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REVIEW 2 major objections 3 minor 58 references

Innermost stable circular orbit (ISCO) of arbitrary-mass compact binaries with spins at the fourth post-Newtonian order

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper derives a gauge-invariant stability function $C(x)$ that locates the innermost stable circular orbit of arbitrary-mass, aligned-spin compact binaries through fourth post-Newtonian order, and shows it recovers the Kerr ISCO in…

desk verdict First 4PN ISCO criterion for spinning comparable-mass binaries, carefully derived and well checked, but the spin-mode decoupling assumption needs to be justified before I'd take C(x)=0 as the stability boundary. read the letter →

arxiv 2608.07756 v1 pith:35IUL5QB submitted 2026-08-07 gr-qc

Luc Blanchet , David Langlois , Etienne Ligout This is my paper · ORCID
classification gr-qc
keywords innermoststablecircularorbitpost-NewtonianapproximationspinningcompactbinariesalignedspinsKerrISCOgravitationalself-forcespin-orbitcouplingspin-spin
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 aims to locate the innermost stable circular orbit (ISCO) of a compact binary when the two bodies have arbitrary mass and spins aligned with the orbital angular momentum, using the post-Newtonian approximation through fourth order (4PN). The authors derive a single gauge-invariant function $C(x)$ of the invariant orbital-frequency parameter $x=(Gm\Omega/c^3)^{2/3}$ such that circular orbits are stable when $C(x)>0$ and the ISCO sits at the smallest root $C(x)=0$. In the test-mass limit the function reproduces the exact Kerr ISCO to 4PN order, and at first order in the mass ratio it reproduces numerical gravitational-self-force shifts of the ISCO frequency to within about 10 percent for retrograde and moderately prograde spins. The same criterion also gives the ISCO shift produced by a small body's own spin, matching the exact Mathisson-Papapetrou-Dixon prediction for a spinning particle in Kerr over the same spin range. If correct, this gives a parameter-free, analytic ISCO condition usable for comparable-mass binaries and for neutron stars once internal-structure coefficients are supplied.

What carries the argument

The central object is the invariant stability function $C(x)$, a renormalised version of the determinant condition coming from the $3\times 3$ orbital block of the linearised system, written in terms of $x=(Gm\Omega/c^3)^{2/3}$ and the dimensionless aligned-spin variables $\chi_S$ and $\chi_\Sigma$. It is computed by perturbing the 4PN effective-field-theory Hamiltonian and, independently, the EFT equations of motion around a circular orbit; the Hamiltonian route reduces to the condition $C=\pi_0\sigma_0-\rho_0\theta_0>0$, built from second functional derivatives of the Hamiltonian, while the equations-of-motion route uses the matrix $M_0$ with coefficients $\alpha_0$, $\beta_0$, $gamma_0$. The spins enter as non-dynamical spectators, an assumption explicitly checked at 1PN and then carried to 4PN. For black holes all spin-induced deformability coefficients are unity, producing the explicit polynomial (3.3); for other compact objects the general expression (A1) retains the parameters $\kappa_a$, $\lambda_a$, $\iota_a$.

What would settle it

Compute the full $9\times 9$ linearised matrix at 4PN for aligned spins using the paper's ancillary expressions and test whether any mode with $\delta S_n$, $\delta S_\lambda$, $\delta\Sigma_n$, or $\delta\Sigma_\lambda$ acquires $\sigma^2<0$ at an $x$ smaller than the smallest root of $C(x)=0$; if such a mode exists, the ISCO is not at that root. A direct numerical alternative is to measure the conservative ISCO frequency of an equal-mass aligned-spin binary, for example with $\chi_1=\chi_2=0.6$, and compare it with the root of Eq. (3.3).

Watch

Extended reading notes

Core claim

In the paper's own terms, the central discovery is that the 4PN gauge-invariant ISCO criterion for aligned-spin compact binaries is the explicit function $C(x)$ of Eq. (3.3) for black holes and Eq. (A1) for arbitrary compact objects, with the ISCO at the smallest root of $C(x)=0$. The paper derives this function twice, once from the 4PN EFT Hamiltonian and once from the EFT equations of motion, and verifies the 3.5PN part in harmonic coordinates. In the test-mass limit it reduces to the 4PN expansion of the exact Kerr stability criterion, and the first-order-in-$\nu$ ISCO-frequency shift agrees with numerical conservative gravitational-self-force calculations within about 10 percent for retrograde spins and moderately prograde spins ($\chi_2\lesssim 0.2$). For a spinning test particle the criterion reproduces the MPD prediction of the ISCO shift to within a few percent for $\chi_2\lesssim 0.3$. The paper also claims that corotating black-hole binaries are more stable than irrotational ones with the same masses and orbital frequency.

Load-bearing premise

The load-bearing premise is that the spin components can be treated as non-dynamical spectators when the circular orbit with aligned spins is perturbed, so that stability is decided by the 3-by-3 orbital block alone; this decoupling is explicitly checked only at 1PN and is assumed to persist at 4PN.

Editorial extensions

If this is right

  • The smallest root of $C(x)=0$ gives the 4PN ISCO frequency for any mass ratio with aligned spins, so the ISCO can be found without interpolation or fitting.
  • In the extreme-mass-ratio limit the criterion recovers the Kerr ISCO to 4PN order, extending the known Schwarzschild recovery to the spinning case.
  • The first-order-in-$\nu$ ISCO shift matches conservative self-force numerics within about 10 percent for retrograde spins and moderate prograde spins, and the failure near maximal prograde spin is explained by the ISCO approaching the horizon.
  • For neutron stars or other compact objects, inserting the appropriate deformability coefficients $\kappa_a$, $\lambda_a$, $\iota_a$ into Eq. (A1) yields the ISCO criterion without a new derivation.
  • Corotating black-hole binaries are predicted to be more stable than irrotational ones at fixed orbital frequency and mass.

Reading between the lines

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

  • A concrete testable extension is to evaluate the full $9\times 9$ linearised matrix at 4PN for aligned spins, using the paper's ancillary expressions, and check whether any transverse spin mode develops an instability before the root of $C(x)=0$; if one does, the 3-by-3 reduction would fail and the ISCO would shift.
  • The close agreement with self-force data for retrograde spins suggests the PN criterion could serve as a cheap surrogate for conservative self-force ISCO frequencies in waveform modeling of extreme-mass-ratio inspirals with retrograde spins.
  • For equal-mass neutron-star binaries the deformability parameters are not unity, so the paper's general formula gives a concrete prediction that future numerical-relativity or tidal-disruption calculations could test.
  • Because the criterion is equivalent to the inverse square of the periastron-precession factor for circular orbits, future higher-order precession calculations could be converted into ISCO estimates without repeating the full perturbation analysis.
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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

2 major / 3 minor

Summary. The paper derives a 4PN-order gauge-invariant stability criterion C(x) for circular orbits of compact binaries with spins aligned with the orbital angular momentum. The criterion is quoted in Eq. (3.3) for black holes and in Eq. (A1) for general compact objects, with the ISCO defined as the smallest root of C(x)=0. The derivation linearizes the EFT equations of motion and the EFT Hamiltonian about aligned-spin circular orbits, and is checked by recovering the spinless 4PN criterion, by reproducing the Kerr ISCO in the test-mass limit, and by comparing first-order-in-mass-ratio ISCO shifts with numerical GSF data and with MPD results for a spinning test particle. The paper also treats corotating black-hole binaries.

Significance. If correct, this is the first complete 4PN ISCO criterion for spinning, arbitrary-mass-ratio binaries and a natural extension of the authors' spinless 4PN work; it provides a gauge-invariant, parameter-free prediction that can be tested against GSF and EOB results. The manuscript has substantial strengths: the criterion is derived by two independent routes (Hamiltonian and EoM), the spinless limit and the 3.5PN harmonic-coordinate checks are reproduced, the test-mass limit exactly matches the Kerr expansion through 4PN, and the GSF/MPD comparisons are favorable in the regime where PN theory is expected to work (retrograde and moderately prograde spins). The ancillary file and appendices make the calculation transparent. The main reservation is a structural gap in the linearization argument, detailed in the major comments, which does not by itself invalidate the result but needs to be addressed.

major comments (2)
  1. [Section II.C, Eqs. (2.19)-(2.26)] The reduction of the linear stability problem to the 3×3 orbital matrix in Eq. (2.20) is load-bearing for the central claim, but the only justification offered is the 1PN eigenvalue statement around Eq. (2.19). Since the leading spin-orbit interaction starts at 1.5PN and spin-spin at 2PN, that 1PN check is silent about the six spin/tilt modes. The Hamiltonian reduction in Eqs. (2.22)-(2.26) simply declares S_ℓ and Σ_ℓ to be spectator variables and never includes δS_⊥, δΣ_⊥, while Appendices D and E contain couplings between transverse spin components and orbital variables at 2.5PN and higher. If any of the six spin modes has σ²=0 at an x smaller than the smallest root of Eq. (3.3), then Eq. (3.3) is not the first instability and the central claim fails. Please either prove the decoupling (for example, by showing that at the aligned-spin background the linearized spin block has purely imaginary eigenvalues and does not feed back into the orbital block) or compute and report the relevant part of the full 9×9 spectrum at 4PN.
  2. [Section III.C and Table I] The favorable agreement with GSF data and with the MPD result tests the value of the first root of Eq. (3.3), but it does not test the assumption that this root is the first instability of the full linearized system. The numerical data are for the ISCO frequency itself; if a spin/tilt mode were to become unstable at smaller x, the comparison would not reveal it. The text should state this limitation explicitly or otherwise supply evidence from the full spectrum that the orbital-root criterion is the first zero of the complete system.
minor comments (3)
  1. [Section II.B, after Eq. (2.19)] The eigenvalue counting is unclear as stated: for the nine variables in Eq. (2.9), the text reports three zero modes and three nonzero modes, which accounts for only six eigenvalues; please clarify the counting or explain how the remaining modes are treated.
  2. [Section III.E, Eqs. (3.41)-(3.42)] The statement that the negative x^4 coefficient in C_corot - C_non-spin is "sub-dominant" would be more convincing if quantified at the relevant ISCO values (x ≈ 0.2) for the mass-ratio range 0 ≤ ν ≤ 1/4, rather than inferred purely from PN-order counting.
  3. [Appendix A] The notation κ_+, κ_-, λ_+, λ_-, ι_+, ι_- is defined only in the appendix; since the main text's Eq. (3.3) already uses the BH values κ=λ=ι=1, a one-line glossary near Eq. (3.3) would help readers who do not go through Appendix A.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 4PN criterion is derived from independent EFT Hamiltonians and validated against external Kerr, GSF, and MPD results.

full rationale

The central criterion (3.3) is obtained by perturbing the 4PN EFT Hamiltonian and EoM, with spin terms taken from the independent Levi-Steinhoff EFT results; the spinless sector is rederived in EFT coordinates in Appendix C from ADM/harmonic results, so the recovery of [9] is a consistency check, not a fitted input. The test-mass limit is compared term-by-term with the exact Kerr criterion (3.5), and the first-order-in-nu ISCO shifts are compared with independent GSF numerics [36] and MPD analytics; no parameters are fitted to these benchmarks. The self-citations ([9], [42], [52]) serve as consistency checks or as independent prior inputs (e.g., the corotation condition from the first law), and the corotating section is itself checked against the exact Kerr corotation criterion (3.43). The paper's weakest point—the 1PN eigenvalue justification for ignoring transverse spin perturbations at 4PN—is a potential correctness gap, not a circular reduction: it does not make Eq. (3.3) equivalent by construction to its inputs. Accordingly, no circular step is exhibited and the circularity score is 0.

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

The central criterion depends on no fitted numerical parameters. It inherits the physical content of the EFT Hamiltonian for spinning binaries (mass ratio, spins, and deformability coefficients), treats the stability definition and the aligned-spin decoupling as domain assumptions, and relies on the validity of the PN expansion in the stated parameter regime. No invented entities appear.

assumptions (6)
  • domain assumption The 4PN EFT Hamiltonian for spinning binaries, including all spin terms up to NNLO spin-orbit and spin-spin and LO cubic/quartic, as derived by Levi and Steinhoff, is correct.
    The paper adopts these expressions (Appendices D and E, refs [25-30]) without re-deriving them; the 4PN spin part of the criterion inherits their validity.
  • domain assumption The stability of a circular orbit follows from the linearized evolution matrix: the orbit is stable iff all eigenfrequencies satisfy sigma_p^2 > 0, and the ISCO is the zero of the resulting criterion C(x).
    This is the definition used in Sec. II.B-II.C, extending the spinless analysis of [9]; it equates ISCO with the loss of linear stability of exactly circular conservative orbits.
  • domain assumption For aligned spins, perturbations of the transverse spin components decouple from the orbital instability; the spins act as non-dynamical spectators.
    Justified at 1PN by the eigenvalue computation in Eq. (2.19); assumed to remain true at 4PN when deriving the criterion via the Hamiltonian (Eqs. (2.22)-(2.26)).
  • domain assumption The post-Newtonian series converges well enough at the ISCO for the parameter ranges where the paper claims agreement with GSF and MPD results.
    Used throughout Sec. III; the paper itself notes the breakdown for chi2 near +1, so the assumption is restricted to retrograde and moderate prograde spins.
  • domain assumption The corotation condition (3.39), taken from Blanchet-Buonanno-Le Tiec [52], correctly fixes the spins of corotating BHs in terms of the orbital frequency.
    Used in Sec. III.E to obtain the corotating criterion (3.41) and the stability comparison with irrotational binaries.
  • standard math The non-local 4PN tail term is consistently perturbed around circular orbits using the Hadamard Partie finie regularization.
    Same treatment as in the authors' previous spinless work [9]; the paper states it is 'crucial' and must be perturbed.

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Cite this review

Pith. "Pith review of Innermost stable circular orbit (ISCO) of arbitrary-mass compact binaries with spins at the fourth post-Newtonian order." pith.science (2026). https://pith.science/paper/35IUL5QB

@misc{pith2026260807756,
  author       = {Pith},
  title        = {Pith review of: Innermost stable circular orbit (ISCO) of arbitrary-mass compact binaries with spins at the fourth post-Newtonian order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35IUL5QB}},
  note         = {Machine review of arXiv:2608.07756}
}
read the original abstract

We compute, up to 4th post-Newtonian (4PN) order, the gauge-invariant stability criterion determining the innermost stable circular orbit (ISCO) for arbitrary-mass compact binaries with spins aligned with the orbital angular momentum. Our calculation, a perturbation analysis, incorporates spin-orbit and spin-spin contributions, both up to next-to-next-to-leading order, as well as leading-order cubic and quartic spin contributions, all calculated in prior works within the effective-field-theory (EFT) Hamiltonian formalism. In the test-mass limit, we recover the Kerr ISCO truncated at 4PN order. Estimating the ISCO shift at first order in the mass ratio and comparing our results with numerical gravitational self-force (GSF) calculations, we find good agreement for retrograde spin (the smallest discrepancy being obtained for maximal retrograde spin) and moderate prograde spin. For nearly maximal prograde spin, the PN approach fails to determine the ISCO, for it is close to the black hole (BH) horizon. We also estimate the ISCO shift due to the test-particle spin and compare our result with the exact prediction from the Mathisson-Papapetrou-Dixon (MPD) equations for a spinning particle orbiting a Kerr black hole. Finally, we discuss the stability and the ISCO of corotating black-hole binaries.

Figures

Figures reproduced from arXiv: 2608.07756 by the authors.

Figure 1
Figure 1. FIG. 1: Comparison of the ISCO shift coefficient [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: ISCO shift coefficient [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Comparison between the ISCO shift coefficients [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Linear-perturbation coefficients [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.