Pith. sign in

REVIEW 3 major objections 4 minor 57 references

Determining parameters of Kerr-Newman black holes by shadow observation from finite distance and spatial infinity

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper proves that the shadow of a Kerr-Newman black hole, observed from spatial infinity, uniquely fixes the dimensionless spin, charge, and inclination angle, while finite-distance shadows are degenerate and cannot do the same.

desk verdict Real new results—KN shadow uniqueness at infinity and an RN finite-distance degeneracy—but the uniqueness proof has a parameterization gap and the abstract overstates the finite-distance claim. read the letter →

arxiv 2411.08486 v2 pith:WLEZ3XOE submitted 2024-11-13 gr-qc astro-ph.HE

Kenta Hioki , Umpei Miyamoto This is my paper · ORCID
classification gr-qcastro-ph.HE MSC 83C5783C10 PACS 04.70.-s04.70.Bw
keywords blackholeshadowKerr-NewmanuniquenessparameterdeterminationBardeencoordinatessphericalphotonorbitsprincipalcomponentanalysis
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 a single shadow image can completely determine the physical parameters of a charged, rotating black hole. The authors prove that for a Kerr-Newman black hole observed from spatial infinity the answer is yes: the contour of the shadow uniquely fixes the dimensionless spin $a/M$, charge $Q/M$, and inclination angle $i$. The proof shows that the shadow contour in Bardeen coordinates is described by irreducible rational functions of the photon-orbit radius, and that the coefficients of these functions determine the parameters. The authors also prove the opposite for finite-distance observations: in the spherically symmetric limit, different combinations of distance and charge produce congruent circular shadows, so the parameters cannot be recovered unambiguously up close. If correct, the result gives future black hole imaging a concrete route to measuring charge and spin from the shadow's size and shape alone.

What carries the argument

The central objects are the Bardeen coordinates ($b_x$, $b_y$), the impact parameters that trace the shadow contour seen by an observer at spatial infinity, obtained as the leading terms of the screen coordinates at large observer distance. The argument is carried by showing that $b_x(r_*; a_*, Q_*, i)$ and $b_y^2(r_*; a_*, Q_*, i)$ are irreducible rational functions of the unstable spherical photon orbit radius $r_*$, with irreducibility of each numerator-denominator pair certified by the non-vanishing of the resultants (determinants of Sylvester matrices) of the two polynomials. Because two identically equal irreducible rational functions must have equal coefficients, comparing coefficients of matching contours forces the dimensionless parameters to coincide, proving injectivity of the map from the quotient parameter space to the apparent-shape library. The companion construction is the observable map: eleven Fourier coefficients of the centered contour, reduced by principal component analysis to the three observables size $z_1$, primary distortion $z_2$, and secondary distortion $z_3$, which the paper shows is one-to-one with $(a/M, Q/M, i)$.

What would settle it

Carry out a dense numerical scan over the dimensionless parameter triples $(a/M, Q/M, i)$, compute the Bardeen-coordinate contour $(b_x(r_*), b_y(r_*))$, and test whether any two contours for distinct triples coincide as unparameterized point sets up to translation, rotation, and reflection; if any pair does, the coefficient-comparison proof has missed a reparameterization degeneracy. The same scan evaluated on the observables $(z_1, z_2, z_3)$ would also settle the numerical injectivity claim, which the paper currently supports with isosurface intersections on a coarse grid.

Watch

Extended reading notes

Core claim

The paper's central claim is that the apparent shape of a bare Kerr-Newman black hole on the Bardeen coordinates ($b_x$, $b_y$), the impact parameters read off by an observer at spatial infinity, uniquely determines the dimensionless parameters $(a/M, Q/M, i)$. The paper states this conclusion directly: the apparent shape of the Kerr-Newman black hole on the Bardeen coordinates is unique, where uniqueness means that no two congruent shadow contours arise from distinct dimensionless parameter values. The proof writes the contour as the pair of rational functions $b_x(r_*)$ and $b_y^2(r_*)$ of the unstable spherical photon orbit radius $r_*$, verifies that these functions are irreducible, and compares polynomial coefficients to force equality of the parameters. The complementary result is that on the screen coordinates of an observer at finite distance $r_o$ the apparent shape is not unique over the full parameter space: in the zero-spin (Reissner-Nordström) sector the shadow is a circle whose radius is governed jointly by $r_o/M$ and $Q/M$, leaving an infinite family of parameter pairs with congruent shadows; for nonzero spin the screen-coordinate contour is unique, so the finite-distance degeneracy rests entirely on the spherically symmetric sector. A concrete extraction recipe accompanies the proof: the first eleven Fourier coefficients of the contour, orthogonally transformed by principal component analysis into the size $z_1$, primary distortion $z_2$, and secondary distortion $z_3$, give a one-to-one correspondence with $(a/M, Q/M, i)$, so the parameters can be read off from the shadow's size and shape alone.

Load-bearing premise

The uniqueness proof assumes that two shadow contours that look identical when drawn are also identical point-for-point as functions of the photon-orbit radius, so that comparing polynomial coefficients is legitimate; the paper does not show that every way of matching two congruent contours preserves that parameterization.

Editorial extensions

If this is right

  • Observing the shadow of a Kerr-Newman black hole from a distant vantage point determines the dimensionless spin $a/M$, charge $Q/M$, and inclination $i$ without degeneracy; knowing any one of $M$, $a$, or $Q$ from other data then fixes all four physical parameters.
  • The uniqueness result is independent of the observer family: Carter's observers and zero-angular-momentum observers see the same contour shapes up to an origin shift of the Bardeen coordinates, which the shape analysis explicitly ignores.
  • At finite distance the shadow cannot be a complete parameter probe: in the spherically symmetric sector, infinitely many pairs $(r_o/M, Q/M)$ produce congruent circular shadows, so the map from the full parameter space to the shadow library is not injective.
  • The observable construction gives a systematic injectivity test for any black hole model with three or more dimensionless parameters, extending the earlier two-parameter analysis of the Kerr case to charged solutions.
  • In images with accretion structure, the same Fourier and principal-component machinery can be applied to the photon ring or to half-peak-brightness contours of time-averaged simulated images, carrying the parameter-determination logic beyond the bare critical curve.

Reading between the lines

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

  • A direct geometric test of the proof's key assumption would be to compare Bardeen-coordinate contours as unparameterized point sets under rigid motions over a dense parameter grid, which is the one check that could expose a reparameterization-induced degeneracy that coefficient comparison would miss.
  • For small spins the shadow is nearly circular, so the distance-charge trade-off seen exactly at $a=0$ should reappear as a near-degeneracy at finite distance, meaning practical charge measurements from close-up images will need an independent distance prior.
  • Since the critical curve is the asymptotic inner boundary of the photon ring, the uniqueness at infinity suggests that a sufficiently resolved photon ring, rather than the shadow interior, could in principle carry the same parameter information.
  • The Fourier-coefficient plus principal-component pipeline is model-agnostic, so applying it to other charged or hairy black hole families, or to horizonless ultracompact objects, would reveal whether the one-to-one parameter map survives beyond the Kerr-Newman family.
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.

Referee Report

3 major / 4 minor

Summary. The paper studies the apparent shadow shape (critical curve) of a bare Kerr-Newman black hole, for an observer at either finite distance or spatial infinity, under a simplified model with a distant spherical light source and a Carter observer. The main claims are: (i) for an observer at spatial infinity, the shadow contour in Bardeen coordinates uniquely determines the dimensionless parameters (a/M, Q/M, i); (ii) for an observer at finite distance, the shadow is generically non-unique. The authors prove the spatial-infinity uniqueness by expressing the Bardeen coordinates (b_x, b_y^2) as irreducible rational functions of the spherical photon orbit radius r_* and comparing polynomial coefficients. They also construct observables (size, primary distortion, secondary distortion) via Fourier coefficients and principal component analysis, and demonstrate parameter extraction on a test case. The finite-distance non-uniqueness is exhibited through the Reissner-Nordström (a=0) subfamily, where infinitely many (r_o,Q) pairs yield the same circular shadow radius.

Significance. If the uniqueness theorem is valid, the paper makes a significant contribution: it would be the first injectivity result for the shadow-to-parameter map of the three-parameter Kerr-Newman family at spatial infinity, extending earlier two-parameter Kerr results. The resultant-based irreducibility technique is a useful algebraic tool, and the proposed observable construction is concrete and reproducible, with supplemental data for the PCA transformation matrix. The finite-distance degeneracy for a=0 is a clear and interesting counterexample. However, the central proof currently has a gap in passing from geometric congruence of curves to pointwise equality of the r_*-parameterized rational functions, so the headline uniqueness result is not yet established as written.

major comments (3)
  1. [Sec. IV B, Eqs. (56)-(58); also Sec. III B after Eq. (47)] This is the main load-bearing gap. Please provide a justification that equality of the shadow as a set forces equality of the parameterized functions for the same r_*, or reformulate the uniqueness statement and proof accordingly.
  2. [Abstract and Sec. III B] This is a substantive misstatement of the paper's own result, not merely a wording issue.
  3. [Sec. IV D-E and Appendix A] This distinction matters because the paper's stated goal is to determine parameters from shadow observations, and a non-injective observable map would break the method even if the shadow map itself is injective.
minor comments (4)
  1. [Appendix A]
  2. [Eq. (59)]
  3. [Sec. III B, Eq. (50)]
  4. [Throughout]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniqueness proof is derived from the geodesic equations and coefficient comparison of irreducible rational functions, and the PCA-based inversion is self-consistent forward modeling rather than a circular derivation.

full rationale

The central claim, uniqueness of the Kerr-Newman shadow in Bardeen coordinates, is derived in Sec. IV B from the null-geodesic integrals (Eqs. 51-52), the explicit rational functions b_x(r_*) and b_y^2(r_*) (Eqs. 56 and 58), and a coefficient-comparison argument that relies on irreducibility verified by resultants. This derivation is self-contained: it does not assume the conclusion or fit the parameters to the shadow being predicted. The self-citations to [27] and [29] supply proof strategy and context for the Kerr case, but the Kerr-Newman irreducibility analysis and coefficient comparisons are carried out here, so the argument does not reduce to a self-citation chain. The PCA-based observables in Sec. IV D-E and Appendix A are calibrated on the same theoretical shadow library and the Appendix B demonstration recovers parameters from a synthetic shadow of the same model; this is an internal consistency check, not a circular derivation, and the analytical uniqueness theorem does not depend on the PCA calibration. One weakness is that the proof passes from congruence of two contours to pointwise equality of the parameterized functions b_x(r_*) and b_y^2(r_*) without justifying that congruence fixes the r_* parameterization; this is a logical gap in the proof rather than a circularity, because it weakens the theorem instead of importing the target result as an input. No fitted parameter is renamed as a prediction, and no load-bearing conclusion is justified solely by an author-overlapping citation. Therefore, no significant circularity is found.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The analytical theorem relies on standard geodesic equations and algebraic irreducibility; no new physical entities are introduced. The numerical invertibility of the observable map depends on the fitted PCA matrix and the sampled grid, which are methodological assumptions rather than physical axioms.

free parameters (2)
  • PCA transformation matrix A = 11x11 matrix (Supplemental Material [46])
    Derived from principal component analysis of 125,000 sampled theoretical shadows; defines the observables z1, z2, z3.
  • Number of Fourier coefficients retained = 11 (c0..c10)
    Chosen by trial: the paper states one-to-one correspondence with the three parameters was first confirmed using eleven coefficients; this is a hand-tuned choice.
assumptions (6)
  • domain assumption The observed black hole is described by the Kerr-Newman solution of Einstein-Maxwell theory.
    Assumed in Sec. I and II A; the entire analysis is within this spacetime.
  • domain assumption The observer is a Carter's observer with the tetrad given by Eqs. (15)-(18).
    Assumed in Sec. II B; the coordinate and screen definitions depend on this observer. The paper argues degeneracy results extend to general observers via conformal transformations, but the main proof uses this choice.
  • domain assumption The light source is a uniformly emitting sphere of radius r_e > r_o, and the black hole is bare (no accretion).
    Assumed in Sec. II B; the shadow is defined as the critical curve of unstable photon orbits in this setting.
  • domain assumption The apparent shape is the critical curve formed by unstable spherical photon orbits and the principal null geodesic.
    Defined in Sec. II E; the proof of uniqueness concerns this critical curve, not a realistic image with accretion.
  • ad hoc to paper Congruence of two apparent shapes implies pointwise equality of the parameterized rational functions (sin^2 alpha, cos^2 beta) or (b_x, b_y^2) as functions of r*.
    Used in Sec. III B and IV B; this assumption is not proven and is the weakest step in the uniqueness proof.
  • standard math The resultant criterion (Sylvester matrix determinant) correctly characterizes common factors of the relevant polynomials over the real parameter domain.
    Standard algebra, invoked in Sec. III B and IV B.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Determining parameters of Kerr-Newman black holes by shadow observation from finite distance and spatial infinity." pith.science (2026). https://pith.science/paper/WLEZ3XOE

@misc{pith2026241108486,
  author       = {Pith},
  title        = {Pith review of: Determining parameters of Kerr-Newman black holes by shadow observation from finite distance and spatial infinity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLEZ3XOE}},
  note         = {Machine review of arXiv:2411.08486}
}
abstract

We present a method for determining the physical parameters of a Kerr-Newman black hole through shadow observation. In a system comprising a Kerr-Newman black hole, an observer, and a light source, the relevant parameters are mass $M$, specific angular momentum $a$, electric charge $Q$, inclination angle $i$, and distance $r_o$. We consider the cases where the observer is at either a finite distance or spatial infinity. Using our method, the dimensionless parameters $(a/M, Q/M, i)$ can be determined by observing the shadow contour of the Kerr-Newman black hole from spatial infinity. We analytically prove that the shadow contour of the Kerr-Newman black hole observed from spatial infinity is unique, where uniqueness is defined as the absence of two congruent shadow contours for distinct sets of dimensionless parameter values. This method is versatile and can be applied to a range of black hole solutions with charge. Additionally, we show analytically that the shadow contour of a Kerr-Newman black hole observed from a finite distance $r_o$ is not unique, meaning that the parameters of a Kerr-Newman black hole at finite distance cannot be determined from shadow observations. This result reveals a new challenge and provides a clear direction for further research on black hole shadows.

Figures

Figures reproduced from arXiv: 2411.08486 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (a) shows that the size of the apparent shape is almost independent of the inclination angle i. When the specific angular momentum is zero, the intersection of the isosurfaces with the Q∗-i plane is parallel to the i-axis. This is because when a∗ = 0, the 25 [PITH_FUL…
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p033_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p033_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 18 canonical work pages

  1. [1]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L1 (2019) doi:10.3847/2041-8213/ab0ec7 [arXiv:1906.11238 [astro -ph.GA]]

    arXiv 2019

  2. [2]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L2 (2019) doi:10.3847/2041-8213/ab0c96 [arXiv:1906.11239 [astro -ph.IM]]

    arXiv 2019

  3. [3]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L3 (2019) doi:10.3847/2041-8213/ab0c57 [arXiv:1906.11240 [astro -ph.GA]]

    arXiv 2019

  4. [4]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L4 (2019) doi:10.3847/2041-8213/ab0e85 [arXiv:1906.11241 [astro -ph.GA]]

    arXiv 2019

  5. [5]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L6 (2019) doi:10.3847/2041-8213/ab1141 [arXiv:1906.11243 [astro -ph.GA]]

    arXiv 2019

  6. [6]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 910, no.1, L12 (2021) doi:10.3847/2041-8213/abe71d [arXiv:2105.01169 [astro -ph.HE]]

    arXiv 2021

  7. [7]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 910, no.1, L13 (2021) doi:10.3847/2041-8213/abe4de [arXiv:2105.01173 [astro -ph.HE]]

    arXiv 2021

  8. [8]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L12 (2022) doi:10.3847/2041-8213/ac6674

    doi:10.3847/2041-8213/ac6674 2022

Show all 57 references
  1. [9]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L13 (2022) doi:10.3847/2041-8213/ac6675

    2022 doi

  2. [10]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L14 (2022) doi:10.3847/2041-8213/ac6429

    2022 doi

  3. [11]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L15 (2022) doi:10.3847/2041-8213/ac6736

    2022 doi

  4. [12]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L16 (2022) doi:10.3847/2041-8213/ac6672. 37

    2022 doi

  5. [13]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L17 (2022) doi:10.3847/2041-8213/ac6756

    2022 doi

  6. [14]

    de Vries, Class

    A. de Vries, Class. Quant. Grav. 17, no.1, 123-144 (1999) doi:10.1088/0264-9381/17/1/309

    1999 doi

  7. [15]

    The gravity field of a particle,

    C. Darwin, in “The gravity field of a particle,” Proceedi ngs of the Royal Society of London. Series A.249, no.1257 (Jan., 1959) 180-194

    1959

  8. [16]

    Black Holes,

    J. M. Bardeen, in “Black Holes,” ed. C. DeWitt and B. DeWi tt, New York, Gordon & Breach (1973)

    1973

  9. [17]

    Hioki and U

    K. Hioki and U. Miyamoto, Phys. Rev. D 78, 044007 (2008), [arXiv:0805.3146 [gr-qc]]

    2008 arXiv

  10. [18]

    Bambi and N

    C. Bambi and N. Yoshida, Class. Quant. Grav. 27, 205006 (2010) doi:10.1088/0264- 9381/27/20/205006 [arXiv:1004.3149 [gr-qc]]

    2010 arXiv

  11. [19]

    Amarilla, E

    L. Amarilla, E. F. Eiroa and G. Giribet, Phys. Rev. D 81, 124045 (2010) doi:10.1103/PhysRevD.81.124045 [arXiv:1005.0607 [gr-q c]]

    2010 arXiv

  12. [20]

    Amarilla and E

    L. Amarilla and E. F. Eiroa, Phys. Rev. D 87, no.4, 044057 (2013) doi:10.1103/PhysRevD.87.044057 [arXiv:1301.0532 [gr-q c]]

    2013 arXiv

  13. [21]

    S. W. Wei and Y. X. Liu, JCAP 11, 063 (2013) doi:10.1088/1475-7516/2013/11/063 [arXiv:1311.4251 [gr-qc]]

    2013 arXiv

  14. [22]

    Papnoi, F

    U. Papnoi, F. Atamurotov, S. G. Ghosh and B. Ahmedov, Phy s. Rev. D 90, no.2, 024073 (2014) doi:10.1103/PhysRevD.90.024073 [arXiv:1407.083 4 [gr-qc]]

    2014 doi

  15. [23]

    S. W. Wei, P. Cheng, Y. Zhong and X. N. Zhou, JCAP 08, 004 (2015) doi:10.1088/1475- 7516/2015/08/004 [arXiv:1501.06298 [gr-qc]]

    2015 arXiv

  16. [24]

    B. P. Singh and S. G. Ghosh, Annals Phys. 395, 127-137 (2018) doi:10.1016/j.aop.2018.05.010 [arXiv:1707.07125 [gr-qc]]

    2018 arXiv

  17. [25]

    Stuchl ´ ık and J

    Z. Stuchl ´ ık and J. Schee, Eur. Phys. J. C 79, no.1, 44 (2019) doi:10.1140/epjc/s10052-019- 6543-8

    2019 doi

  18. [26]

    Tsukamoto and R

    N. Tsukamoto and R. Kase, [arXiv:2404.06414 [gr-qc]]

    arXiv

  19. [27]

    Hioki and K

    K. Hioki and K. i. Maeda, Phys. Rev. D 80, 024042 (2009) doi:10.1103/PhysRevD.80.024042 [arXiv:0904.3575 [astro-ph.HE]]

    2009 arXiv

  20. [28]

    Hioki and U

    K. Hioki and U. Miyamoto, Phys. Rev. D 107, no.4, 044042 (2023) doi:10.1103/PhysRevD.107.044042 [arXiv:2210.02164 [gr -qc]]

    2023 arXiv

  21. [29]

    Hioki and U

    K. Hioki and U. Miyamoto, Phys. Rev. D 109, no.4, 044030 (2024) doi:10.1103/PhysRevD.109.044030 [arXiv:2311.16802 [gr -qc]]. 38

    2024 arXiv

  22. [30]

    C. F. Paganini and M. A. Oancea, Class. Quant. Grav. 35, no.6, 067001 (2018) doi:10.1088/1361-6382/aaaa5b [arXiv:1710.02403 [gr-qc ]]

    2018 arXiv

  23. [31]

    M. Mars, C. F. Paganini and M. A. Oancea, Class. Quant. Gr av. 35, no.2, 025005 (2018) doi:10.1088/1361-6382/aa97ff [arXiv:1710.02402 [gr-qc] ]

    2018 arXiv

  24. [32]

    H. C. D. Lima, Junior., L. C. B. Crispino, P. Cunha, V.P. a nd C. A. R. Herdeiro, Phys. Rev. D 103, no.8, 084040 (2021) doi:10.1103/PhysRevD.103.084040 [a rXiv:2102.07034 [gr-qc]]

    2021 arXiv

  25. [33]

    E. T. Newman, R. Couch, K. Chinnapared, A. Exton, A. Prak ash and R. Torrence, J. Math. Phys. 6, 918-919 (1965) doi:10.1063/1.1704351

    1965 doi

  26. [34]

    V. P. Frolov and I. D. Novikov, doi:10.1007/978-94-011 -5139-9

    doi

  27. [35]

    The mathematical theory of black ho les

    S. Chandrasekhar, “The mathematical theory of black ho les”, Oxford Univ. Press (1992)

    1992

  28. [36]

    Kocherlakota et al

    P. Kocherlakota et al. [Event Horizon Telescope], Phys. Rev. D 103, no.10, 104047 (2021) doi:10.1103/PhysRevD.103.104047 [arXiv:2105.09343 [gr -qc]]

    2021 arXiv

  29. [37]

    Vagnozzi, R

    S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A . Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and P. S. Joshi, et al. Class. Quant. Grav. 40, no.16, 165007 (2023) doi:10.1088/1361-6382/acd97b [arXiv:2205.07787 [gr-qc ]]

    2023 arXiv

  30. [38]

    Grenzebach, V

    A. Grenzebach, V. Perlick and C. L¨ ammerzahl, Phys. Rev . D 89, no.12, 124004 (2014) doi:10.1103/PhysRevD.89.124004 [arXiv:1403.5234 [gr-q c]]

    2014 arXiv

  31. [39]

    Chang and Q

    Z. Chang and Q. H. Zhu, Phys. Rev. D 102, no.4, 044012 (2020) doi:10.1103/PhysRevD.102.044012 [arXiv:2006.00685 [gr -qc]]

    2020 arXiv

  32. [40]

    Takahashi, Publ

    R. Takahashi, Publ. Astron. Soc. Jap. 57, 273 (2005) doi:10.1093/pasj/57.2.273 [arXiv:astro- ph/0505316 [astro-ph]]

    2005

  33. [41]

    Using algebraic geomet ry

    D. Cox, J. Little and D. O’shea, “Using algebraic geomet ry”, Springer Science & Business Media (2005)

    2005

  34. [42]

    Penrose and W

    R. Penrose and W. Rindler, Cambridge Univ. Press, 2011, ISBN 978-0-521-33707-6, 978-0- 511-86766-8, 978-0-521-33707-6 doi:10.1017/CBO9780511 564048

    2011 doi

  35. [43]

    Perlick and O

    V. Perlick and O. Y. Tsupko, Phys. Rept. 947, 1-39 (2022) doi:10.1016/j.physrep.2021.10.004 [arXiv:2105.07101 [gr-qc]]

    2022 arXiv

  36. [44]

    A. F. Zakharov, F. De Paolis, G. Ingrosso and A. A. Nucita , Astron. Astrophys. 442, no.3, 795-799 doi:10.1051/0004-6361:20053432 [arXiv:astro-p h/0505286 [astro-ph]]

  37. [45]

    Principal component a nalysis: a review and recent devel- opments,

    Jolliffe, Ian T and Cadima, Jorge, “Principal component a nalysis: a review and recent devel- opments,” Phil. Trans. R. Soc. A. 374, 20150202 (2016). 39

    2016

  38. [46]

    See the Supplemental Material for table data on the tran sformation matrix A

  39. [47]

    See the Supplemental Material for table data on the dime nsionless parameters, their corre- sponding Fourier coefficients, and principal components

  40. [48]

    G. V. Kraniotis, Gen. Rel. Grav. 46, no.11, 1818 (2014) doi:10.1007/s10714-014-1818-8 [arXiv:1401.7118 [gr-qc]]

    2014 arXiv

  41. [49]

    Tsukamoto, Phys

    N. Tsukamoto, Phys. Rev. D 97, no.6, 064021 (2018) doi:10.1103/PhysRevD.97.064021 [arXiv:1708.07427 [gr-qc]]

    2018 arXiv

  42. [50]

    Y. W. Hsiao, D. S. Lee and C. Y. Lin, Phys. Rev. D 101, no.6, 064070 (2020) doi:10.1103/PhysRevD.101.064070 [arXiv:1910.04372 [gr -qc]]

    2020 arXiv

  43. [51]

    S. V. M. C. B. Xavier, P. V. P. Cunha, L. C. B. Crispino and C . A. R. Herdeiro, Int. J. Mod. Phys. D 29, no.11, 2041005 (2020) doi:10.1142/S0218271820410059 [a rXiv:2003.14349 [gr-qc]]

    2020 arXiv

  44. [52]

    C. Y. Wang, D. S. Lee and C. Y. Lin, Phys. Rev. D 106, no.8, 084048 (2022) doi:10.1103/PhysRevD.106.084048 [arXiv:2208.11906 [gr -qc]]

    2022 arXiv

  45. [53]

    S. Guo, Y. X. Huang, E. W. Liang, Y. Liang, Q. Q. Jiang and K . Lin, Astrophys. J. 975, no.2, 237 (2024) doi:10.3847/1538-4357/ad7d85 [arXiv:24 11.07914 [astro-ph.HE]]

    2024 doi

  46. [54]

    Tsukamoto, Z

    N. Tsukamoto, Z. Li and C. Bambi, JCAP 06, 043 (2014) doi:10.1088/1475-7516/2014/06/043 [arXiv:1403.0371 [gr-qc]]

    2014 arXiv

  47. [55]

    X. M. Kuang, Z. Y. Tang, B. Wang and A. Wang, Phys. Rev. D 106, no.6, 064012 (2022) doi:10.1103/PhysRevD.106.064012 [arXiv:2206.05878 [gr -qc]]

    2022 arXiv

  48. [56]

    G´ omez and P

    G. G´ omez and P. Valageas, Phys. Rev. D 109, no.10, 103038 (2024) doi:10.1103/PhysRevD.109.103038 [arXiv:2403.08988 [as tro-ph.CO]]

    2024 arXiv

  49. [57]

    Medeiros, D

    L. Medeiros, D. Psaltis and F. ¨Ozel, Astrophys. J. 896, no.1, 7 (2020) doi:10.3847/1538- 4357/ab8bd1 [arXiv:1907.12575 [astro-ph.HE]]. 40

    2020 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.