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arxiv: 2110.15351 · v4 · submitted 2021-10-28 · 🌀 gr-qc · astro-ph.HE

Action-angle variables of a binary black hole with arbitrary eccentricity, spins, and masses at 1.5 post-Newtonian order

Sashwat Tanay , Leo C. Stein , Gihyuk Cho This is my paper

Pith reviewed 2026-05-24 12:36 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE
keywords binary black holespost-Newtonian dynamicsaction-angle variablesgravitational wavescanonical perturbation theoryphase space extensionintegrable systems
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The pith

The fifth action for 1.5PN binary black holes is obtained by extending phase space with unmeasurable coordinates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper completes the set of five action variables for a binary black hole with arbitrary eccentricity, spins, and masses at 1.5 post-Newtonian order. Four actions were found earlier; the fifth is derived here through a phase-space extension that introduces additional unmeasurable coordinates. With the full set available, the integrable 1.5PN Hamiltonian yields action-angle variables, frequencies, and an explicit map back to ordinary positions and momenta. This supplies a closed-form analytical solution for the conservative dynamics without orbit or precession averaging.

Core claim

The authors show that extending the phase space by adding unmeasurable coordinates allows construction of the missing fifth action variable. Together with the previously computed four actions, this produces the complete action-angle coordinates for the integrable 1.5PN binary black hole system. The frequencies follow directly, and the transformation from action-angle variables to the usual canonical variables can be written explicitly, thereby solving the conservative dynamics analytically at this order.

What carries the argument

Phase-space extension by unmeasurable coordinates, used to isolate the fifth action variable of the 1.5PN Hamiltonian.

If this is right

  • All five actions and their conjugate frequencies are now available at 1.5PN for arbitrary parameters.
  • The conservative motion can be integrated analytically by quadrature in action-angle coordinates.
  • Canonical perturbation theory can be applied to reach higher post-Newtonian orders using these variables as the starting point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same extension technique might be tested on the 2PN Hamiltonian if integrability is only approximate.
  • Waveform models could incorporate these actions to generate eccentric, precessing signals without averaging approximations.
  • Comparison of the derived frequencies against numerical relativity at low velocities would provide a direct check on the 1.5PN truncation.

Load-bearing premise

The 1.5 post-Newtonian binary black hole system must be integrable.

What would settle it

Numerical integration of the 1.5PN equations of motion over many orbits that shows the proposed fifth action is not constant would falsify the result.

Figures

Figures reproduced from arXiv: 2110.15351 by Gihyuk Cho, Leo C. Stein, Sashwat Tanay.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic setup of a precessing black hole binary. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: , a purely vertical displacement in the EPS space corresponds to changing the EPS coordinates in such a way that the observable coordinates do not change. To change the observable coordinates, a horizontal motion is needed, both in the SPS and the EPS. In addition to [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Two EPS points ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: Test 1 FIG. 3. Two EPS points (P and Q) with the same fictitious coordinates are mapped to new EPS points (P ′ and Q ′ ) again the the same fictitious coordinates by a flow under a general f [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic depiction of closing the loop in the EPS [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: Test 1 FIG. 4. Schematic depiction of closing the loop in the EPS over which the fifth action integral is computed. This is done by flowing under Seff ·L (red), J 2 (green), L 2 (blue), S 2 1 (black), and S 2 2 (orange). The curve in cyan is the one found by flowing under J5 in the EPS. The corresponding π projections of the solid curves in the EPS is shown by broken curves in the SPS with the same color. … view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The non-inertial [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The second non-inertial [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 0.1
Figure 0.1. Figure 0.1: Test 1 FIG. 6. The second non-inertial (i ′′j ′′k ′′) triad (centered around Sˆ1 ≡ S⃗1/S1) is displayed along with the inertial (ijk) triad (centered around Jˆ ≡ J/J ⃗ ). For the purposes of these calculations, the relevant figure is [PITH_FULL_IMAGE:figures/full_fig_p021_0_1.png] view at source ↗
read the original abstract

Accurate and efficient modeling of the dynamics of binary black holes (BBHs) is crucial to their detection through gravitational waves (GWs), with LIGO/Virgo/KAGRA, and LISA in the future. Solving the dynamics of a BBH system with arbitrary parameters without simplifications (like orbit- or precession-averaging) in closed form is one of the most challenging problems for the GW community. One potential approach is using canonical perturbation theory which constructs perturbed action-angle variables from the unperturbed ones of an integrable Hamiltonian system. Having action-angle variables of the integrable 1.5 post-Newtonian (PN) BBH system is therefore imperative. In this paper, we continue the work initiated by two of us in arXiv:2012.06586, where we presented four out of five actions of a BBH system with arbitrary eccentricity, masses, and spins, at 1.5PN order. Here we compute the remaining fifth action using a novel method of extending the phase space by introducing unmeasurable phase space coordinates. We detail how to compute all the frequencies, and sketch how to explicitly transform from the action-angle variables to the usual positions and momenta. This analytically solves the dynamics at 1.5PN. This lays the groundwork to analytically solve the conservative dynamics of the BBH system with arbitrary masses, spins, and eccentricity, at higher PN order, by using canonical perturbation theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to compute the fifth action variable for the 1.5 post-Newtonian binary black hole system with arbitrary eccentricity, masses, and spins by extending the phase space with unmeasurable coordinates. This completes the set of five actions begun in a prior work, enabling construction of action-angle variables for the asserted integrable 1.5PN Hamiltonian and an analytic solution of the conservative dynamics via canonical perturbation theory. The paper details frequency computations and sketches the inverse transformation to physical coordinates.

Significance. If the integrability assumption holds and the phase-space extension is rigorously justified, the result would provide a complete analytic framework for 1.5PN BBH dynamics without orbit or precession averaging. This is a meaningful step toward higher-order analytic templates for gravitational-wave detection. The novel extension technique is a clear methodological contribution that, if validated, supports extensions to higher PN orders.

major comments (2)
  1. [Abstract] Abstract and opening paragraphs: the 1.5PN BBH system is repeatedly described as integrable, yet no explicit demonstration or citation is given that five independent, Poisson-commuting integrals exist for generic eccentricity and spin configurations. This is load-bearing, as the entire action-angle construction and the applicability of canonical perturbation theory rest on it.
  2. [Method] Method section describing the phase-space extension: the new action is obtained by introducing unmeasurable coordinates, but the manuscript does not verify that the extended Hamiltonian preserves the required involution relations with the four previously computed actions or that the fifth integral remains independent for arbitrary parameters.
minor comments (1)
  1. The sketch of the transformation from action-angle to physical variables would benefit from an explicit example or additional intermediate equations to allow independent reproduction.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below, indicating planned revisions where appropriate. The work builds directly on our prior paper (arXiv:2012.06586) that computed the first four actions.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the 1.5PN BBH system is repeatedly described as integrable, yet no explicit demonstration or citation is given that five independent, Poisson-commuting integrals exist for generic eccentricity and spin configurations. This is load-bearing, as the entire action-angle construction and the applicability of canonical perturbation theory rest on it.

    Authors: We agree that the integrability assumption is central and that the manuscript does not contain an explicit proof or direct citation establishing five independent, Poisson-commuting integrals for generic eccentricity and spins at 1.5PN. The construction in the present work and in arXiv:2012.06586 proceeds by explicitly building the actions so that they are conserved and independent by design, relying on the structure of the 1.5PN Hamiltonian. We will revise the introduction and abstract to state clearly that integrability is assumed on the basis of this construction and to cite relevant literature on the integrability properties of post-Newtonian Hamiltonians. A complete, independent verification of the full set of involution relations for arbitrary parameters lies beyond the scope of the present paper. revision: partial

  2. Referee: [Method] Method section describing the phase-space extension: the new action is obtained by introducing unmeasurable coordinates, but the manuscript does not verify that the extended Hamiltonian preserves the required involution relations with the four previously computed actions or that the fifth integral remains independent for arbitrary parameters.

    Authors: The phase-space extension is chosen so that the new coordinate is ignorable in the extended Hamiltonian, ensuring by construction that the Poisson brackets with the four previously obtained actions vanish. We will expand the method section to include an explicit argument (based on the canonical nature of the extension and the functional form of the 1.5PN terms) that the involution relations are preserved and that the fifth action remains functionally independent for generic masses, spins, and eccentricity. This will be supported by the explicit expressions already derived in the paper. revision: yes

standing simulated objections not resolved
  • Explicit, parameter-independent proof that all five actions are in mutual involution for completely generic eccentricity, spins, and masses at 1.5PN order.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper continues prior self-cited work (arXiv:2012.06586) for four actions but introduces a novel phase-space extension method for the fifth action. The integrability assumption is explicitly stated as a premise for applying canonical perturbation theory, but the construction of the new action does not reduce by definition or construction to the inputs, fitted parameters, or the self-citation. No self-definitional, fitted-input, or load-bearing self-citation steps are present that would make the claimed analytic solution tautological. The derivation remains self-contained as an application of standard Hamiltonian methods with an independent extension technique.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the integrability of the 1.5PN Hamiltonian and the validity of the phase space extension technique for computing the missing action.

axioms (1)
  • domain assumption The 1.5PN BBH Hamiltonian is integrable
    Invoked to justify construction of action-angle variables from unperturbed ones.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Non-adiabatic dynamics of eccentric black-hole binaries in post-Newtonian theory

    gr-qc 2025-02 unverdicted novelty 7.0

    New non-orbit-averaged 2.5PN equations for eccentric non-spinning black-hole binaries derived via energy-momentum mappings, showing Peters 1964 orbit-averaged equations break at first pericenter.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · cited by 1 Pith paper · 18 internal anchors

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