REVIEW 4 major objections 4 minor 44 references
The no-hair theorems at work in M87$^\ast$
T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read The no-hair theorems are fully at work in M87*: this paper shows that including the Kerr mass quadrupole moment alongside Lense-Thirring precession lets a circular test-particle disk model reproduce the observed jet precession time series.
desk verdict Worth a look for the quadrupole effect on the M87* parameter space, but the claimed quantitative match to the precession time series is asserted, not demonstrated. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The working core is the orbit-averaged precession velocity of the orbital angular momentum, $\boldsymbol{\Omega}_d^{\mathrm{NH}} = \boldsymbol{\Omega}_d^{\mathrm{LT}} + \boldsymbol{\Omega}_d^{Q_2}$, where the Lense-Thirring term is $\boldsymbol{\Omega}_d^{\mathrm{LT}} = \frac{2GJ}{c^2 a^3 (1-e^2)^{3/2}}\hat{k}$ and the quadrupole term is $\boldsymbol{\Omega}_d^{Q_2} = -\frac{3}{2} n_K J_2 (R/p)^2 (\hat{k}\cdot\hat{h})\hat{k}$. Here $\hat{k}$ is the spin axis of the hole, $\hat{h}$ the orbital angular momentum direction, $J_2 = -Q_2/(M R^2)$, and $Q_2$ is the no-hair quadrupole. These averaged rates are used to map allowed $(a^*, r_0)$ regions against the measured precession speed $|\omega_p^{\mathrm{exp}}| = 0.56\pm 0.02$ rad yr$^{-1}$, and to integrate the time series of the inclination and node angles.
What would settle it
Extend the VLBI time series of M87$^*$'s jet precession angles well beyond the current span; if the measured $\eta(t)$ and $\phi(t)$ drift out of phase relative to the predicted curves for $a^*=0.98$, $r_0=14.1\,R_g$ (for example, if the retrograde solution or a quadrupole-free solution fits better), the paper's central claim would be contradicted.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is that the mass quadrupole moment of M87$^*$, fixed by the no-hair theorems to $Q_2 = -J^2/(c^2 M)$, has a non-negligible dynamical effect on the predicted disk-jet precession and must be added to the Lense-Thirring precession. With both effects included, numerically integrating the orbit-averaged equations for the angles $\eta$ and $\phi$ for $a^* = +0.98$ and $r_0 = 14.1\,R_g$ reproduces the measured time series; the alternative $a^* = -0.95$, $r_0 = 16\,R_g$ keeps the amplitudes but is out of phase. This is taken as direct evidence that the Kerr no-hair relation governs the spacetime around the supermassive black hole.
Load-bearing premise
The load-bearing assumption is that the accretion disk can be replaced by one test particle on a circular orbit of radius $r_0$ that stays rigidly aligned with the jet, so the orbit-averaged first-post-Newtonian equations describe the whole precessing system.
Editorial extensions
If this is right
- The no-hair quadrupole should be included in any first-post-Newtonian analysis of jet or disk precession around a rotating black hole; omitting it biases the allowed ranges of spin and disk radius.
- Prograde and retrograde spin solutions can be distinguished by the phase of the precession time series, not only by its amplitude, because the two predictions are out of phase.
- For M87$^*$ the preferred configuration is prograde rotation with $a^*\approx 0.98$ and an effective disk radius near $14\,R_g$.
- The agreement provides an astrophysical, non-gravitational-wave test of the Kerr no-hair relation in a supermassive black hole.
Reading between the lines
- A natural extension would apply the same quadrupole-plus-Lense-Thirring machinery to other galaxies with measured jet precession, using longer VLBI monitoring to break degeneracies that a single system leaves open.
- If the single-orbit model is taken literally, sharper tests would come from resolving radial structure in the disk; a finite-width disk would smear the precession and could be fit with a two- or three-orbit ensemble.
- The phase asymmetry between prograde and retrograde fits suggests that future observations may only need to resolve the sign of the phase evolution, rather than its amplitude, to determine the spin direction of M87$^*$.
- One could test the model's internal consistency by computing the spin-orbit tilt angle's time variation at the next post-Newtonian order, where the quadrupole may contribute.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper extends a previous 1pN Lense-Thirring model for the jet precession of M87* by including the Kerr mass quadrupole moment Q2 = -J^2/(c^2 M) required by the no-hair theorems. It derives averaged rates of change of the orbital inclination and node for the LT and quadrupole perturbations (Eqs. 4-13), imposes the observational precession magnitude 0.54-0.58 rad/yr on the combined precession vector, and obtains allowed regions in the (a*, r0) plane. It claims that numerical integration of the averaged equations with a* = +0.98 and r0 = 14.1 Rg reproduces the measured eta(t) and phi(t) time series both qualitatively and quantitatively, while a retrograde solution with a* = -0.95 and r0 = 16 Rg is out of phase. The paper concludes that the no-hair theorems are fully at work in M87*.
Significance. If the quantitative claim can be substantiated, the paper is a valuable consistency test of the Kerr multipole structure: the quadrupole term breaks the prograde/retrograde symmetry of the allowed parameter regions, and the failure of the retrograde branch provides a falsifiable asymmetry. The analytic equations are standard, the use of the no-hair Q2 rather than a fitted quadrupole is a strength, and the allowed-region calculation is transparent. At present, however, the central comparison with the observed time series is only asserted; there are no residuals, model-selection statistics, uncertainties, or code, so the paper demonstrates consistency rather than a measured detection of the quadrupole effect.
major comments (4)
- [Section 3, after Eq. (22)] The claim that numerically integrating Eqs. (12)-(13) with a* = +0.98 and r0 = 14.1 Rg reproduces the measured eta(t) and phi(t) time series "both quantitatively and qualitatively" is the load-bearing step of the paper, but no quantitative comparison is shown. No residuals, chi-square or RMS values, phase errors, or uncertainty intervals are reported, and the predicted time series are not plotted. Please add a quantitative goodness-of-fit measure for the prograde solution, the retrograde solution, and a reference LT-only model, and state explicitly how the parameter point was selected inside the allowed region.
- [Section 3, Eq. (22)] The allowed regions in Figure 1 are derived only from the magnitude condition 0.54 <= |Omega_d| <= 0.58 rad/yr, so a point inside such a region is not a fit to the time series. The paper does not demonstrate that no LT-only parameter choice can reproduce the phase and shape of the measured time series. Please scan the (a*, r0) plane and plot contours of a time-series misfit for the LT-only and LT+Q2 models, so that the reader can see whether the quadrupole term is actually required by the data.
- [Section 3, effective test-particle model] The reduction of the accretion disk to a single circular orbit at r0, tightly coupled to the jet, underpins the derived (a*, r0) values and the no-hair conclusion. Finite disk extent, internal stresses, and warp propagation would alter the predicted phase and shape of the time series. Please add a robustness test (for example, allowing r0 to vary across the allowed region, or comparing with a thin annulus of finite width) and discuss how the inferred parameters and the prograde/retrograde asymmetry depend on this assumption.
- [Data Availability statement] The statement "No new data were generated or analysed in support of this research" is difficult to reconcile with the use of the published VLBI time series of Cui et al. (2023) and with the numerical integrations reported in Section 3. Moreover, no code or integration output is provided, so the central quantitative claim cannot be independently verified. Please clarify the data provenance and make the analysis artifacts (code and generated time series) available.
minor comments (4)
- [Section 2, Eqs. (4)-(13)] The symbol a is used for both the orbital semimajor axis and the dimensionless spin parameter; although the conflict is acknowledged in the text, a consistent notation (for example, chi for the spin parameter and a_orb for the semimajor axis) would reduce confusion.
- [Section 3] The predicted time series are not plotted anywhere in the paper; adding a panel with the overlaid measured and modeled eta(t) and phi(t) would allow the reader to assess the claimed visual agreement directly.
- [Equation (11)] The equatorial radius R introduced in Eq. (11) cancels when J2 is expressed through Q2, because J2 (R/p)^2 = -Q2/(M p^2); stating this cancellation explicitly would avoid any appearance of dependence on an arbitrary radius.
- [References] Iorio (2024b) is cited as an arXiv preprint; if a peer-reviewed version exists, the reference should be updated accordingly.
Circularity Check
Partial circularity: the precession speed used to select a* and r0 is derived from the same VLBI time series that the model is then said to reproduce; the phase/sign test retains independent content.
-
fitted input called prediction
[Section 3, Eqs. (21)–(22) and the paragraph after Eq. (22)]
"By considering the total NH precession velocity, given by Equations (15)–(17), as a function of a∗ and the effective radius r0 of the precessing accretion disk, it is possible to plot its absolute value by imposing the condition that its graph is comprised within the upper an lower measured values ... |ωexp p | = 0.56± 0.02 rad yr−1, (21), i.e., 0.54 rad yr−1≤ |ΩNH d (a∗, r0)|≤ 0.58 rad yr−1. (22) ... by adopting the values a∗ = +0.98, r0 = 14.1 Rg ... It turns out that the NH prograde signatures agree well with their measured counterparts ... both qualitatively and quantitatively."
The model parameters (a*, r0) are not independently predicted: they are selected so that the norm of the precession vector, |Ω_NH|, lies inside the observed band 0.54–0.58 rad/yr, and that band is derived from the same VLBI time series whose eta(t) and phi(t) are later said to be reproduced. Since Ω_NH drives the time evolution of the angles, the dominant frequency of the generated time series is fixed by Eq. (22); reporting the curves as a quantitative prediction is therefore partly a restatement of the fit. The comparison is not fully circular because the magnitude constraint does not fix the sign of a* or the relative phase of eta and phi, and the retrograde solution is found to be out of phase.
full rationale
The analytic part of the derivation (Eqs. 4–17) is self-contained and does not reduce to a previous result by construction. The main circular element is that the observed precession speed |ω_p^exp| is a summary of the VLBI time series, and Eq. (22) uses that speed to select the allowed (a*, r0) region; the same time series is then said to be reproduced when integrating with a point from that region, so the fitted frequency is being presented as part of a quantitative prediction. However, the phase and sign content of the time series is not encoded in the magnitude constraint, and the prograde/retrograde comparison provides independent discriminating information. The no-hair value Q2 = -J^2/(c^2 M) is assumed rather than tested against non-Kerr quadrupoles, so the conclusion that the no-hair theorems are 'fully at work' overstates what is actually a consistency check; this is underdetermination, not circularity. Self-citations to Iorio (2024a,b) are not load-bearing because the present paper derives the relevant equations and describes the integration itself. The absence of residuals, uncertainty propagation, and code makes the quantitative claim unverifiable, but that is a verification concern rather than an additional circular step.
Assumptions & free parameters
free parameters (2)
- dimensionless spin parameter a* =
+0.98 (prograde solution), -0.95 (retrograde solution)
- effective disk radius r0 =
14.1 Rg (prograde solution), 16 Rg (retrograde solution)
assumptions (7)
- domain assumption Kerr metric and no-hair relation M_l + iJ_l = M(iJ/(cM))^l, giving Q2 = -J^2/(c^2 M)
- domain assumption First post-Newtonian perturbative treatment
- ad hoc to paper Test particle on circular orbit models accretion disk
- domain assumption Tight coupling between accretion disk and jet
- standard math Averaged orbital equations valid over the precession timescale
- domain assumption Spin axis orientation from Cui et al. 2023 is correct
- domain assumption TISCO radius from Al Zahrani 2024 sets lower bound on r0
Cite this review
Pith. "Pith review of The no-hair theorems at work in M87$^\ast$." pith.science (2026). https://pith.science/paper/MXGA2RFH
@misc{pith2026241111292,
author = {Pith},
title = {Pith review of: The no-hair theorems at work in M87$^\ast$},
year = {2026},
howpublished = {\url{https://pith.science/paper/MXGA2RFH}},
note = {Machine review of arXiv:2411.11292}
}
abstract
Recently, a perturbative calculation to the first post-Newtonian order has shown that the analytically worked out Lense-Thirring precession of the orbital angular momentum of a test particle following a circular path around a massive spinning primary is able to explain the measured features of the jet precession of the supermassive black hole at the centre of the giant elliptical galaxy M87. It is shown that also the hole's mass quadrupole moment $Q_2$, as given by the no-hair theorems, has a dynamical effect which cannot be neglected, as, instead, done so far in the literature. New allowed regions for the hole's dimensionless spin parameter $a^\ast$ and the effective radius $r_0$ of the accretion disk, assumed tightly coupled with the jet, are obtained by including both the Lense-Thirring and the quadrupole effects in the dynamics of the effective test particle modeling the accretion disk. One obtains that, by numerically integrating the resulting averaged equations for the rates of change of the angles $\eta$ and $\phi$ characterizing the orientation of the orbital angular momentum with $a^\ast = +0.98$ and $r_0=14.1$ gravitational radii, it is possible to reproduce, both quantitatively and qualitatively, the time series for them recently measured with the Very Long Baseline Interferometry technique. Instead, the resulting time series produced with $a^\ast = -0.95$ and $r_0=16$ gravitational radii turn out to be out of phase with respect to the observationally determined ones, while maintaining the same amplitudes.
Figures
Reference graph
Works this paper leans on
- [1]
- [2]
- [3]
- [4]
-
[5]
Bertotti B., Farinella P., Vokrouhlick\' y D., 2003, Physics of the Solar System . Kluwer
work page 2003
-
[6]
A., 1991, Essential Relativistic Celestial Mechanics
Brumberg V. A., 1991, Essential Relativistic Celestial Mechanics . Adam Hilger
work page 1991
-
[7]
Carter B., 1971, Phys. Rev. Lett., 26, 331
work page 1971
-
[8]
Cui Y., Hada K., Kawashima T., et al. , 2023, Nature, 621, 711
work page 2023
Show all 44 references
-
[9]
Cui Y., Lin W., 2024, arXiv e-prints, arXiv:2410.10965
2024 arXiv
-
[10]
S., Fish V
Doeleman S. S., Fish V. L., Schenck D. E., et al. , 2012, Science, 338, 355
2012
-
[11]
I., 2023, Astronomy, 2, 141
Dokuchaev V. I., 2023, Astronomy, 2, 141
2023
-
[12]
Feng J., Wu Q., 2017, Mon. Not. Roy. Astron. Soc., 470, 612
2017
-
[13]
Phys.--Berlin, 532, 1900480
Garofalo D., 2020, Ann. Phys.--Berlin, 532, 1900480
2020
-
[14]
Second Edition
Goldstein H., 1980, Classical Mechanics. Second Edition . Addison Wesley
1980
-
[15]
Cambridge University Press
Iorio L., 2024 a , General Post-Newtonian Orbital Effects From Earth's Satellites to the Galactic Center . Cambridge University Press
2024
-
[16]
Iorio L., 2024 b , arXiv e-prints, arXiv:2411.08686
2024 arXiv
-
[17]
Rev., 164, 1776
Israel W., 1967, Phys. Rev., 164, 1776
1967
-
[18]
P., 1963, Phys
Kerr R. P., 1963, Phys. Rev. Lett., 11, 237
1963
-
[19]
Kissler-Patig M., Gebhardt K., 1998, Astron J., 116, 2237
1998
-
[20]
M., Efroimsky M., Kaplan G., 2011, Relativistic Celestial Mechanics of the Solar System
Kopeikin S. M., Efroimsky M., Kaplan G., 2011, Relativistic Celestial Mechanics of the Solar System . Wiley
2011
-
[21]
, 2009, Astrophys
Li Y.-R., Yuan Y.-F., Wang J.-M., et al. , 2009, Astrophys. J., 699, 513
2009
-
[22]
, 2018, Mon
Liska M., Hesp C., Tchekhovskoy A., et al. , 2018, Mon. Not. Roy. Astron. Soc., 474, L81
2018
-
[23]
D., 2013, Science, 339, 49
McKinney J., Tchekhovskoy A., Blandford R. D., 2013, Science, 339, 49
2013
-
[24]
Meng X.-C., Wang C.-H., Wei S.-W., 2024, arXiv e-prints, arXiv:2411.07481
2024 arXiv
-
[25]
D., Dermott S
Murray C. D., Dermott S. F., 1999, Solar System Dynamics . Cambridge University Press
1999
-
[26]
Nemmen R., 2019, Astrophys. J. Lett., 880, L26
2019
-
[27]
T., Couch E., Chinnapared K., et al
Newman E. T., Couch E., Chinnapared K., et al. , 1965, J. Math. Phys., 6, 918
1965
-
[28]
T., Janis A
Newman E. T., Janis A. I., 1965, J. Math. Phys., 6, 915
1965
-
[29]
E., Gurvits L
Nokhrina E. E., Gurvits L. I., Beskin V. S., et al. , 2019, Mon. Not. Roy. Astron. Soc., 489, 1197
2019
-
[30]
Astrophys
Penrose R., 1999, J. Astrophys. Astron., 20, 233
1999
-
[31]
Penrose R., 2002, Gen. Relativ. Gravit., 7, 1141
2002
-
[32]
M., 2014, Gravity
Poisson E., Will C. M., 2014, Gravity. Newtonian, Post--Newtonian, Relativistic . Cambridge University Press
2014
-
[33]
C., 1975, Phys
Robinson D. C., 1975, Phys. Rev. Lett., 34, 905
1975
-
[34]
E., 2005, Orbital Motion
Roy A. E., 2005, Orbital Motion. Fourth Edition . IOP Publishing
2005
-
[35]
L., Teukolsky S
Shapiro S. L., Teukolsky S. A., 1986, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects . Wiley
1986
-
[36]
L., Teukolsky S
Shapiro S. L., Teukolsky S. A., 1991, Phys. Rev. Lett., 66, 994
1991
-
[37]
N., 2018, Mon
Sob'yanin D. N., 2018, Mon. Not. Roy. Astron. Soc., 479, L65
2018
-
[38]
H., 1989, Relativity in Astrometry, Celestial Mechanics and Geodesy
Soffel M. H., 1989, Relativity in Astrometry, Celestial Mechanics and Geodesy. Springer
1989
-
[39]
H., Han W.-B., 2019, Applied General Relativity , Astronomy and Astrophysics Library
Soffel M. H., Han W.-B., 2019, Applied General Relativity , Astronomy and Astrophysics Library . Springer
2019
-
[40]
G., 1985, Celestial Mechanics: A Computational Guide for the Practitioner
Taff L. G., 1985, Celestial Mechanics: A Computational Guide for the Practitioner . Wiley
1985
-
[41]
Tamburini F., Thid \'e B., Della Valle M., 2020, Mon. Not. Roy. Astron. Soc., 492, L22
2020
-
[42]
A., 2015, Class
Teukolsky S. A., 2015, Class. Quantum Gravit., 32, 124006
2015
-
[43]
Wang J.-M., Li Y.-R., Wang J.-C., Zhang S., 2008, Astrophys. J. Lett., 676, L109
2008
-
[44]
Yodzis P., Seifert H.-J., M \"u ller Zum Hagen H., 1973, Communications in Mathematical Physics, 34, 135
1973
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.