metric for all photons, but the Hadamard discontinuity analysis of Eqs. (9)-(12) yields two photon propagation modes: the extraordinary mode with g̃_{αβ}=L_F g_{αβ}-L_{FF}F_α^λ F_{λβ}, and an ordinary mode propagated by the conformally related metric L_F g_{αβ}, whose null geodesics coincide with those of the Bardeen metric. The emission model in Section 5 is polarization-independent, and no argument is given that only the extraordinary mode is radiated. If both modes are emitted with comparable intrinsic intensity, the ordinary-mode contribution has frequency shifts in the range of the 'spacetime geometry' columns of Table 1 and, because of the g^3 factor in Eq. (39), dominates the total specific flux. The abstract and Conclusions claims of >10x red shift and 15-50x narrowing therefore do not apply to an unpolarized line profile unless a polarization-selection assumption is explicitly stated and justified.
[§5, Eq. (36); §7, Eqs. (47)-(49)] The frequency shift is not a property of the effective metric alone. In a static spacetime, the time component k_t of the wave 1-form is conserved along the ray, and the frequency measured by a physical observer with 4-velocity U is -k_μ U^μ; for a static observer U=(1/√f,0,0,0), the static redshift is √(f_e/f_o), independent of L_F. The factor L(o)/L(e) in Eq. (47) can only arise if the observer's frequency is evaluated using g̃ instead of the physical Bardeen metric, but the paper does not define or justify such an effective-metric frequency. The same problem enters Eq. (36) through the prefactor L(e)/L(o). Since Eq. (36) is used to construct Table 1 and every plotted profile, the numerical predictions are not established as observer-frame line shifts. If the authors intend a different, nonstandard definition of frequency, it must be stated explicitly and its observational relevance demonstrated.

minor comments (5)

[§4-5] The central transfer-function expression in Eq. (36) and the ray-tracing framework are referred to the submitted manuscripts [59] and [73]. The paper should be self-contained: at least a derivation of Eq. (36) and the definitions of the variables entering the Jacobian in Eq. (44) should be included or summarized in an appendix.
[§6.3, Fig. 7] The right column of Fig. 7 is described as the Reissner-Nordström spacetime, whereas Table 1 compares the effective Bardeen result with the Bardeen spacetime geometry. The text should be explicit that these are two different 'spacetime geometry' baselines, so the reader can compare like with like.
[§3.1, Eqs. (9)-(12)] The sentence 'Putting (11) and (12) into (9) and (9)' should read 'into (9) and (10)'.
[Abstract and §6.1] The wording 'for all values of the magnetic charge and the inclination angle' overstates the three discrete values q_m/M=0.05, 0.5, 0.768 and θ_o=30°, 60°, 85° actually computed; a phrase such as 'for the studied values' would be more precise.
[§3.2, Eqs. (20)-(35)] The notation for covariant and contravariant latitudinal momentum components is not consistently defined, and Eq. (35) contains a typo ('where is u_in'). A short list of symbols or a clarification of the index placement would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the NED redshift, narrowing, and flux suppression follow from the in-paper effective-geometry derivation, not from fitted inputs or load-bearing self-citation.

full rationale

The derivation is self-contained. Sec. 3.1 obtains the effective metric (13) from the Hadamard discontinuity conditions (11)-(12), and the Bardeen Lagrangian (8) is stated explicitly. The central redshift ratio R = g_B/g_NLB = L(e)/L(o) (Eq. 49) follows directly from the time-time components of the effective and background metrics (Eqs. 47-48), with no parameter fitted to the target line profiles. The numerical line profiles use only this effective-geometry photon propagation plus a standard disk emissivity ansatz (Eq. 40); no normalization or fit is tuned to produce the claimed >10x red-edge shift, 15-50x narrowing, or 10^6 flux reduction. The authors' submitted self-citations [59,73] supply the ray-tracing framework, but the needed equations (13)-(46) are restated in the paper, so those citations are not load-bearing. The broken placeholder citation in Sec. 3 is a referencing defect, not a circular step, since the effective-geometry condition is rederived in the text. The skeptic's point about the unmodeled ordinary photon mode is a physics-assumption concern about the NED propagation prescription, not a circularity in the paper's own derivation chain.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No parameter is fitted to observed line profiles; the central redshift factor R = L(e)/L(o) is fixed by the Lagrangian and the emitter/observer radii. The ledger contains model parameters and domain assumptions rather than fitted constants.

free parameters (3)

Magnetic charge q_m = 0.05M, 0.5M, 0.768M
Scanned as three representative values, not fitted to data; the abstract's claim of behavior for 'all values' of magnetic charge samples only three points.
Thermal broadening parameter gamma = 10^3
Chosen fixed in Eq. (40); affects the width of the locally emitted line but does not drive the main redshift ratio.
Emissivity power-law index p = not stated in text
Appears in the disk emissivity law in Eq. (40) and affects the extended disk profile shapes and flux, but is never assigned a numerical value.

assumptions (4)

domain assumption Photon rays in NED obey the effective geometry g~_αβ = L_F g_αβ - L_FF F_α^λ F_λβ (Eq. 13) from the Hadamard discontinuity method.
This is the central assumption separating photon motion from spacetime geometry; the paper invokes it at §3.1 without independent verification beyond standard NED references.
domain assumption The Bardeen NED Lagrangian L(F) = (3/(2 q_m^3)) (sqrt(2 q_m^2 F)/(1+sqrt(2 q_m^2 F)))^(5/2) (Eq. 8) generates the Bardeen metric f(r) (Eq. 7).
Taken from Bardeen and Ayon-Beato/Garcia; all computations depend on this specific Lagrangian form.
domain assumption Uncharged orbiting matter follows circular geodesics of the spacetime metric with Keplerian frequency Ω^2 = (r^2 - 2 q_m^2)/(r^2+q_m^2)^(5/2) (Eq. 37), while photons follow the effective geometry.
The two-geometry picture is the physical basis for constructing ring emission and redshift; stated in the Introduction and used in §5.
domain assumption Line emissivity follows I_e = ε_0 r^{-p} exp[-γ(ν_o/g - ν_0)^2] (Eq. 40) with γ=10^3.
Standard disk line profile modeling; p is never specified, which under-specifies the extended disk results.

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Cite this review

Pith. "Pith review of Profiled spectral lines of Keplerian rings orbiting in the regular Bardeen black hole spacetimes." pith.science (2026). https://pith.science/paper/NZ6CMINH

@misc{pith2026190807197,
  author       = {Pith},
  title        = {Pith review of: Profiled spectral lines of Keplerian rings orbiting in the regular Bardeen black hole spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NZ6CMINH}},
  note         = {Machine review of arXiv:1908.07197}
}

read the original abstract

Considering the regular Bardeen black hole spacetimes, we test the observational effects of the general relativistic solutions coupled to non-linear electrodynamics (NED) by studying the photon motion in the effective geometry governed by the spacetime geometry and the NED Lagrangian. We focus our attention to the observationally important case of profiled spectral lines generated by rings radiating in a fixed frequency and orbiting the black hole along circular geodesics of the Bardeen spacetime. Such profiled spectral lines are observed in active galactic nuclei and in microquasars, giving sufficient data for the test of regular black holes. We expect that such radiating rings could arise around the Galaxy central supermassive black hole SgrA*, and the related profiled spectral lines could give important additional information to those obtained by direct observations due to the Event Horizon (GRAVITY) Telescope. We demonstrate that the profiled spectral lines of the radiating rings predict strong signatures of the NED effects on the photon motion -- namely the frequency shift to the red edge of the spectrum, and narrowing of the profile, by more than one order in comparison with the case of the profiles generated purely by the spacetime geometry, for all values of the magnetic charge and the inclination angle of the observer. The specific flux is substantially suppressed and for extended Keplerian disks even the shape of the profiled line is significantly modified due to the NED effect.

Figures

Figures reproduced from arXiv: 1908.07197 by the authors.

**Figure 2.** Ring profiled spectral line generated for three represen [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Ring profiled spectral line generated for three represen [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Ring profiled spectral line generated. Illustration of obse [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Ring profiled spectral line generated. Illustration of obse [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Ring profiled spectral line generated. Illustration of obse [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Profile of spectral lines of radiation with local energy 6 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Plot of the ratio R(qm) to illustrate the effect of magnetic field non-linearity of frequency shift. Emitter is at re = 10M and observer is at ro = 100M. From the plot of R(qm) one can read (see Fig.8) that the ratio is of order of 10 and significantly shifts the energ…

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