Circular-orbit dynamics and QPO constraints in static Einstein--scalar--Gauss--Bonnet black holes

Anvar Kurbaniyazov; Bekzod Rahmatov; Faisal Javed; Javlon Rayimbaev; Otabek Umarov; Sardor Murodov

REVIEW 2 major objections 6 minor 102 references

Circular-orbit dynamics and QPO constraints in static Einstein--scalar--Gauss--Bonnet black holes

T0 review · 2 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Twin-peak QPO data from four black-hole candidates are compatible with static Einstein–scalar–Gauss–Bonnet gravity, but the EsGB deformation parameter is not measured: its posterior mirrors the prior.

desk verdict An honest and useful negative result: QPO data do not constrain the EsGB deformation parameter p, though the quantitative claim inherits the continued-fraction metric's accuracy limits near large p. read the letter →

arxiv 2607.19805 v1 pith:7BGTJ3VX submitted 2026-07-22 gr-qc

Sardor Murodov , Bekzod Rahmatov , Otabek Umarov , Anvar Kurbaniyazov , Faisal Javed , Javlon Rayimbaev This is my paper · ORCID

classification gr-qc MSC 83C5783C1083D05 PACS 04.70.-s

keywords Einstein-scalar-Gauss-BonnetgravityblackholescircularorbitsepicyclicfrequenciesHF-QPOsrelativisticprecessionmodelcontinued-fractionmetricstrong-field

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 studies timelike circular motion and high-frequency quasi-periodic oscillations (HF-QPOs) around static black holes in Einstein–scalar–Gauss–Bonnet (EsGB) gravity, using a one-parameter continued-fraction metric labelled by deformation p. It shows that the model's orbital and radial epicyclic frequencies can reproduce the twin-peak QPO pairs of XTE J1550–564, GRO J1655–40, GRS 1915+105, and M82 X-1 within their uncertainties. However, a controlled prior-sensitivity analysis finds that the marginal posterior of p nearly coincides with the adopted prior (posterior-to-prior width ratios R68 ≈ 0.985–0.998, median shifts |Sp| < 0.03, and a flat QPO profile likelihood). The paper therefore concludes that these data do not measure p; the reported intervals are model-dependent compatibility regions. This clarifies what a static EsGB QPO analysis can and cannot claim, and sets a baseline for rotating and multi-observable tests.

What carries the argument

The central object is a continued-fraction metric, N(x) = x A(x) and B(x), with coefficients ϵ(p), a_i(p), b_i(p) fitted from numerical EsGB solutions; it reduces the spacetime to the parameter set (M, p, r). The load-bearing mechanism is the relativistic precession model, which maps the upper QPO to the orbital frequency νφ and the lower to the periastron-precession combination νφ − νr. In this static geometry the vertical and orbital frequencies coincide, so the radial epicyclic frequency — fixed by the second derivative of the effective potential — carries the main model-level sensitivity to p.

What would settle it

Measure twin-peak QPO frequencies with high precision for one source together with an independent dynamical mass (for example, from radial-velocity monitoring of GRO J1655–40) and compute the profile likelihood over p. If the profile develops a minimum with Δχ² well above zero inside 0 ≤ p ≤ 1, or if the posterior contracts by substantially more than the R68 ≈ 1 seen here, the paper's conclusion that QPO-only data leave p unconstrained would be contradicted.

Watch

Extended reading notes

Core claim

Within the quadratic-coupling, Schwarzschild-connected branch, the static EsGB black hole can be represented as a controlled one-parameter deformation of Schwarzschild, with the dimensionless parameter p governing the near-horizon geometry. The paper derives the effective potential, circular-orbit energy and angular momentum, characteristic radii (photon sphere, marginally bound orbit, ISCO), and the orbital and radial epicyclic frequencies. Applying the relativistic precession identification νU = νφ and νL = νφ − νr to four observed twin-peak QPO sources, it finds the observed pairs are reproduced within uncertainties, with fitted radii clustered near r/M ≈ 6.7–6.8. But because only two fre

Load-bearing premise

The load-bearing premise is that the fitted continued-fraction metric with coefficients ϵ(p), a_i(p), b_i(p), valid to second order and taken from numerical solutions, accurately represents the true quadratic-coupling EsGB black hole across 0 ≤ p ≤ 1, including near p ≈ 0.8 where convergence is slower; if the parametrization drifts from the actual solution, the constraints apply only to an approximate metric.

Editorial extensions

If this is right

All four observed twin-peak QPO pairs (XTE J1550–564, GRO J1655–40, GRS 1915+105, M82 X-1) can be reproduced within their uncertainties by the static EsGB relativistic-precession model.
The radial epicyclic frequency and its combinations (lower RP frequency, 3:2 resonance condition) are the timing observables most sensitive to the EsGB deformation at the model level.
Current QPO-only data do not prefer any value of p in [0,1]; treating reported p intervals as measurements would overstate the information in the data.
The fitted mass M and radius r/M remain localized and strongly anticorrelated, so joint timing constraints on mass and radius coexist with an unconstrained deformation parameter.
The static analysis supplies a baseline; robust constraints on EsGB gravity will require rotation and independent mass, spin, shadow, or ringdown information.

Reading between the lines

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

If the flat profile likelihood persists with larger QPO samples, then any single-spacetime RP-model fit to two frequencies is degenerate along a valley in (M, p, r); adding one independent mass or radius measurement should slice that valley and may convert compatibility into a p constraint.
The same prior-tracking pattern probably afflicts other parametrized-metric QPO studies that fit three parameters to two frequencies; prior-sensitivity tests like the R68 ratio presented here could be applied routinely elsewhere.
The nonmonotonic shifts of the ISCO and photon sphere near p ≈ 0.8 hint that the strong-coupling regime may produce qualitatively different signatures once the metric parametrization is validated there; this is worth probing with an exact or higher-order numerical solution.
Switching the QPO identification from relativistic precession to the 3:2 epicyclic resonance will change which radius is inferred; since νθ = νφ in the static case, the resonance model provides a directly testable alternative prediction for the same sources.

Share X Bluesky LinkedIn Reddit HN

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 careful, honest paper whose main result is negative—twin-peak QPO data from four sources, interpreted through the relativistic precession model, do not constrain the EsGB deformation parameter p. The claim is well supported by the uniform-prior MCMC and the flat profile likelihood (max Δχ² ~ 2×10⁻⁶), and the posterior-to-prior width ratios R68 ≈ 0.985–0.998 make the point quantitatively. I believe the conclusion.

What's new: a source-by-source MCMC with prior-sensitivity and profile-likelihood diagnostics applied to the continued-fraction static EsGB metric of Konoplya et al. The derivations in Secs. IV–V are standard but clean; Schwarzschild limits reproduce correctly, and the frequency plots are useful. The paper is explicitly cautious about interpreting its own intervals as compatibility regions rather than measurements, which is refreshing.

Where it gets soft: 1. The "controlled" prior-sensitivity comparison in Sec. VI.B uses Gaussian localization priors on M and r/M whose centers and widths are derived from the same data via a preliminary frequency-matching scan. So that comparison is partly tautological—it demonstrates that a prior built from the data is followed by the posterior. The uniform-prior run is the real evidence, and it does the job. 2. The metric itself is an inherited continued-fraction fit from Ref. [102]. The stress-test concern is legitimate: the paper notes in Secs. III and V.E that the fit converges slowly near p→1, and the uniform-prior posterior samples over the full [0,1] interval. A few-percent error in the metric functions at p≳0.7 could shift the RP track by more than the observational uncertainties, which means the max Δχ² ~ 2×10⁻⁶ number is conditioned on an unvalidated fit. The qualitative degeneracy of three parameters vs. two frequencies should survive for any nearby smooth metric, so the broad conclusion is probably robust; but the quantitative flatness and compatibility regions are claims about the fitted representation, not directly about EsGB gravity. 3. No code or data are released, which makes it harder to verify the MCMC details, though the diagnostics reported are thorough.

Who it's for: people fitting QPOs to parametrized black-hole metrics, especially modified-gravity models. It's a useful cautionary baseline and a good example of why posterior contraction must be checked against the prior. It deserves a serious referee—the methodology is sound, the negative result is important for the subfield, and the limitations are honestly stated. I'd recommend engaging with it; the revision should address the near-p→1 accuracy issue and clarify that the controlled-prior comparison is not independent of the data.

Referee Report

2 major / 6 minor

Summary. The paper studies circular timelike geodesics and high-frequency quasi-periodic oscillations (HF-QPOs) in static Einstein–scalar–Gauss–Bonnet (EsGB) black holes, using the continued-fraction metric parametrization of Ref. [102] with a single deformation parameter p on the quadratic-coupling, Schwarzschild-connected branch. It derives the effective potential, circular-orbit energy and angular momentum, characteristic radii, and the orbital and radial epicyclic frequencies, then applies the relativistic precession (RP) model to twin-peak QPO data from four sources. A source-by-source MCMC analysis shows that the observed frequency pairs can be reproduced within their uncertainties, but that the marginal posterior of p closely follows the adopted prior for both uniform and truncated-Gaussian choices: R68≈0.985–0.998 and max Δχ²_prof≲2×10⁻⁶. The paper concludes that the QPO data provide compatibility regions rather than an independent measurement of the EsGB deformation.

Significance. If the inherited continued-fraction metric is accurate, the paper's central negative result is a useful and honest contribution: it demonstrates quantitatively that twin-peak QPO data, under the RP identification, do not statistically identify p in this static EsGB model, and it makes the prior dependence explicit through controlled sensitivity runs and a flat profile likelihood. The geodesic derivations are standard, the Schwarzschild limits are correctly recovered, and the MCMC diagnostics (acceptance fractions, autocorrelation times, burn-in checks) are thorough. The main strength is that the paper does not overclaim a measurement; it explicitly labels the intervals as model-dependent compatibility regions. The significance is limited by the fact that all quantitative claims are computed within the fitted metric representation of Ref. [102] and within ad hoc source-specific priors, so the headline numbers are conditional on those choices.

major comments (2)

[II, Eqs. (16)–(20); V.E; VI.E] The entire quantitative inference is computed inside the continued-fraction metric inherited from Ref. [102]. The paper itself notes (Secs. III and V.E) that convergence is slower near p→1 and that large couplings are likely unstable, yet the uniform-prior analysis places posterior mass over the full [0,1] interval. The headline values R68≈0.985–0.998 and max Δχ²_prof≲2×10⁻⁶ are therefore conditional on the accuracy of this fitted metric in the strong-coupling regime. A few-percent error in N(r) or B(r) at p≳0.7 could shift the RP track ν_L(ν_U) by more than the 3–5 Hz observational uncertainties. Please validate the fitted coefficients against the numerical EsGB solutions, propagate the fitting error into the profile statistic, or restrict the physical domain to the range where the fit is verified. Without this, the flat-profile conclusion is a statement about an approximate metric, not
[VI.B, Table IV] The Gaussian localization priors on M and r/M are centered on values obtained from a preliminary frequency-matching scan, and the top-hat supports in Table IV are not anchored to independent dynamical mass measurements (e.g., XTE J1550–564: M∈[6.48,6.88] M☉; GRO J1655–40: M∈[4.02,4.16] M☉). This makes the statement that all four observed pairs are reproduced within their uncertainties partly tautological: the nuisance position is tuned to the data before the fit. The negative conclusion about p is likely robust because three parameters are fit to two frequencies, but the quantitative compatibility regions and the flat profile are measured over these chosen supports. Please repeat the analysis with priors based on published mass estimates, or explicitly report how the required effective masses compare with independent mass measurements. The text acknowledges the issue in Sec. VI.B but doe

minor comments (6)

[IV.A, Eq. (38)] The reduction from Eq. (6) to Eq. (38) uses a0=b0=0; this should be stated explicitly to avoid confusion.
[V.C] The epicyclic-resonance model is introduced and Table II is presented, but this model is not used in the observational analysis. If it is included only for completeness, say so explicitly.
[Figure 6] The upper-left panel has a formatting artifact in the axis label ('+1.199e1') and the tick labels are hard to read; please clean up the figure.
[Table IV] The notation N[0,1](0.495,0.296²) for the truncated Gaussian prior is not defined in the text; please define it.
[Table VII] The Gaussian-prior posterior medians for GRS 1915+105 and M82 X-1 are both reported as 0.4860; check whether this is a rounding artifact or a real coincidence.
[VI.E] The claim max Δχ²_prof≲2×10⁻⁶ should be accompanied by a plot or table of χ²_prof(p) and the grid resolution used, so that the flatness can be inspected directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper explicitly demonstrates the underdetermination of p, and its load-bearing metric input is an external fitted parametrization, not a self-cited or self-defined result.

full rationale

The paper's central claim is negative: the two observed QPO frequencies leave the EsGB deformation parameter p unconstrained. This claim is supported by an explicit prior-sensitivity analysis (uniform and truncated-Gaussian priors), posterior-to-prior width ratios R68 = 0.985-0.998, and a flat QPO-only profile likelihood with max Delta chi^2_prof < 2e-6 (Sec. VI.E). None of these diagnostics is circular: the likelihood (Eqs. 67-68) depends on the observed frequencies and the geodesic RP model; the uniform prior is not derived from the data; and the flat profile is a genuine non-identifiability result rather than a fitted prediction. The metric coefficients epsilon(p), a_i(p), b_i(p) (Eqs. 16-20) are taken transparently from external Ref. [102], which fitted them to numerical EsGB solutions; this is an accuracy/validity assumption, and the paper itself flags slower convergence near p->1 (Secs. III, V.E), but it is not a circular reduction because the QPO data and the EsGB field equations are independent inputs. The Gaussian-prior sensitivity run is weakened by the fact that the Gaussian prior's mean and width were obtained from a preliminary frequency-matching scan over the same QPO data (Sec. VI.B), but the paper discloses this and bases its conclusion primarily on the data-independent uniform prior and profile likelihood. No load-bearing self-citation, no uniqueness claim imported from the authors, and no renamed empirical pattern were found; the paper consistently describes its intervals as 'model-dependent compatibility regions' rather than measurements.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the trustworthiness of the fitted continued-fraction metric (an external numerical fit), on the RP-model identification, and on the chosen priors. There are no new particles or forces; the only fitted quantities are (M, p, r) plus the localization hyperparameters.

free parameters (4)

p (EsGB deformation parameter) = Posterior median ≈0.49–0.50, 68% interval ≈0.16–0.84 for uniform prior; values essentially track prior
Fitted to QPO data; the paper shows the posterior is flat over [0,1] and does not identify a preferred value.
M (black-hole mass) = e.g., 6.67 M_sun (XTE J1550–564), 4.09 M_sun (GRO J1655–40), 11.01 M_sun (GRS 1915+105), 358 M_sun (M82 X-1)
Fitted with a Gaussian localization prior centered via a preliminary frequency-matching scan; effectively the prior sets the value.
r/M (orbital radius) = Posterior median ≈6.73–6.82 for the four sources
Fitted with a Gaussian localization prior; strongly anticorrelated with M.
Gaussian localization hyperparameters for M and r/M = Listed per source in Table IV (e.g., XTE J1550–564: μ_M=6.669, σ_M=0.090; μ_r=6.754, σ_r=0.053)
Chosen/fitted from a preliminary frequency-matching scan over the same QPO data; they act as regularization rather than independent astrophysical measurements.

assumptions (6)

domain assumption The continued-fraction coefficients ϵ(p), a1(p), a2(p), b1(p), b2(p) of Eqs. (16)–(20), taken from Ref. [102], accurately represent the numerical quadratic-coupling EsGB black-hole solutions.
All metric functions, geodesics, and frequencies are computed from these fitted rational functions; the paper does not re-derive or validate them, and notes convergence issues near p≈1.
domain assumption The relativistic precession identification ν_U = ν_ϕ, ν_L = ν_ϕ − ν_r (Eq. 58) correctly maps observed twin-peak QPOs to geodesic frequencies.
Standard in the QPO literature but not independently justified; the paper acknowledges the physical mechanism of HF-QPOs is debated.
domain assumption The observed QPO frequencies and uncertainties in Table III are correct and arise from the relativistic-precession mechanism.
Data are taken from published references and used as phenomenological inputs.
domain assumption The Schwarzschild-connected quadratic-coupling branch (f(ϕ)=ϕ^2) is the relevant EsGB model.
Restricts to one specific branch; other coupling families or branches would give different metric coefficients.
domain assumption Asymptotic flatness with β=γ=1 (a0=b0=0) holds for the chosen EsGB family.
Inherited from Ref. [102]; the paper uses this to fix a0=b0=0 and the asymptotic mass relation.
standard math Standard geodesic/effective-potential formalism for static spherically symmetric spacetimes.
The derivations of Veff, E_c, L_c, and epicyclic frequencies are standard and verified by Schwarzschild limits.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Circular-orbit dynamics and QPO constraints in static Einstein--scalar--Gauss--Bonnet black holes." pith.science (2026). https://pith.science/paper/7BGTJ3VX

@misc{pith2026260719805,
  author       = {Pith},
  title        = {Pith review of: Circular-orbit dynamics and QPO constraints in static Einstein--scalar--Gauss--Bonnet black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BGTJ3VX}},
  note         = {Machine review of arXiv:2607.19805}
}

abstract

Einstein--scalar--Gauss--Bonnet (EsGB) gravity provides a physically motivated framework for testing strong-field deviations from the Schwarzschild geometry through scalar hair. We study neutral-particle circular motion and high-frequency quasi-periodic oscillations (HF-QPOs) in static EsGB black holes described by a continued-fraction metric with a single dimensionless deformation parameter \(p\) on the Schwarzschild-connected quadratic-coupling branch. We determine the effective potential, circular-orbit energy and angular momentum, characteristic radii, and orbital and radial epicyclic frequencies, and apply the relativistic precession model to twin-peak QPO data from XTE J1550--564, GRO J1655--40, GRS 1915+105, and M82 X-1. A source-by-source Markov chain Monte Carlo analysis shows that the observed frequency pairs can be reproduced within their uncertainties and that the radial epicyclic frequency carries the main model-level sensitivity to \(p\). However, a controlled prior-sensitivity analysis using uniform and truncated Gaussian priors finds that the marginal posterior of \(p\) closely follows the adopted prior for all four sources. This reflects the intrinsic underconstraint of fitting three correlated parameters \((M,p,r)\) to two measured frequencies. The inferred intervals therefore represent model-dependent compatibility regions rather than an independent measurement or preferred value of the EsGB deformation. The static results provide a baseline for future rotating and multi-observable tests.

Figures

Figures reproduced from arXiv: 2607.19805 by the authors.

**Figure 1.** FIG. 1. Radial behaviour of the metric functions in the EsGB black hole spacetime and their deviations from the Schwarzschild [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** FIG. 2. Effective potential for timelike equatorial motion [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Specific energy [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Dependence of the specific energy [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. Characteristic radii of the static EsGB black hole as functions of the deformation parameter [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Orbital and epicyclic frequency profiles, RP-model tracks, and the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7. Marginalized one- and two-dimensional posterior distributions of [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: FIG. 8. Controlled prior-sensitivity analysis for the EsGB deformation parameter [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

102 extracted references · 3 linked inside Pith

[102]

R. A. Konoplya, T. D. Pappas, and A. Zhidenko, Phys- ical Review D101, 044054 (2020)

2020
[1]

Einstein, Sitzungsberichte der K"oniglich Preussis- chen Akademie der Wissenschaften , 844 (1915)

A. Einstein, Sitzungsberichte der K"oniglich Preussis- chen Akademie der Wissenschaften , 844 (1915)

1915
[2]

Schwarzschild, Sitzungsberichte der K"oniglich Preussischen Akademie der Wissenschaften , 189 (1916)

K. Schwarzschild, Sitzungsberichte der K"oniglich Preussischen Akademie der Wissenschaften , 189 (1916)

1916
[3]

R. P. Kerr, Physical Review Letters11, 237 (1963)

1963
[4]

Carter, Physical Review174, 1559 (1968)

B. Carter, Physical Review174, 1559 (1968)

1968
[5]

J. M. Bardeen, W. H. Press, and S. A. Teukolsky, in Black Holes(Gordon and Breach, 1973) pp. 241–289

1973
[6]

Chandrasekhar,The Mathematical Theory of Black Holes(Oxford University Press, 1983)

S. Chandrasekhar,The Mathematical Theory of Black Holes(Oxford University Press, 1983)

1983
[7]

R. M. Wald,General Relativity(University of Chicago Press, 1984)

1984

Show all 102 references

[8]

C. M. Will, Living Reviews in Relativity17, 4 (2014)

2014
[9]

Bertiet al., Classical and Quantum Gravity32, 243001 (2015)

E. Bertiet al., Classical and Quantum Gravity32, 243001 (2015)

2015
[10]

Cardoso and P

V. Cardoso and P. Pani, Living Reviews in Relativity 22, 4 (2019)

2019
[11]

Lovelock, Journal of Mathematical Physics12, 498 (1971)

D. Lovelock, Journal of Mathematical Physics12, 498 (1971)

1971
[12]

D. G. Boulware and S. Deser, Physical Review Letters 55, 2656 (1985)

1985
[13]

Zwiebach, Physics Letters B156, 315 (1985)

B. Zwiebach, Physics Letters B156, 315 (1985)

1985
[14]

D. J. Gross and J. H. Sloan, Nuclear Physics B291, 41 (1987)

1987
[15]

R. R. Metsaev and A. A. Tseytlin, Nuclear Physics B 293, 385 (1987)

1987
[16]

Nojiri, S

S. Nojiri, S. D. Odintsov, and M. Sasaki, Physical Re- view D71, 123509 (2005)

2005
[17]

Nojiri and S

S. Nojiri and S. D. Odintsov, Physics Reports505, 59 (2011)

2011
[18]

Kanti, N

P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, Physical Review D54, 5049 (1996)

1996
[19]

S. O. Alexeev and M. V. Pomazanov, Physical Review D55, 2110 (1997)

1997
[20]

T. P. Sotiriou and S.-Y. Zhou, Physical Review Letters 112, 251102 (2014). 20

2014
[21]

T. P. Sotiriou and S.-Y. Zhou, Physical Review D90, 124063 (2014)

2014
[22]

Pani and V

P. Pani and V. Cardoso, Physical Review D79, 084031 (2009)

2009
[23]

Kleihaus, J

B. Kleihaus, J. Kunz, S. Mojica, and M. Zagermann, Physical Review D93, 064077 (2016)

2016
[24]

Kleihaus, J

B. Kleihaus, J. Kunz, and E. Radu, Phys. Rev. Lett. 106, 151104 (2011), arXiv:1101.2868 [gr-qc]

2011 arXiv
[25]

D. D. Doneva and P. Kanti, Physical Review D98, 084053 (2018)

2018
[26]

H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Physical Review Letters120, 131104 (2018)

2018
[27]

Antoniou, A

G. Antoniou, A. Bakopoulos, and P. Kanti, Physical Review Letters120, 131102 (2018)

2018
[28]

J. L. Blázquez-Salcedo, D. D. Doneva, B. Kleihaus, J. Kunz, and S. S. Yazadjiev, Physical Review D98, 084011 (2018)

2018
[29]

Minamitsuji and T

M. Minamitsuji and T. Ikeda, Physical Review D99, 104069 (2019)

2019
[30]

P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Runarsson, Physical Review Letters115, 211102 (2015)

2015
[31]

L. G. Collodel, B. Kleihaus, J. Kunz, and E. Berti, Clas- sical and Quantum Gravity37, 075018 (2020)

2020
[32]

A. Dima, E. Barausse, N. Franchini, and T. P. Sotiriou, Physical Review Letters125, 231101 (2020)

2020
[33]

C. A. R. Herdeiro, E. Radu, H. O. Silva, T. P. Sotiriou, and N. Yunes, Physical Review Letters126, 011103 (2021)

2021
[34]

W. E. East and J. L. Ripley, Physical Review D103, 044040 (2021)

2021
[35]

Witek, L

H. Witek, L. Gualtieri, P. Pani, and T. P. Sotiriou, Physical Review D99, 064035 (2019)

2019
[36]

D. D. Doneva and S. S. Yazadjiev, International Journal of Modern Physics D27, 1842007 (2018)

2018
[37]

Rezzolla and A

L. Rezzolla and A. Zhidenko, Physical Review D90, 084009 (2014)

2014
[39]

Johannsen and D

T. Johannsen and D. Psaltis, Physical Review D83, 124015 (2011)

2011
[40]

Johannsen, Physical Review D88, 044002 (2013)

T. Johannsen, Physical Review D88, 044002 (2013)

2013
[41]

Glampedakis and S

K. Glampedakis and S. Babak, Classical and Quantum Gravity23, 4167 (2006)

2006
[42]

Cardoso, P

V. Cardoso, P. Pani, and J. Rico, Physical Review D 89, 064007 (2014)

2014
[43]

Bambi, Reviews of Modern Physics89, 025001 (2017)

C. Bambi, Reviews of Modern Physics89, 025001 (2017)

2017
[44]

Psaltis, Living Reviews in Relativity11, 9 (2008)

D. Psaltis, Living Reviews in Relativity11, 9 (2008)

2008
[45]

Yunes and F

N. Yunes and F. Pretorius, Physical Review D80, 122003 (2009)

2009
[46]

Event Horizon Telescope Collaboration, The Astrophys- ical Journal Letters875, L1 (2019)

2019
[47]

Event Horizon Telescope Collaboration, The Astrophys- ical Journal Letters875, L6 (2019)

2019
[48]

Event Horizon Telescope Collaboration, The Astrophys- ical Journal Letters930, L12 (2022)

2022
[49]

Vagnozziet al., Classical and Quantum Gravity (2023)

S. Vagnozziet al., Classical and Quantum Gravity (2023)

2023
[50]

Perlick and O

V. Perlick and O. Y. Tsupko, Physics Reports947, 1 (2022)

2022
[51]

P. V. P. Cunha and C. A. R. Herdeiro, General Relativ- ity and Gravitation50, 42 (2018)

2018
[52]

Heydari-Fard and H

M. Heydari-Fard and H. R. Sepangi, Physics Letters B 816, 136276 (2021)

2021
[53]

Paul, The European Physical Journal C84, 256 (2024)

P. Paul, The European Physical Journal C84, 256 (2024)

2024
[54]

B. C. Lütfüoğlu, J. Rayimbaev, S. Murodov, J. Kur- banov, and M. Matyoqubov, Annals of Physics491, 170514 (2026)

2026
[55]

I. D. Novikov and K. S. Thorne, inBlack Holes(Gordon and Breach, 1973) pp. 343–450

1973
[56]

D.N.PageandK.S.Thorne,TheAstrophysicalJournal 191, 499 (1974)

1974
[57]

N. I. Shakura and R. A. Sunyaev, Astronomy and As- trophysics24, 337 (1973)

1973
[58]

J. E. Pringle, Annual Review of Astronomy and Astro- physics19, 137 (1981)

1981
[59]

M. A. Abramowicz and P. C. Fragile, Living Reviews in Relativity16, 1 (2013)

2013
[60]

Rahmatov, I

B. Rahmatov, I. Egamberdiev, O. Umarov, M. Vapayev, S. Karshiboev, Y. Turaev, and S. Murodov, Nuclear Physics B1022(2026)

2026
[61]

Rahmatov, S

B. Rahmatov, S. Murodov, J. Rayimbaev, Y. Turaev, I. Egamberdiev, K. Badalov, S. Ahmedov, and S. Us- anov, Annals of Physics488(2026)

2026
[62]

Umarov, S

B.Rahmatov, I.Nishonov, S.Murodov, I.Egamberdiev, O. Umarov, S. Karshiboev, M. Vapayev, and M. Maty- oqubov, Physics of the Dark Universe52(2026)

2026
[63]

Meliyeva, O

L. Meliyeva, O. Xoldorov, O. Tursunboyev, S. Karshi- boev, S. Murodov, I. Nishonov, and B. Rahmatov, Chi- nese Physics C49(2025)

2025
[64]

Rahmatov, I

B. Rahmatov, I. Egamberdiev, S. Murodov, J. Ray- imbaev, I. Ibragimov, E. Davletov, and S. Djumanov, Physics of the Dark Universe50(2025)

2025
[65]

Jumaniyozov, S

S. Jumaniyozov, S. Murodov, J. Rayimbaev, I. Ibrag- imov, B. Madaminov, S. Urinbaev, and A. Abdujab- barov, European Physical Journal C85(2025)

2025
[66]

Saydullayev, I

S. Saydullayev, I. Nishonov, M. Dusaliyev, O. Xoldorov, S. Murodov, S. Karshiboev, S. Urinov, and B. Rahma- tov, European Physical Journal C85(2025)

2025
[67]

Rahmatov, S

B. Rahmatov, S. Murodov, J. Rayimbaev, S. Muminov, I. Ibragimov, and R. Eshburiev, Physics of the Dark Universe50(2025)

2025
[68]

Nishonov, S

I. Nishonov, S. Murodov, B. Ahmedov, S. U. Khan, J. Rayimbaev, I. Ibragimov, and S. Sabirov, European Physical Journal C85(2025)

2025
[69]

R. M. Zulqarnain, A. Ashraf, A. Bouzenada, E. Demir, E. Gudekli, S. Murodov, and F. Atamurotov, Interna- tional Journal of Geometric Methods in Modern Physics (2026)

2026
[70]

B. C. Lütfüoğlu, J. Rayimbaev, B. Rahmatov, F. Shay- imov, and I. Davletov, Physics Letters B876, 140392 (2026)

2026
[71]

Turimov, S

B. Turimov, S. Usanov, and Y. Khamroev, Physics of the Dark Universe48, 101876 (2025), arXiv:2502.11185 [gr-qc]

2025 arXiv
[72]

Rakhimova, B

G. Rakhimova, B. Puliçe, E. Ghorani, F. Atamurotov, andA. Abdujabbarov,EuropeanPhysicalJournalC86, 311 (2026), arXiv:2603.28181 [gr-qc]

2026
[73]

Oteev, Z

T. Oteev, Z. Stuchlík, J. Rayimbaev, I. Ibragimov, M. Sharibaev, and A. Abdujabbarov, European Physi- cal Journal C85, 953 (2025)

2025
[74]

Rahmatov, B

B. Rahmatov, B. Turimov, S. Murodov, K. Khak- nazarova, S.Usanov, M.Vapayev,andZ.Avezmuratova, Nuclear Physics B1024(2026)

2026
[75]

Banerjee, I

A. Banerjee, I. Karar, J. Rayimbaev, I. Ibragimov, S. Murodov, S. Muminov, and S. Jumaniyozov, Chinese 21 Physics C49(2025)

2025
[76]

Banerjee, B

A. Banerjee, B. Dayanandan, J. Rayimbaev, S. Murodov, I. Ibragimov, S. Muminov, and I. Davletov, European Physical Journal C85(2025)

2025
[77]

J. Guo, F. Javed, S. Murodov, A. Shermatov, J. Rayim- baev,andM.Matyoqubov,PhysicsoftheDarkUniverse 52(2026)

2026
[78]

J. Guo, A. Eid, A. Waseem, S. Murodov, J. Rayim- baev, N. Mustapha, A. Seytov, and O. Sirajiddin, Nu- clear Physics B1025(2026)

2026
[79]

S. Khan, S. Murodov, J. Rayimbaev, I. Davletov, I. Ibragimov, and S. Muminov, Nuclear Physics B1022 (2026)

2026
[80]

Q. U. Asadov, K. K. Sabirov, and J. R. Yusupov, EPL (Europhysics Letters)153, 52002 (2026), arXiv:2510.17819 [nlin.PS]

2026
[81]

van der Klis, inCompact Stellar X-ray Sources (Cambridge University Press, 2006) pp

M. van der Klis, inCompact Stellar X-ray Sources (Cambridge University Press, 2006) pp. 39–112

2006
[82]

R. A. Remillard and J. E. McClintock, Annual Review of Astronomy and Astrophysics44, 49 (2006)

2006
[83]

T. M. Belloni and L. Stella, Space Science Reviews183, 43 (2014)

2014
[84]

Stella and M

L. Stella and M. Vietri, The Astrophysical Journal Let- ters492, L59 (1998)

1998
[85]

Stella and M

L. Stella and M. Vietri, Physical Review Letters82, 17 (1999)

1999
[86]

M. A. Abramowicz and W. Kluźniak, Astronomy and Astrophysics374, L19 (2001)

2001
[87]

Kluźniak and M

W. Kluźniak and M. A. Abramowicz, Acta Physica Polonica B32, 3605 (2001)

2001
[88]

Török, M

G. Török, M. A. Abramowicz, W. Kluźniak, and Z. Stuchlík, Astronomy and Astrophysics436, 1 (2005)

2005
[89]

Bursa, M

M. Bursa, M. A. Abramowicz, V. Karas, and W. Kluź- niak, inProceedings of RAGtime 4/5(2004) pp. 21–25, arXiv:astro-ph/0406586

2004 arXiv
[90]

Rezzolla, S

L. Rezzolla, S. Yoshida, T. J. Maccarone, and O. Zan- otti, Monthly Notices of the Royal Astronomical Society 344, L37 (2003)

2003
[91]

Kato, Publications of the Astronomical Society of Japan53, 1 (2001)

S. Kato, Publications of the Astronomical Society of Japan53, 1 (2001)

2001
[92]

Stuchlík, A

Z. Stuchlík, A. Kotrlová, and G. Török, Astronomy and Astrophysics552, A10 (2013)

2013
[93]

E. H. Morgan, R. A. Remillard, and J. Greiner, The Astrophysical Journal482, 993 (1997)

1997
[94]

R. A. Remillard, M. P. Muno, J. E. McClintock, and J. A. Orosz, The Astrophysical Journal580, 1030 (2002)

2002
[95]

T. E. Strohmayer, The Astrophysical Journal Letters 552, L49 (2001)

2001
[96]

D. R. Pasham, T. E. Strohmayer, and R. F. Mushotzky, Nature513, 74 (2014)

2014
[97]

Shermatov, J

A. Shermatov, J. Rayimbaev, S. Murodov, B. Majeed, I. Ibragimov, B. Shermatov, E. Davletov, and I. Davle- tov, Nuclear Physics B1022(2026)

2026
[98]

Donmez, S

O. Donmez, S. Murodov, and J. Rayimbaev, Annals of Physics486(2026)

2026
[99]

Shermatov, J

A. Shermatov, J. Rayimbaev, S. Murodov, B. C. Lut- fuoglu, B. Ahmedov, M. Zahid, I. Ibragimov, and B. Shermatov, Physics of the Dark Universe50(2025)

2025
[100]

Jumaniyozov, M

S. Jumaniyozov, M. Zahid, M. Alloqulov, I. Ibragi- mov, J. Rayimbaev, and S. Murodov, European Physi- cal Journal C85(2025)

2025
[101]

Murodov, A

S. Murodov, A. H. Bokhari, J. Rayimbaev, and B. Ahmedov, European Physical Journal C85(2025)

2025
[103]

Foreman-Mackey, D

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Publications of the Astronomical Society of the Pacific125, 306 (2013)

2013

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [102]

R. A. Konoplya, T. D. Pappas, and A. Zhidenko, Phys- ical Review D101, 044054 (2020)

2020

[2] [1]

Einstein, Sitzungsberichte der K"oniglich Preussis- chen Akademie der Wissenschaften , 844 (1915)

A. Einstein, Sitzungsberichte der K"oniglich Preussis- chen Akademie der Wissenschaften , 844 (1915)

1915

[3] [2]

Schwarzschild, Sitzungsberichte der K"oniglich Preussischen Akademie der Wissenschaften , 189 (1916)

K. Schwarzschild, Sitzungsberichte der K"oniglich Preussischen Akademie der Wissenschaften , 189 (1916)

1916

[4] [3]

R. P. Kerr, Physical Review Letters11, 237 (1963)

1963

[5] [4]

Carter, Physical Review174, 1559 (1968)

B. Carter, Physical Review174, 1559 (1968)

1968

[6] [5]

J. M. Bardeen, W. H. Press, and S. A. Teukolsky, in Black Holes(Gordon and Breach, 1973) pp. 241–289

1973

[7] [6]

Chandrasekhar,The Mathematical Theory of Black Holes(Oxford University Press, 1983)

S. Chandrasekhar,The Mathematical Theory of Black Holes(Oxford University Press, 1983)

1983

[8] [7]

R. M. Wald,General Relativity(University of Chicago Press, 1984)

1984