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Comparing effective-one-body and Mathisson-Papapetrou-Dixon results for a spinning test particle on circular equatorial orbits around a Kerr black hole

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For a spinning test particle on circular equatorial orbits, the effective-one-body and Mathisson-Papapetrou-Dixon formalisms give identical gravitational-wave fluxes for a Schwarzschild primary, and differ for Kerr primaries, with the…

desk verdict The new u_MPD(x) relation for Kerr is solid and the paper is a useful benchmark, but the headline flux comparison is contaminated by feeding linearized-in-sigma orbits into an unlinearized Teukolsky solver, so the finite-sigma differences should not be taken as exact EOB-vs-MPD benchmarks. read the letter →

arxiv 2412.16077 v1 pith:P2PJSQO6 submitted 2024-12-20 gr-qc

Angelica Albertini , Viktor Skoupý , Georgios Lukes-Gerakopoulos , Alessandro Nagar This is my paper · ORCID
classification gr-qc MSC 83C5783C1083C25 PACS 04.30.-w04.70.-s
keywords effective-one-bodyMathisson-Papapetrou-DixonTulczyjew-DixonspinsupplementaryconditionKerrblackholecircularequatorialorbitsTeukolskyequationgravitational-wavefluxesspin-orbitcoupling
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 effective-one-body (EOB) approach and the Mathisson-Papapetrou-Dixon (MPD) description of a spinning test particle predict the same gravitational-wave emission for circular equatorial orbits around a Kerr black hole. It derives the radius–frequency relation, energy, and angular momentum at linear order in the secondary spin $\sigma$, then feeds these to a frequency-domain Teukolsky solver. The central finding is that for a Schwarzschild primary the EOB and MPD fluxes agree exactly, while for a Kerr primary EOB fluxes are larger than MPD for negative $\sigma$ and smaller for positive $\sigma$, with the gap growing at high primary spin. This matters for building EOB radiation-reaction models for extreme-mass-ratio inspirals with a spinning secondary.

What carries the argument

The comparison runs through the linear-in-$\sigma$ radius–frequency relation $u(x)$, where $x = \Omega^{2/3}$ is the orbital-frequency parameter. For EOB this relation follows from imposing circular conditions on the effective Hamiltonian, whose spin-orbit sector uses the gyrogravitomagnetic functions $G_S$ and $G_{S*}$ with the complete zeroth-order self-force expression and an NNLO centrifugal radius; for MPD it follows from expanding the orbital-frequency formula of the MPD equations under the Tulczyjew-Dixon condition. These $u(x)$ relations convert both formalisms to the same gauge-invariant frequency parameter, allowing direct comparison of energy, angular momentum, and last-stable-orbit location, and providing the input for the frequency-domain Teukolsky equation solver that computes the fluxes.

What would settle it

Compute the same Teukolsky fluxes using fully nonlinear-in-$\sigma$ EOB and MPD orbital data at $\sigma=0.9$ for a Schwarzschild primary; if the EOB and MPD fluxes no longer coincide, the reported Schwarzschild equality is an artifact of the linear-in-$\sigma$ truncation.

Watch

Extended reading notes

Core claim

At linear order in the secondary spin $\sigma$, and using the Tulczyjew-Dixon spin supplementary condition, the paper establishes that the test-mass EOB Hamiltonian and the MPD equations produce the same circular-orbit dynamics when the central black hole is Schwarzschild: the $u(x)$ relations, energies, and angular momenta coincide, so a Teukolsky-based flux computation returns identical infinity and horizon fluxes for every $x$ and $\sigma$ considered. For a Kerr background, the formalisms differ: the EOB energy and angular momentum are larger/smaller than MPD for negative/positive $\sigma$, the EOB asymptotic and horizon fluxes are correspondingly larger/smaller, and the angular-momentum difference begins at 3PN order, consistent with the EOB spin-orbit sector being complete only through 2.5PN. The paper also shows that an earlier reported EOB/MPD flux difference on Schwarzschild came from not linearizing the dynamics in $\sigma$.

Load-bearing premise

The comparison is made only to first order in the secondary particle's spin $\sigma$, yet fluxes are evaluated for $\sigma$ up to 0.9, where quadratic-in-spin effects are sizable and incomplete.

Editorial extensions

If this is right

  • For a nonspinning primary, EOB and MPD (Tulczyjew-Dixon) give the same asymptotic and horizon energy fluxes at linear order in the secondary spin, providing a benchmark for EOB radiation reaction in the Schwarzschild limit.
  • On Kerr, the flux mismatch is systematic: EOB fluxes exceed MPD for $\sigma<0$ and fall below for $\sigma>0$, so the choice of formalism visibly changes predicted gravitational-wave emission from spinning extreme-mass-ratio inspirals.
  • The 3PN start of the angular-momentum difference pinpoints the EOB spin-orbit sector's known truncation at 2.5PN as the source of the flux gap for Kerr primaries.
  • The linearized EOB last-stable-orbit behavior for large positive $\hat a$ and negative $\sigma$ hides a genuine absence of a last stable orbit in the full non-linearized EOB dynamics, a limitation for templates in that parameter region.
  • The results extend the previous Schwarzschild-only comparison to Kerr and, unlike that earlier work, explain the Schwarzschild equality as a consequence of consistent linearization in $\sigma$.

Reading between the lines

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

  • If the linear-order agreement on Schwarzschild persists at higher order in $\sigma$, it would mean the EOB test-mass spin-orbit sector and the Tulczyjew-Dixon MPD description are aligned in the non-spinning-primary limit, and the Kerr discrepancy is primarily a spin-orbit truncation effect rather than a fundamental formalism mismatch.
  • A natural testable extension is to recompute both the dynamics and the fluxes without linearizing in $\sigma$; the paper's own Appendix B suggests that at $\sigma=0.5$ the nonlinear EOB and MPD energies already disagree on Schwarzschild, so the equality likely degrades as spin grows.
  • The horizon-flux results hint that spin-spin effects dominate horizon absorption near the last stable orbit; feeding the Teukolsky solver with fully nonlinear orbital data for $\{\hat a,\sigma\} = \{0.9,0.9\}$ would separate spin-orbit from spin-spin contributions.
  • Extending the same comparison to eccentric equatorial orbits or to second order in the secondary spin would test whether the EOB/MPD flux difference observed here is a robust feature of the spin sector or a peculiarity of circular orbits.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper compares, at linear order in the secondary spin σ, the circular equatorial orbit dynamics of the effective-one-body (EOB) Hamiltonian of Ref. [15] and the Mathisson-Papapetrou-Dixon (MPD) formalism under the Tulczyjew-Dixon spin supplementary condition around a Kerr black hole. It derives radius-frequency relations u_EOB(x) and u_MPD(x), compares energies and angular momenta, discusses last stable orbits, and then feeds the linearized orbital data into a frequency-domain Teukolsky solver to obtain gravitational-wave energy fluxes at infinity and into the horizon. The main reported results are: the angular-momentum difference between EOB and MPD starts at 3PN, consistent with the EOB spin-orbit sector being complete through 2.5PN; for Schwarzschild primaries the EOB and MPD fluxes coincide; and for Kerr primaries EOB fluxes are larger than MPD for negative σ and smaller for positive σ.

Significance. If fully established, the comparison would be a useful benchmark for improving the radiation-reaction sector of EOB models for spinning test particles in Kerr, which is a timely step for EMRI waveform modeling. The paper contains several clean and verifiable analytical results: the PN expansions in Eqs. (49)-(50) explicitly show the expected 2.5PN limitation of the EOB spin-orbit sector, the NNLO centrifugal-radius modification is shown to matter, and the linear-order Schwarzschild equality is a transparent consistency check. The authors are also candid about the need for the anti-DJS spin gauge and about the absence of an EOB LSO for large positive primary spins. However, the significance of the finite-σ flux differences is weakened by a truncation inconsistency: the Teukolsky solver is not linearized in σ, while the orbital inputs are, so the plotted flux differences at σ = ±0.9 are not established as differences of the full EOB and MPD models.

major comments (3)
  1. [Sec. VI and Figs. 5-7] The central flux comparison is affected by an inconsistent perturbative truncation. Section VI explicitly states that the Teukolsky solver is not linearized in the secondary spin, but the orbital data fed into it are linearized in σ via the u(x) and energy/angular-momentum expressions of Secs. IIB, III C and Eq. (48). Since the gravitational-wave flux is quadratic in the source, a solver that is not linearized will include some O(σ²) source contributions while omitting the O(σ²) orbital corrections. Consequently, the flux differences shown in Figs. 5 and 6 for σ up to 0.9 are not clean differences between the full EOB and full MPD dynamics. This is not merely a cosmetic issue: for the Schwarzschild case, the equality in Fig. 7 is guaranteed by construction once both inputs are linearized, because u_EOB(x) and u_MPD(x) coincide at linear order. Appendix B, Table II, shows that the full, non-linearized EOB and MPD energies do not coincide (e.g., at r=4, σ=0.5: 0.950686 vs 0.946587), while both linearized values equal 0.937500. The authors acknowledge incomplete quadratic-in-spin flux contributions for the horizon flux at {a,σ}={0.9,0.9}, but the same caveat applies to the headline asymptotic-flux differences for all large-σ cases. The paper should either restrict the flux conclusions to spin magnitudes where O(σ²) contributions are demonstrably negligible, or supply a consistent linearized Teukolsky computation, or include nonlinear-in-σ orbital data.
  2. [Sec. IIB and Ref. [27]] The central EOB radius-frequency relation u_EOB(x) is not shown in the text: the reader is told that 'for practical reasons we do not present the final formula here' and is referred to the supplementary material, but reference [27] contains only 'URL-will-be-inserted' and the supplementary notebook is not available in the arXiv version. Since u_EOB(x) is the input to every subsequent EOB energy, angular-momentum and flux evaluation, the paper is currently not independently reproducible at the point where its EOB results begin. The formula should be included in an appendix, or at least a machine-readable expression should be made available with the submission.
  3. [Eq. (60) and Appendix A] The claimed linear-in-σ horizon-flux behavior rests on a numerical fit: the statement 'we have found numerically that E_H^1 = ...' appears without a derivation or an error estimate. This is used to infer that the linear-in-σ part starts 3/2 PN orders higher than the nonspinning term, and it feeds the qualitative discussion in Appendix A. As a fitted auxiliary relation it does not by itself invalidate the main conclusions, but the paper should present the fitting procedure, the range and number of data points, and the residual quality, so that the reader can judge whether Eq. (60) is a PN result or an empirical interpolation.
minor comments (3)
  1. [Throughout] There are several typographical and wording issues: 'liner order' in the Introduction, 'normaliszd' in the caption of Fig. 8, 'Schwarzchild' in Sec. VII, 'if he had not chosen' in Sec. III (should be 'if we had not chosen'), and inconsistent capitalization of sigma (σ vs lowercase 'sigma') in several places.
  2. [Sec. VI and Fig. 5] In the text describing Fig. 5 the sentence 'As for â = −0.9, we actually consider the fluxes only up to the largest LSO value, which is ∼ 0.135' appears twice in slightly different forms; this should be condensed to avoid duplication.
  3. [Sec. VII] The conclusion that 'EOB spin-orbit interaction is stronger than the MPD one' is presented as an explanation of the flux ordering, but the paper also notes that spin-spin contributions play a role and that the ordering changes with parameters. The wording should be softened to reflect the fact that the flux ordering is not uniquely determined by the spin-orbit term alone.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: EOB and MPD are compared as independent formalisms; the Schwarzschild flux equality is an explicitly derived consistency check, and the only fitted expression is auxiliary and disclosed as a fit.

full rationale

The central comparison is not circular. The EOB Hamiltonian is an externally defined model (with self-citations to Refs. [5,26] supplying the specific spin-orbit gauge, but no uniqueness theorem is invoked to force the result), and the MPD equations are independent first-principles equations of motion. The constants of motion for both formalisms are derived analytically as functions of the frequency parameter x, then fed separately to a Teukolsky solver; no flux difference is fitted or imposed. The Schwarzschild equality is explicitly derived rather than assumed: the paper shows that the linearized-in-sigma EOB and MPD energy expressions agree, so the same dynamics is fed to the solver and the flux difference vanishes. This is a self-consistency check, not a hidden input. The only numerical fit in the paper is Eq. (60), the small-frequency horizon-flux scaling, which is clearly labeled as obtained by numerical differentiation and fitting; it is auxiliary and does not drive the main EOB-vs-MPD comparison, nor is it presented as a prediction. The linearization-in-sigma versus non-linear-Teukolsky-solver issue identified in Sec. VI is a correctness/robustness concern about whether the finite-sigma flux differences represent the full models, not a circularity: the outputs are computed, not equivalent to the inputs by construction. Self-citations appear, but they supply model choices and numerical tools rather than load-bearing justifications for the claimed result.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim has no fitted free parameters; the only fitted object is an auxiliary horizon-flux expression. The comparison relies on standard MPD and EOB inputs from prior literature plus two explicit modeling choices (the complete GSF GS* and linearization in sigma). No new particles, fields, or conserved quantities are introduced.

free parameters (1)
  • linear-in-sigma horizon flux coefficient = -a(1+3a^2)/2 x^{3/2} (numerically fitted)
    Eq. (60) is obtained by computing l=m=2 Teukolsky fluxes at small x, taking a numerical derivative in sigma, and fitting the dependence on the primary spin. It is auxiliary to the central EOB/MPD comparison and is labeled as a numerical finding.
assumptions (4)
  • domain assumption MPD pole-dipole equations with Tulczyjew-Dixon SSC (Eqs. 27-28) and the Ehlers-Rudolph momentum-velocity relation (Eq. 34).
    Used to define circular equatorial orbits, constants of motion E and J_z, and the u_MPD(x) relation in Sec. III.
  • domain assumption EOB Hamiltonian of Damour-Nagar 2014 (Ref. [15]) in test-mass limit with GS and GS* from Eqs. (10)-(11) and NNLO centrifugal radius Eq. (9).
    Central input; the paper notes this Hamiltonian's spin-orbit couplings are reproduced only through 2.5PN, which is what the comparison is designed to expose.
  • ad hoc to paper The complete zeroth-order GSF spin-orbit coupling from Bini-Damour-Geralico (Ref. [24], Eq. 2.21) is the correct GS* function.
    This is a modeling choice imported from the authors' anti-DJS gauge work [26], replacing the standard leading-order (3/2)u^3 expression; it is not derived in this paper.
  • ad hoc to paper All quantities are linearized in the secondary spin sigma, including the orbital data fed into a Teukolsky solver that is not itself linearized.
    The restriction to O(sigma) is explicit in Secs. IV-VI and Appendix B; the authors flag that quadratic contributions to the fluxes may be incomplete for sigma=0.9.

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Cite this review

Pith. "Pith review of Comparing effective-one-body and Mathisson-Papapetrou-Dixon results for a spinning test particle on circular equatorial orbits around a Kerr black hole." pith.science (2026). https://pith.science/paper/P2PJSQO6

@misc{pith2026241216077,
  author       = {Pith},
  title        = {Pith review of: Comparing effective-one-body and Mathisson-Papapetrou-Dixon results for a spinning test particle on circular equatorial orbits around a Kerr black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2PJSQO6}},
  note         = {Machine review of arXiv:2412.16077}
}
read the original abstract

We consider a spinning test particle around a rotating black hole and compare the Mathisson-Papapetrou-Dixon (MPD) formalism under the Tulczyjew-Dixon spin supplementary condition to the test-mass limit of the effective-one-body (EOB) Hamiltonian of [Phys. Rev. D.90, 044018(2014)], with enhanced spin-orbit sector. We focus on circular equatorial orbits: we first compare the constants of motion at their linear in secondary spin approximation and then we compute the gravitational-wave (GW) fluxes using a frequency domain Teukolsky equation solver. We find no difference between the EOB and MPD fluxes when the background spacetime is Schwarzschild, while the difference for a Kerr background is maximum for large, positive spins. Our work could be considered as a first step to improve the radiation reaction of the EOB model, in view of the needs of the next-generation of GW detectors.

Figures

Figures reproduced from arXiv: 2412.16077 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. Behavior of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The EOB Hamiltonian evaluated at [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Gravitational wave energy fluxes at infinity (left) and at the horizon (right) for evaluated with the FD TE solver [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Gravitational wave energy fluxes into the horizon for [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Gravitational wave energy fluxes into the horizon for [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

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Reference graph

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