Observational constraints on the Kerr and its several single-parameter modified spacetimes using quasi-periodic oscillation data

Jianbo Lu; Jingyang Xu; Mou Xu; Shining Yang; Wenmei Li

REVIEW 3 major objections 6 minor 1 cited by

Observational constraints on the Kerr and its several single-parameter modified spacetimes using quasi-periodic oscillation data

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read QPO data from three microquasars exclude the Kerr limit at 68% confidence for eight of the nine single-parameter modified Kerr spacetimes tested.

desk verdict A useful catalog of epicyclic frequencies and an honest model-comparison table, but the headline Kerr-exclusion claim is contradicted by the paper's own chi-square values. read the letter →

arxiv 2505.08409 v2 pith:FIQQLI6W submitted 2025-05-13 gr-qc astro-ph.CO

Shining Yang , Jianbo Lu , Wenmei Li , Mou Xu , Jingyang Xu This is my paper · ORCID

classification gr-qcastro-ph.CO

keywords quasi-periodicoscillationsmicroquasarsblackholesmodifiedKerrspacetimesrelativisticprecessionmodelKerr-Newmanepicyclicfrequencies

topics Dark Matter

open problems Dark Matter

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

The paper asks whether the spacetime around stellar-mass black holes is exactly Kerr or one of nine single-parameter modifications, using the quasi-periodic oscillation (QPO) frequencies of three microquasars as the observable. Under the relativistic precession model, each observed peak is mapped to a combination of the orbital and epicyclic frequencies of a test particle, and a $\chi^2$ fit constrains the modification parameter, spin, and mass of each geometry. Its central finding is that only the Kerr-Newman charge interval contains zero, while the fitted intervals for the other eight modified spacetimes are positive at 68% confidence, which the authors read as a statistical deviation of the Kerr solution. The conclusion is qualified by model selection: Bayes factors mildly prefer five modified geometries, while the Akaike criterion keeps Kerr as the best model.

What carries the argument

The load-bearing object is the relativistic precession model of QPOs, supplied by reference [71] and encoded in Eq. (27). For each of the ten metrics, the paper computes the orbital frequency $\omega_\phi$ and the radial and latitudinal epicyclic frequencies $\omega_r$, $\omega_\theta$ from linearized geodesic deviations on equatorial circular orbits; under the RP mapping these become functions of the black hole mass, spin, modification parameter, and the orbital radius of the emitting ring. A single global $\chi^2$ compares four observed QPO frequency sets with these model predictions, leaving the four orbital radii as free nuisance parameters. The modification parameters enter through the metric functions, such as the $\Delta$ factors and mass functions that differ from Kerr, so the fit simultaneously constrains the geometry and the microquasar properties.

What would settle it

Inject a known Kerr spacetime with the same masses, spins, and noise levels as the real data, generate four QPO frequency sets under the RP model, and run the paper's $\chi^2$ pipeline. If the best-fit modification parameters for the eight non-KN geometries routinely land away from zero at 68% confidence when the input geometry is exactly Kerr, then the reported exclusion is an artifact of the fitting procedure; if they center on zero, the observed exclusions carry real geometric information.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that QPO data, interpreted through the relativistic precession mapping $\nu_u = \nu_\phi$, $\nu_l = \nu_\phi + \nu_r$, and $\nu_c = \nu_\phi - \nu_\theta$, exclude the Kerr limit for eight of the nine tested single-parameter deviations. The fitted 68% values are $b^* = 0.229^{+0.045}_{-0.034}$ for Bardeen, $Q_1^* = 0.360^{+0.012}_{-0.011}$ for ABG, $l^* = 0.295 \pm 0.019$ for Hayward, $n^* = 0.257^{+0.054}_{-0.025}$ for Kerr-Taub-NUT, $q^* = 0.152 \pm 0.019$ for the braneworld Kerr, $\alpha^* = 0.795 \pm 0.011$ for Kerr-MOG, $Q_3^* = 0.361^{+0.028}_{-0.024}$ for Kerr-Sen, and $k^* = 0.019^{+0.002}_{-0.003}$ for the perfect-fluid dark matter geometry, all strictly positive. Only the Kerr-Newman charge, $Q_2^* = -0.004 \pm 0.078$, spans negative and positive values and therefore contains the Kerr case at zero. The same fits yield masses for the three microquasars that are mostly consistent with optical/near-infrared dynamical measurements, while the inferred spins are systematically lower than iron-line/continuum-fitting values, a discrepancy the paper leaves open.

Load-bearing premise

The load-bearing assumption is that each observed QPO peak is produced by the relativistic precession of a test particle on an equatorial circular orbit, with the specific identifications $\nu_u=\nu_\phi$, $\nu_l=\nu_\phi+\nu_r$, and $\nu_c=\nu_\phi-\nu_\theta$; if the peaks have a different physical origin, or the orbits are not equatorial circular test-particle orbits, the fitted modification parameters lose their geometric meaning.

Editorial extensions

If this is right

If the RP interpretation is right, microquasar QPO data currently favor a non-Kerr geometry in eight of nine tested one-parameter extensions, with Kerr-Newman as the sole exception that keeps the Kerr limit inside its 68% interval.
The fitted masses for the three microquasars agree with independent dynamical measurements in most models, which supports the internal consistency of the radii and frequency assignments used in the fit.
The systematic gap between QPO-inferred spins and iron-line/continuum-fitting spins indicates that at least one of these spin diagnostics is model-dependent; the paper notes this could mean the continuum method overestimates spin or the QPO model is incomplete.
Bayes factors and AIC give opposite rankings, with a slight preference for five modified models in the Bayes analysis but Kerr as the best model in the AIC analysis, so current QPO data cannot decisively choose among the ten geometries.

Reading between the lines

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

Beyond the paper: the shared positive offset across eight unrelated geometries suggests the RP model's radial-frequency prediction may carry a systematic bias, so the geometric conclusion should be tested by injecting synthetic Kerr data into the same pipeline before treating it as physical.
Beyond the paper: the paper quotes all exclusions at the 68% level; the 95% contours already plotted in Figures 3 and 4 would show how many of the eight exclusions survive a stricter threshold.
Beyond the paper: because the four orbital radii are free nuisance parameters, the model has built-in flexibility; adding a prior on the radii or fitting more QPO sets per source could either sharpen or dissolve the positive-parameter pattern.

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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

Short version: this is a carefully built catalog of epicyclic frequencies for Kerr and nine single-parameter generalizations, fitted to four QPO datasets, and the comparison tables are honest. But the headline claim—that eight of nine models exclude Kerr at 68% CL—is contradicted by the paper's own chi-square values. That claim should not survive review.

What the paper does well: the Appendix works out explicit omega_phi, omega_r, omega_theta expressions for all ten spacetimes; that is tedious and useful. The simultaneous fit of ten models to the same QPO data with a consistent relativistic-precession mapping is a reasonable way to compare models. The spin and mass results for Kerr line up with earlier QPO-based estimates, and the cited comparisons to Bardeen, KN, KTN, BK, MOG, and PFDM constraints are real. The AIC and Bayes tables are reported straightforwardly, and the AIC column actually keeps Kerr as the best model.

The soft spot is load-bearing. Equation (27) has 11 fitted parameters for 11 data points, so the reported chi-square-minimum values are not evidence of fit quality. More decisively, every modified model reduces to Kerr at the modification parameter Delta*=0. The paper's own Table II gives chi2_min(Kerr)=0.976 and chi2_min(model) between 0.424 and 0.712. For a single parameter of interest, a 68% profile-likelihood interval is the set of Delta* with chi2(Delta*)-chi2_min <= 1. Since chi2(Delta*=0)=0.976, the difference for every model is below 1, so zero lies inside the 68% region for every model. The positive-definite intervals in Table II are not standard profile-likelihood intervals; they look like slices with other parameters fixed at the modified best fit. The abstract's "statistical deviation of the Kerr solution" is unsupported by the paper's own numbers and is also in tension with its own Bayesian and AIC results, all of which are inconclusive.

The RP-model assumption is another limitation, though the authors are upfront about it. If the QPO peaks come from a different mechanism or from non-circular orbits, the fitted parameters lose their geometric meaning.

Bottom line: the audience is QPO phenomenology and modified-gravity constraints, and the frequency catalog will be useful. But the paper deserves a serious referee, not publication in this form. I would send it to peer review rather than desk reject, with the expectation of major revision: replace the interval construction with a proper profile likelihood, state the degree-of-freedom situation honestly, and soften the conclusion to "no significant preference." I would not cite the Kerr-exclusion claim.

Referee Report

3 major / 6 minor

Summary. The paper derives test-particle orbital and epicyclic frequencies in Kerr and nine single-parameter modified Kerr spacetimes (Bardeen, ABG, Hayward, KN, KTN, BK, Kerr-MOG, Kerr-Sen, PFDM), then uses the relativistic precession model and QPO measurements from three microquasars to fit each spacetime through the chi-squared function in Eq. (27). It reports 68% confidence intervals for the modification parameter of each model, the spins and masses of the three sources, and the four orbital radii. The central conclusion, repeated in the abstract and Section VI, is that all modified spacetimes except Kerr-Newman require a positive-definite modification parameter at 68% CL, 'demonstrating statistical deviation of the Kerr solution.' A Bayes-factor and AIC comparison is also performed; the AIC actually identifies Kerr as the best model, and all Bayes factors are inconclusive.

Significance. If the 68% exclusion claim were correct, it would be a significant observational challenge to the Kerr paradigm and a useful demonstration that QPO data can discriminate among black hole spacetimes. The paper also provides a convenient compilation of epicyclic-frequency formulas in Appendix A and a transparent statement of the data and fitting function, which are useful for follow-up work. However, the central inference is not reliable: the fit has zero degrees of freedom, and the claimed 68% intervals are inconsistent with the profile-likelihood values implied by the paper's own Table II and with its own model-selection results. The significance of the paper therefore rests on a statistical artifact rather than on evidence.

major comments (3)

[IV, Eq. (27), Table I] The chi-squared function in Eq. (27) contains 11 measured frequencies in Table I but also 11 fitted parameters: the modification parameter Delta*, the three spins a*_p, the three masses M*_p, and the four radii r1, r1', r2, r3. The fit therefore has zero degrees of freedom, and the reported chi2_min values between 0.4 and 1.0 are a measure of the model's flexibility, not of its predictive success. In this regime the 68% intervals in Table II and the associated statements of 'stringent constraints' in Section IV are not statistically meaningful; a saturated fit cannot yield valid confidence intervals through the conventional asymptotic arguments used here.
[Table II and Section VI] For every modified metric, Delta*=0 reduces the model exactly to Kerr. Hence the minimum of chi2(Delta*) over the remaining parameters at Delta*=0 is the Kerr value chi2_min = 0.976 in Table II. Using the paper's own numbers, the improvement over Kerr is 0.404 (Bardeen), 0.548 (ABG), 0.534 (Hayward), 0.547 (KN), 0.552 (KTN), 0.479 (BK), 0.496 (Kerr-MOG), 0.277 (Kerr-Sen), and 0.264 (PFDM). For a single parameter of interest, the 68% profile-likelihood interval is defined by Delta-chi2 <= 1, so Delta*=0 lies inside the 68% interval for every model. The positive-definite intervals reported in Table II therefore cannot be profile-likelihood intervals; they appear to be conditional slices taken at the best-fit values of the nuisance parameters. The conclusion in the abstract and Section VI that eight spacetimes 'mandate positive-definite parameters at 68% CL' is not supported and is in direct tension with the Bayes factors (all |ln R| < 0.11 in Table IV) and the AIC analysis in Table V, which keeps Kerr as the best model.
[Section IV before Eq. (27)] The analysis assumes the relativistic precession mapping nu_u = nu_phi, nu_l = nu_phi + nu_r, and nu_c = nu_phi - nu_theta exactly, while the four radii are free nuisance parameters. Under this assumption the fitted Delta* values are only interpretable as spacetime parameters; if the QPO peaks have a different physical origin, or if the frequencies do not correspond to equatorial circular test-particle orbits, every reported modification-parameter constraint loses its geometric meaning. Given the ongoing debate on QPO mechanisms, the paper should at least demonstrate robustness to alternative QPO models or explicitly frame all constraints as conditional on the RP model. This limitation is load-bearing for the abstract's physical claim, but the paper does not address it.

minor comments (6)

[Section V title] The heading 'Model camparison' should be 'Model comparison'.
[Section II.D title] The heading 'Rotating BHs modiffed by dark matter' should be 'Rotating BHs modified by dark matter'.
[Figure 1] The label 'Badeen' in the first panel should be 'Bardeen'.
[Section IV, discussion near [143]] The text 'GR0 J1655-40' should be 'GRO J1655-40'.
[Table III] The best-fit radii are listed without uncertainties, so the reader cannot assess whether the four radii are pinned by the data or merely absorbed by the saturated fit.
[Section V.A, Eq. (29)] The 'marginal likelihood' values in Table II are not reproducible because the priors used in Eq. (29) are not specified; the Bayes factors in Table IV depend on those priors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QPO fits and model comparisons are data-driven and do not reduce to their inputs.

full rationale

The paper's derivation chain is: define ten stationary axisymmetric metrics (each reducing to Kerr when the modification parameter vanishes), derive the orbital and epicyclic frequencies from the geodesic deviation equations in Appendix A, impose the relativistic precession mapping in Eq. (27), fit the modification parameters, spins, masses, and orbital radii to the four QPO data sets, and then report confidence intervals and Bayes/AIC model comparisons. No step uses the target conclusion as an input. The modification parameters are free parameters fitted to the data, not quantities defined by the data labels; the radii in Table III are fitted nuisance parameters, not imposed from the observed QPO frequencies; and the Bayes factor and AIC are computed from the same fitted likelihoods, which is standard model comparison rather than circular prediction. The self-citations in the reference list are contextual and do not carry the central derivation, and the relativistic precession framework is taken from prior external literature. The apparent tension between the positive-definite 68% parameter intervals and the inconclusive Bayes factors (all |ln R| < 0.11) or the AIC preference for Kerr is a statistical consistency issue about how the intervals were constructed, not a circularity in which the fitted result is equivalent to the input. Because no specific reduction (Eq. X = Eq. Y by construction, or fitted parameter renamed as a prediction) can be exhibited from the paper's own equations, the appropriate circularity score is 0.

Assumptions & free parameters 12 free parameters · 4 assumptions · 0 invented entities

The fit introduces one global modification parameter per spacetime, three spins, three masses, and four orbital radii, for eleven fitted numbers against eleven data points. No new particles or fields are postulated; the modified spacetimes and the RP model are taken from prior literature. The most consequential free parameters for the central claim are the nine modification parameters, whose 68% intervals drive the 'deviation from Kerr' statement.

free parameters (12)

b* (Bardeen monopole charge) = 0.229 (+0.045, -0.034)
Global modification parameter of the rotating Bardeen metric; fitted to QPO data via Eq. (27); the 68% interval excludes zero.
Q1* (ABG charge) = 0.360 (+0.012, -0.011)
Global modification parameter of the rotating ABG metric; fitted to QPO data.
l* (Hayward length) = 0.295 +/- 0.019
Global modification parameter of the rotating Hayward metric; fitted to QPO data.
Q2* (Kerr-Newman charge) = -0.004 +/- 0.078
Global modification parameter of the Kerr-Newman metric; includes zero within its 68% interval.
n* (Kerr-Taub-NUT parameter) = 0.257 (+0.054, -0.025)
Global modification parameter of the Kerr-Taub-NUT metric; fitted to QPO data.
q* (Braneworld Kerr tidal charge) = 0.152 +/- 0.019
Global modification parameter of the Braneworld Kerr metric; fitted to QPO data.
alpha* (Kerr-MOG deformation) = 0.795 +/- 0.011
Global modification parameter of the Kerr-MOG metric; fitted to QPO data.
Q3* (Kerr-Sen charge parameter) = 0.361 (+0.028, -0.024)
Global modification parameter of the Kerr-Sen metric; fitted to QPO data.
k* (PFDM dark matter parameter) = 0.019 (+0.002, -0.003) in Table II; text repeats 0.019 (+0.024, -0.003)
Global modification parameter of the PFDM metric; fitted to QPO data.
a*_1, a*_2, a*_3 (microquasar spins) = Kerr best fit: 0.287, 0.283, 0.150; values shift per spacetime in Table II
Each source has a free spin parameter used in the QPO frequency prediction.
M*_1, M*_2, M*_3 (microquasar masses) = Kerr best fit: 5.307, 9.745, 7.774 solar masses; values shift per spacetime in Table II
Each source has a free mass parameter used in the QPO frequency prediction.
r1*, r1'*, r2*, r3* (QPO orbital radii) = Kerr: 5.685, 5.582, 5.662, 6.895; values in Table III
Four free orbital radii correspond to the four observed QPO sets in Eq. (27).

assumptions (4)

domain assumption Circular equatorial geodesic motion, with small perturbations, gives the epicyclic frequencies via Eqs. (23)-(25).
Used for all ten spacetimes; standard test-particle approximation for QPOs, but neglects fluid pressure, magnetic fields, and disk effects.
domain assumption Relativistic precession mapping: HFQPO upper equals nu_phi, lower equals nu_phi + nu_r, and LFQPO equals nu_phi - nu_theta.
Load-bearing identification adopted in Section IV; invalid if the QPO mechanism differs.
domain assumption The rotating Bardeen, ABG, Hayward, and other modified metrics are treated as valid black hole spacetimes.
Section II; several are generated via non-complexification and are not proven exact solutions of the underlying field theories.
domain assumption The statistical model uses a Gaussian likelihood with chi-square from Eq. (27), and Bayesian evidence is computed with implicit priors.
Sections IV and V; priors and integration method are not stated, and asymmetric data errors are not propagated.

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Cite this review

Pith. "Pith review of Observational constraints on the Kerr and its several single-parameter modified spacetimes using quasi-periodic oscillation data." pith.science (2026). https://pith.science/paper/FIQQLI6W

@misc{pith2026250508409,
  author       = {Pith},
  title        = {Pith review of: Observational constraints on the Kerr and its several single-parameter modified spacetimes using quasi-periodic oscillation data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FIQQLI6W}},
  note         = {Machine review of arXiv:2505.08409}
}

abstract

This paper investigates the dynamical effects of particles moving in the Kerr spacetime and its nine single-parameter modified spacetimes, including Bardeen, Ayon-Beato and Garcia (ABG), Hayward, Kerr-Newman (KN), Kerr-Taub-NUT (KTN), Braneworld Kerr (BK), Kerr-MOG, Kerr-Sen, and Perfect Fluid Dark Matter (PFDM) black holes. Using quasi-periodic oscillation (QPO) observational data, we constrain the free parameters of the ten spacetimes through $\chi^2$ analysis under the relativistic precession model of QPO. We constrain the modification parameters for the nine single-parameter modified spacetimes and provide the spin and mass ranges of three microquasars within the ten spacetime models (including Kerr) at the $68\%$ confidence level (CL). The results demonstrate that, at the $68 \%$ CL, the QPO data impose stringent constraints on the free parameters, as evidenced by the narrow confidence intervals. Among them, only the KN spacetime yields a modification parameter constraint spanning both negative and positive values (encompassing the Kerr case at zero). In contrast, all other tested geometries mandate positive-definite parameters at $68 \%$ CL, demonstrating statistical deviation of the Kerr solution. This highlights the significance of exploring modifications to the Kerr spacetime. Finally, we evaluate the spacetime models using the Bayes factor and the Akaike Information Criterion (AIC). Based on the current QPO observational data, the Bayesian factor analysis indicates that the ABG, Hayward, KN, BK, and Kerr-MOG spacetime have a slight advantage over the Kerr solution, while the Bardeen, KTN, Kerr-Sen, and PFDM spacetime are somewhat inferior to the Kerr model. In contrast, the AIC analysis shows that the Kerr spacetime remains the optimal model under the current QPO data.

Figures

Figures reproduced from arXiv: 2505.08409 by the authors.

**Figure 2.** FIG. 2: The parameter plots of the spin parameters [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3: The parameter plots of the modification parameters and the spin parameters [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: The parameter plots of the modification parameters and the spin parameters [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: FIG. 5: In the Kerr spacetime and its nine single-parameter modified spacetimes, the constraint ranges for the spin [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Constraints on extra charges in dyonic Kerr-Newman-Kasuya-Taub-NUT black hole from the observations of quasi-periodic oscillations
astro-ph.HE 2025-07 conditional novelty 4.0 of 10

Using QPO data from five X-ray binaries, the authors place upper limits on electric, magnetic, and NUT charges of a dyonic Kerr black hole, with a tentative nonzero NUT parameter in GRS 1915+105.

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