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REVIEW 3 major objections 5 minor 87 references

Gravitational radiation reaction for compact binary systems at the fourth-and-a-half post-Newtonian order in harmonic coordinates

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper derives the 4.5PN radiation-reaction acceleration of compact binaries in the unique global harmonic gauge and proves the flux-balance laws and Lorentz invariance.

desk verdict The first harmonic-coordinate 4.5PN radiation-reaction acceleration, with solid cross-checks and a real open question about the DR pole—worth serious refereeing. read the letter →

arxiv 2601.06743 v3 pith:J6XCQWKN submitted 2026-01-11 gr-qc

Luc Blanchet , Guillaume Faye , Emeric Seraille , David Trestini This is my paper
classification gr-qc PACS 04.25.Nx04.30.-w
keywords post-Newtoniantheoryradiationreactionharmoniccoordinatescompactbinariesflux-balancelawsdimensionalregularizationLorentzinvariancemultipolemoments
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 derives the gravitational radiation-reaction force for a compact binary at 4.5PN order — a factor of 1/c^9 beyond Newton's acceleration — but expressed in harmonic coordinates rather than the simpler Burke-Thorne gauge. To do this it constructs the coordinate shift between the two gauges, computing auxiliary gauge multipole moments with dimensional regularization. One of those moments, the gauge dipole, develops a 1/ε pole, which produces logarithmic and time-nonlocal hereditary terms in the final acceleration. The paper then proves that this acceleration obeys the flux-balance equations for energy, angular momentum, linear momentum, and center-of-mass position, and verifies its manifest Lorentz invariance. This completes the equations of motion of nonspinning compact binaries at 4.5PN order in harmonic coordinates, a coordinate system well suited to comparisons with self-force and post-Minkowskian approaches.

What carries the argument

The contact transformation: a coordinate shift from the extended Burke-Thorne coordinates (where the leading reaction force is a simple multiple of the fifth time-derivative of the quadrupole) to the unique global harmonic coordinates selected by the no-incoming-radiation condition. The shift is built from the linear gauge vector φ(1) = ξ(1) + ζ(1) and the quadratic solution φ(2); the linear vector combines the piece that undoes the earlier gauge with the piece generated by the gauge moments W, X, Y, Z. The workhorse is the 2PN-accurate gauge dipole Y_i, the only moment that needs dimensional regularization at this order; its 1/ε pole generates the pole, logarithmic, and nonlocal hereditary

What would settle it

Recompute the 2PN gauge dipole Y_i for the binary directly in d dimensions with an independent regularization scheme. It is the only moment with a 1/ε pole at this order, and Eq. (4.12) fixes both the pole and the logarithmic finite part; any difference in that expression would propagate through the contact transformation and alter Eqs. (5.6).

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Extended reading notes

Core claim

The central claim is that Eqs. (5.1)–(5.6) give the 4.5PN radiation-reaction acceleration of each body in the unique global harmonic coordinate system, obtained by a contact transformation from the earlier extended Burke-Thorne result. The transformation has a linear piece, built from a gauge vector that undoes the earlier gauge and a second vector built from the four gauge multipole families W, X, Y, Z, plus a quadratic piece solving the wave equation with a source quadratic in the metric and shift. The gauge dipole Y_i, computed to 2PN order in d dimensions, carries a 1/ε pole; this is the origin of the logarithmic and time-nonlocal hereditary terms in the acceleration. The paper shows tha

Load-bearing premise

The load-bearing input is the earlier 4.5PN radiation-reaction acceleration in the extended Burke-Thorne coordinates together with the identification of its canonical multipole moments with the source moments at the required order, which, if off at 4.5PN, would alter the contact transformation and the final harmonic acceleration.

Editorial extensions

If this is right

  • This completes the 4.5PN equations of motion for nonspinning compact binaries in harmonic coordinates when combined with the known 4PN conservative dynamics.
  • The proved flux-balance laws imply that evolution computed from these equations is consistent with the radiated fluxes at infinity to 2PN relative accuracy; the Schott terms are the only gauge-dependent pieces.
  • Dropping the pole part of the gauge dipole Y_i yields an alternative acceleration that is still Lorentz invariant but has no 1/ε pole and no 4.5PN hereditary term, which the paper suggests may be more practical.
  • The twenty gauge parameters of the general 4.5PN parametrized reaction force are fixed uniquely by the harmonic gauge, and the parametrized method is corrected for the semi-hereditary recoil contribution.
  • The result provides a harmonic-coordinate reference against which gravitational self-force and post-Minkowskian effective-field-theory results can be compared.

Reading between the lines

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

  • A direct numerical test is to integrate the equations of motion with this acceleration and compare the orbital phase with the phase obtained from the energy flux; the paper's balance proof implies they agree to 4.5PN relative order, so a mismatch would indicate an implementation error rather than physics.
  • The arbitrary dimensional-regularization scale ℓ0 enters the nonlocal logarithmic terms; checking that physical observables such as the accumulated phase are independent of ℓ0 is a clean way to validate the regularization choice.
  • Because the pole part is separately Lorentz invariant, the split of Y_i into pole and 3D parts defines a family of coordinate prescriptions with identical asymptotic fluxes; comparing them would isolate which intermediate quantities are coordinate artifacts.
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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 / 5 minor

Summary. The paper derives the 4.5PN gravitational radiation-reaction (RR) acceleration for nonspinning compact binaries in harmonic coordinates. Starting from the authors' previous 4.5PN RR result in extended Burke-Thorne coordinates, they construct the coordinate shift to the (claimed unique) harmonic gauge, compute the required source and gauge multipole moments using dimensional regularization, and present the general-frame acceleration in Eqs. (5.1)-(5.6). They verify the flux-balance laws for energy, angular momentum, linear momentum, and CM position, compute the corresponding Schott terms, reduce the result to the CM frame, extract the Iyer-Will-Gopakumar gauge parameters, and check manifest Lorentz invariance (δΛ a1=0 in Eq. 7.8). The paper also reports agreement with independent post-Minkowskian RR results.

Significance. If the central claim is correct, this completes the 4.5PN equations of motion in harmonic coordinates and provides a Lorentz-invariant, general-frame RR force with considerably fewer terms than the BT-gauge expression. The paper has substantial strengths: it includes a machine-readable Supplemental Material file, multiple internal consistency checks (flux balance, CM reduction, Lorentz invariance), and an external cross-check against 2PM results. The derivation is long but organized, and the presentation of the abstract multipole-moment framework in d dimensions is valuable in itself. However, the scheme-dependence of the DR split in Secs. IV-V is a load-bearing issue that must be addressed before the result can be regarded as uniquely defined.

major comments (3)
  1. [Sec. IV after Eq. (4.12); Sec. V after Eq. (5.4)] The split Y_i = Y^3D_i + Y^pole_i is called 'rather arbitrary' (after Eq. 4.12), and the later distribution of finite logarithmic and hereditary terms between Eqs. (5.5) and (5.6) is called 'somewhat arbitrary'. No renormalization condition is given to fix the finite part of Y^pole, and the paper does not prove that the total acceleration (5.1)-(5.6) is independent of this split. A different split, e.g. moving the 3 ln(√q r12/ℓ0) term from Y^pole into Y^3D, would change the individual terms. Moreover, the pole part (5.6b) contains a 1/ε divergence, so the general-frame acceleration of body 1 is singular as ε→0. The paper states that the pole cancels in the relative acceleration, but the central claim is that Eqs. (5.1)-(5.6) give the harmonic-coordinate RR acceleration of each body. This requires a definite prescription or a proof of split-independence, and currently neither is supplied.
  2. [Sec. II, footnote 5] The uniqueness argument for the harmonic gauge is given for a smooth C∞ source. The compact-binary problem treated here uses point particles with dimensional regularization, and the DR pole in Y_i (Eq. 4.12) shows that the singular limit is not smooth. The paper does not extend the uniqueness proof to this singular, ε-dependent setting. Thus the phrase 'the unique global harmonic coordinate system' used in the abstract and Sec. II is not established for the actual problem; the arbitrary split in Sec. IV may correspond to different DR subtraction schemes and hence to different 'harmonic' coordinate prescriptions. The authors should either prove uniqueness for the regularized problem or reformulate the claim with an explicit renormalization/finite-part prescription.
  3. [Sec. III.B and Appendix A] The proof that the homogeneous solution φ_hom does not contribute to RR effects is essential for the quadratic shift φ^i_(2) in Eq. (3.26). The text says 'We have checked that the same reasoning yields the same conclusion for all the contributions' (Sec. III.B) and Appendix A similarly relies on 'a similar analysis' for vanishing commutators. These are not merely cosmetic statements: if any non-zero homogeneous term survived, the coordinate shift and hence the final acceleration would change. Given the length and complexity of the derivation, the authors should provide the full computation or a reproducible ancillary check for these steps rather than leaving them as asserted verifications.
minor comments (5)
  1. [Eq. (4.12)] The notation q̄ is introduced in Eq. (4.12) but the definition appears only later in the text; please define it at first use.
  2. [Sec. V, Eqs. (5.6b)-(5.6d)] The split into 'residue', 'finite', and 'hereditary' parts is not unique. Please document exactly which O(ε^0) terms are assigned to the pole part and what criterion (if any) selects them.
  3. [Sec. VII, Eq. (7.8)] The Lorentz-invariance check is stated as 'After an explicit computation, we obtain δΛ a1 = 0'. Given that this is a central consistency check, it would be helpful to include at least a summary of the computation or to point to the Supplemental Material for the intermediate expressions.
  4. [Footnote 2 and Acknowledgments] The agreement with PM results is reported via private communication. Since this is an important external validation, please include a quantitative comparison or provide the explicit mapping in a table or in the Supplement.
  5. [Throughout] The phrases 'rather arbitrarily' and 'somewhat arbitrarily' (Secs. IV and V) should be replaced by a precise definition of the split and the finite-part prescription; otherwise the reader cannot assess the scheme dependence of the result.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; the derivation is a coordinate transformation of an independent prior BT-coordinate result, with minor self-citation caveats.

full rationale

The central 4.5PN harmonic RR acceleration (5.1)-(5.6) is not obtained by fitting a parameter to the target quantity or by defining the target in terms of itself. It is constructed by applying the contact transformation (3.27)-(3.27c), a_i_RR1|_harm = a_i_RR1|_BT + delta_psi a_i1, to the previously computed BT-coordinate RR acceleration of Ref. [27]. That Ref. [27] is a same-author prior derivation, so there is a load-bearing self-citation, but it is an independent first-principles calculation, not a parameter fitted to the present result. The replacement M_L = I_L + O(c^-5), used after Eq. (3.7), is explicitly stated to affect only 5PN order, so it is not load-bearing at the claimed 4.5PN level. The uniqueness of harmonic coordinates is argued inside the paper (footnote 5) by the Fresnel-Kirchhoff representation and no-incoming-radiation condition, not simply imported as an unexamined self-citation. The paper's own caveats, 'we split Y_i rather arbitrarily' (after Eq. 4.12) and 'somewhat arbitrarily' (before Eq. 5.6), concern the split between 3D and pole parts of one moment and of the acceleration. Since the total Y_i = Y^{3D}_i + Y^{pole}_i and the total 4.5PN acceleration is the sum of the two displayed parts, that split is a bookkeeping/presentation choice rather than a fitted input or a prediction equal to its input by construction. The flux-balance checks (5.7) and the Lorentz-invariance check delta_Lambda a_1 = 0 (7.8) are consistency checks on the total expression, and the comparison with the independent PM calculation in Ref. [37] provides external corroboration. No equation is shown to reduce to itself by construction, so I find no circular step; the score reflects only the self-citation and split-arbitrariness caveats, not a circular derivation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the authors' prior BT 4.5PN calculation [27], the canonical/source moment identification from [66], and the standard MPM-PN/DR machinery. The only new 'free' quantity is the DR scale ℓ0, which enters gauge-dependent expressions but drops out of the final CM acceleration. No new particles, forces, or physical entities are introduced.

free parameters (1)
  • DR length scale ℓ0 = arbitrary; not fitted
    Introduced in (2.4a); inherited by the gauge moment Y_i pole (4.12) and by the 4.5PN acceleration (5.6) and Schott terms (5.13). It cancels in the final CM formulas presented, so it is not a physically fitted constant, but it is a regularization scale that affects intermediate gauge-dependent expressions.
assumptions (5)
  • domain assumption No-incoming-radiation condition at past null infinity selects the unique global harmonic coordinate system.
    Invoked in Sec. II, footnote 5, and used in the Fresnel-Kirchhoff argument to prove uniqueness. If incoming radiation were present, the harmonic metric and the RR force would differ.
  • domain assumption Compact binaries are modeled as point particles with no spins, and dimensional regularization is used to treat UV divergences.
    The whole PN iteration and DR computation (Secs. II-IV) assumes point masses with delta-function stress-energy and no internal-structure corrections at 4.5PN.
  • domain assumption Canonical moments M_L and S_L in the BT metric can be replaced by source moments I_L and J_L with M_L = I_L + O(c^-5), S_L = J_L + O(c^-5).
    Stated in Sec. III.A after Eq. (3.7), citing [66]. This is load-bearing: the BT RR input [27] is transcribed in terms of source moments, and any O(c^-5) correction would propagate through the coordinate shift.
  • domain assumption The d-dimensional MPM-PN matching construction gives the correct gauge moments, including the 2PN-order pole in Y_i, and no other pole is needed at this order.
    Sec. II and IV assert that DR is only needed for the 2PN-accurate moment Y_i among the gauge moments, and that source moments have no poles before 3PN. This justifies the restriction of the d-dimensional calculation.
  • standard math Standard mathematical tools: dimensional regularization, Hadamard partie finie, and distributional (Gel'fand-Shilov) derivatives are applicable.
    Used throughout Secs. II-IV without re-derivation, e.g. in (4.3)-(4.11). These are standard methods in the PN literature.

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Pith. "Pith review of Gravitational radiation reaction for compact binary systems at the fourth-and-a-half post-Newtonian order in harmonic coordinates." pith.science (2026). https://pith.science/paper/J6XCQWKN

@misc{pith2026260106743,
  author       = {Pith},
  title        = {Pith review of: Gravitational radiation reaction for compact binary systems at the fourth-and-a-half post-Newtonian order in harmonic coordinates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6XCQWKN}},
  note         = {Machine review of arXiv:2601.06743}
}
read the original abstract

We derive the gravitational radiation-reaction (RR) force in the harmonic coordinate system at the fourth-and-a-half post-Newtonian (4.5PN) order in the case of compact binary systems. Dimensional regularization is used to treat the ultra-violet divergences which appear at that order. We prove that the RR acceleration implies the known radiation fluxes at infinity associated with energy, angular momentum, linear momentum, and center-of-mass position. As a consistency check, we verify the manifest Lorentz invariance of the RR acceleration in harmonic coordinates. Our result should be useful for comparisons with other approaches such as the gravitational self force (GSF) and the post-Minkowskian (PM) effective field theory.

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Works this paper leans on

87 extracted references · 44 linked inside Pith

  1. [27]

    Jaranowski and G

    P. Jaranowski and G. Sch¨ afer, Radiative 3.5 post-Newtonian ADM Hamiltonian for many body point - mass systems, Phys. Rev. D55, 4712 (1997)

    1997

  2. [1]

    Quadratic-type terms∂P (C) 1 ∂P (C) 2 at relative 1PN order, with commutator □−1 inst,□ (P (C) 1 P (C) 2 ) = 1 c2 3S (1) 1 S (1) 2 +S 1S (2) 2 +S (2) 1 S2 +O 1 c4 .(A34) Since we always haveP (C) 1 =P (C) 2 =Vin our case, the monopolesS A (withA= 1,2) are both equal to the total massmof the binary, hence the commutator vanishes at the 1PN order

  3. [2]

    instantaneous

    This yields ∆0 (2) = 16G UW(3) c8 +O 1 c10 , ∆i (2) = 4G c9 4UY (3) i −2U jI(4) ij −x jUI (5) ij +O 1 c9 ,(A26) where we have truncated consistently with the 4.5PN order required for φα (2) in the contact transformation (3.27b). The solution (3.26) for φα (2) is recovered directly in the near zone from φα (2) =□ −1 inst∆α (2) (see Sec. (4.2) of [27]). To ...

  4. [3]

    Quadratic-type termsx i∂jP (C) 1 ∂P (C) 2 up to the relative 1PN order at most, with commutator □−1 inst,□ (xi∂jP (C) 1 P (C) 2 ) =− δij c2 S (1)S (1) 2 + 2 3 S1S (2) 2 + 1 3 S (2) 1 S2 +O 1 c4 .(A35) Again,P (C) 1 =P (C) 2 =Vfor us and the commutator reduces to zero, neglecting the 2PN remainder

  5. [4]

    Quadratic-type termsx ixj∂P (C) 1 ∂P (C) 2 at leading order, with commutator □−1 inst,□ (xixjP (C) 1 P (C) 2 ) =− 1 3 δijS1S2 +O 1 c2 .(A36) Again, the potentialsP (C) 1 andP (C) 2 are both equal toV. In terms of the form∂P (C) 1 P (C) 2 where the first potential is differentiated, we may directly apply the above equation provided we regardS 1 as referrin...

  6. [5]

    Putting together the four categories of contributions, we obtain □−1 inst,□ Ω 00 (2) = 14G3m 3c11 W (4) ,(A38) while the commutator of the other components of Ωαβ (2) vanishes

    Quadratic-type termsx ixjxk∂lP (C) 1 ∂P (C) 2 at leading order, with commutator □−1 inst,□ (xixjxk∂lP (C) 1 P (C) 2 ) = 1 5 δ(ijδkl)S1S2 .(A37) Those are contracted either with M (5) ij δkl or with M (5) il δjk , yielding zero. Putting together the four categories of contributions, we obtain □−1 inst,□ Ω 00 (2) = 14G3m 3c11 W (4) ,(A38) while the commutat...

  7. [6]

    Chandrasekhar and F

    S. Chandrasekhar and F. Esposito, The 5/2-post-Newtonian equations of hydrodynamics and radiation reaction in general relativity, Astrophys. J.160, 153 (1970)

    1970

  8. [7]

    Burke and K

    W. Burke and K. Thorne, Gravitational radiation damping, inRelativity, edited by M. Carmeli, S. Fickler, and L. Witten (Plenum Press, New York and London, 1970) pp. 209–228

    1970

Show all 87 references
  1. [8]

    Burke, Gravitational radiation damping of slowly moving systems calculated using matched asymptotic expansions, J

    W. Burke, Gravitational radiation damping of slowly moving systems calculated using matched asymptotic expansions, J. Math. Phys.12, 401 (1971)

    1971

  2. [9]

    Misner, K

    C. Misner, K. Thorne, and J. Wheeler,Gravitation(Freeman, San Francisco, 1973)

    1973

  3. [10]

    Miller, The effect of gravitational radiation-reaction on the evolution of the riemann s-type ellipsoids, Astrophys

    B. Miller, The effect of gravitational radiation-reaction on the evolution of the riemann s-type ellipsoids, Astrophys. J. 187, 609 (1974)

    1974

  4. [11]

    Walker and C

    M. Walker and C. Will, The approximation of radiative effects in relativistic gravity — gravitational radiation reaction and energy loss in nearly Newtonian systems, Astrophys. J.242, L129 (1980). 44

    1980

  5. [12]

    Ehlers, Isolated systems in general relativity, Ann

    J. Ehlers, Isolated systems in general relativity, Ann. N.Y. Acad. Sci.336, 279 (1980)

    1980

  6. [13]

    Kerlick, Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity

    G. Kerlick, Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity. i. first post- Newtonian equations, Gen. Relativ. Gravit.12, 467 (1980)

    1980

  7. [14]

    Kerlick, Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity

    G. Kerlick, Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity. ii. radiation reaction and higher-order divergent terms, Gen. Relativ. Gravit.12, 521 (1980)

    1980

  8. [15]

    Papapetrou and B

    A. Papapetrou and B. Linet, Equation of motion including the reaction of gravitational radiation, Gen. Relativ. Gravit. 13, 335 (1981)

    1981

  9. [16]

    Sch¨ afer, The equations of motion for an astrophysical binary with accuracy 1/c 5, Progress of Theoretical Physics68, 2191 (1982)

    G. Sch¨ afer, The equations of motion for an astrophysical binary with accuracy 1/c 5, Progress of Theoretical Physics68, 2191 (1982)

    1982

  10. [17]

    Blanchet and T

    L. Blanchet and T. Damour, Multipolar radiation reaction in general relativity, Phys. Lett. A104, 82 (1984)

    1984

  11. [18]

    Damour, An introduction to the theory of gravitational radiation, inGravitation in Astrophysics

    T. Damour, An introduction to the theory of gravitational radiation, inGravitation in Astrophysics. Carg` ese 1986, edited by B. Carter and J. Hartle (Plenum Press, New York and London, 1986) pp. 3–62

    1986

  12. [19]

    Sch¨ afer, On often used gauge transformations in gravitational radiation reaction calculations, Lett

    G. Sch¨ afer, On often used gauge transformations in gravitational radiation reaction calculations, Lett. Nuovo Cim.36, 105 (1983)

    1983

  13. [20]

    Damour, Probl` eme des deux corps et freinage de rayonnement en relativit´ e g´ en´ erale, C

    T. Damour, Probl` eme des deux corps et freinage de rayonnement en relativit´ e g´ en´ erale, C. R. Acad. Sc. Paris294, 1355 (1982)

    1982

  14. [21]

    Damour, Gravitational radiation reaction in the binary pulsar and the quadrupole formula controvercy, Phys

    T. Damour, Gravitational radiation reaction in the binary pulsar and the quadrupole formula controvercy, Phys. Rev. Lett. 51, 1019 (1983)

    1983

  15. [22]

    Iyer and C

    B. Iyer and C. Will, Post-Newtonian gravitational-radiation reaction for two-body systems, Phys. Rev. Lett.70, 113 (1993)

    1993

  16. [23]

    Iyer and C

    B. Iyer and C. Will, Post-Newtonian gravitational radiation reaction for two-body systems: Nonspinning bodies, Phys. Rev. D52, 6882 (1995)

    1995

  17. [24]

    Gopakumar, B

    A. Gopakumar, B. R. Iyer, and S. Iyer, Second post-Newtonian gravitational radiation reaction for two-body systems: Nonspinning bodies, Phys. Rev. D55, 6030 (1997), gr-qc/9703075

    1997 arXiv

  18. [25]

    Blanchet, Time asymmetric structure of gravitational radiation, Phys

    L. Blanchet, Time asymmetric structure of gravitational radiation, Phys. Rev. D47, 4392 (1993)

    1993

  19. [26]

    Blanchet, Gravitational radiation reaction and balance equations to post-Newtonian order, Phys

    L. Blanchet, Gravitational radiation reaction and balance equations to post-Newtonian order, Phys. Rev. D55, 714 (1997), gr-qc/9609049

    1997 arXiv

  20. [28]

    Pati and C

    M. Pati and C. Will, Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed einstein equations. ii. two-body equations of motion to second post-Newtonian order, and radiation-reaction to 3.5 post- Newtonian order, Phys. Rev. D65, 1...

    2002 arXiv

  21. [29]

    K¨ onigsd¨ orffer, G

    C. K¨ onigsd¨ orffer, G. Faye, and G. Sch¨ afer, The binary black-hole dynamics at the third-and-a-half post-Newtonian order in the ADM-formalism, Phys. Rev. D68, 044004 (2003), astro-ph/0305048

    2003 arXiv

  22. [30]

    Nissanke and L

    S. Nissanke and L. Blanchet, Gravitational radiation reaction in the equations of motion of compact binaries to 3.5 post- Newtonian order, Class. Quant. Grav.22, 1007 (2005), gr-qc/0412018

    2005 arXiv

  23. [31]

    Itoh, Third-and-a-half order post-Newtonian equations of motion for relativistic compact binaries using the strong field point particle limit, Phys

    Y. Itoh, Third-and-a-half order post-Newtonian equations of motion for relativistic compact binaries using the strong field point particle limit, Phys. Rev. D80, 024003 (2009), arXiv:0911.4232 [gr-qc]

    2009 arXiv

  24. [32]

    Blanchet, G

    L. Blanchet, G. Faye, and D. Trestini, Gravitational radiation reaction for compact binary systems at the fourth-and-a-half post-newtonian order, Classical and Quantum Gravity42, 065015 (2025)

    2025

  25. [33]

    A. K. Leibovich, B. A. Pardo, and Z. Yang, Radiation reaction for non-spinning bodies at 4.5 pn in the effective field theory approach, arXiv preprint arXiv:2302.11016 (2023)

    2023 arXiv

  26. [34]

    Blanchet and T

    L. Blanchet and T. Damour, Radiative gravitational fields in general relativity. i. general structure of the field outside the source, Phil. Trans. Roy. Soc. Lond. A320, 379 (1986)

    1986

  27. [35]

    Blanchet, Radiative gravitational fields in general relativity

    L. Blanchet, Radiative gravitational fields in general relativity. ii. asymptotic behaviour at future null infinity, Proc. Roy. Soc. Lond. A409, 383 (1987)

    1987

  28. [36]

    Blanchet and T

    L. Blanchet and T. Damour, Tail-transported temporal correlations in the dynamics of a gravitating system, Phys. Rev. D37, 1410 (1988)

    1988

  29. [37]

    Blanchet and T

    L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D46, 4304 (1992)

    1992

  30. [38]

    Blanchet, On the multipole expansion of the gravitational field, Class

    L. Blanchet, On the multipole expansion of the gravitational field, Class. Quant. Grav.15, 1971 (1998), gr-qc/9801101

    1971 arXiv

  31. [39]

    Poujade and L

    O. Poujade and L. Blanchet, Post-Newtonian approximation for isolated systems calculated by matched asymptotic ex- pansions, Phys. Rev. D65, 124020 (2002), gr-qc/0112057

    2002 arXiv

  32. [40]

    Blanchet, G

    L. Blanchet, G. Faye, and S. Nissanke, Structure of the post-Newtonian expansion in general relativity, Phys. Rev. D72, 044024 (2005)

    2005

  33. [41]

    Schott, On the motion of the lorentz electron, Phil

    G. Schott, On the motion of the lorentz electron, Phil. Mag.29, 49 (1915)

    1915

  34. [42]

    D. Bini, T. Damour, and A. Geralico, Explicit solution of the gravitational two-body problem at the second post- minkowskian order (2024), arXiv:2408.17193 [gr-qc]

    2024 arXiv

  35. [43]

    Jaranowski and G

    P. Jaranowski and G. Sch¨ afer, Towards the fourth post-Newtonian Hamiltonian for two-point-mass systems, Phys. Rev. D 86, 061503(R) (2012), arXiv:1207.5448 [gr-qc]

    2012 arXiv

  36. [44]

    Jaranowski and G

    P. Jaranowski and G. Sch¨ afer, Dimensional regularization of local singularities in the 4th post-Newtonian two-point-mass hamiltonian, Phys. Rev. D87, 081503(R) (2013), arXiv:1303.3225 [gr-qc]

    2013 arXiv

  37. [45]

    Bini and T

    D. Bini and T. Damour, Analytical determination of the two-body gravitational interaction potential at the fourth post- Newtonian approximation, Phys. Rev. D87, 121501(R) (2013), arXiv:1305.4884 [gr-qc]

    2013 arXiv

  38. [46]

    Damour, P

    T. Damour, P. Jaranowski, and G. Sch¨ afer, Non-local-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems, Phys. Rev. D89, 064058 (2014), arXiv:1401.4548 [gr-qc]. 45

    2014 arXiv

  39. [47]

    Jaranowski and G

    P. Jaranowski and G. Sch¨ afer, Derivation of the local-in-time fourth post-Newtonian ADM Hamiltonian for spinless compact binaries, Phys. Rev. D92, 124043 (2015), arXiv:1508.01016 [gr-qc]

    2015 arXiv

  40. [48]

    Damour, P

    T. Damour, P. Jaranowski, and G. Sch¨ afer, Fourth post-Newtonian effective one-body dynamics, Phys. Rev. D91, 084024 (2015), arXiv:1502.07245 [gr-qc]

    2015 arXiv

  41. [49]

    Bernard, L

    L. Bernard, L. Blanchet, A. Boh´ e, G. Faye, and S. Marsat, Fokker action of non-spinning compact binaries at the fourth post-Newtonian approximation, Phys. Rev. D93, 084037 (2016), arXiv:1512.02876 [gr-qc]

    2016 arXiv

  42. [50]

    Bernard, L

    L. Bernard, L. Blanchet, A. Boh´ e, G. Faye, and S. Marsat, Energy and periastron advance of compact binaries on circular orbits at the fourth post-Newtonian order, Phys. Rev. D95, 044026 (2017), arXiv:1610.07934 [gr-qc]

    2017 arXiv

  43. [51]

    Bernard, L

    L. Bernard, L. Blanchet, A. Boh´ e, G. Faye, and S. Marsat, Dimensional regularization of the ir divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order, Phys. Rev. D96, 104043 (2017), arXiv:1706.08480 [gr-qc]

    2017 arXiv

  44. [52]

    Marchand, L

    T. Marchand, L. Bernard, L. Blanchet, and G. Faye, Ambiguity-free completion of the equations of motion of compact binary systems at the fourth post-Newtonian order, Phys. Rev. D97, 044023 (2018), arXiv:1707.09289 [gr-qc]

    2018 arXiv

  45. [53]

    Bernard, L

    L. Bernard, L. Blanchet, G. Faye, and T. Marchand, Center-of-mass equations of motion and conserved integrals of compact binary systems at the fourth post-Newtonian order, Phys. Rev. D97, 044037 (2018), arXiv:1711.00283 [gr-qc]

    2018 arXiv

  46. [54]

    Foffa and R

    S. Foffa and R. Sturani, The dynamics of the gravitational two-body problem in the post-Newtonian approximation at quadratic order in the Newton’s constant, Phys. Rev. D87, 064011 (2013), arXiv:1206.7087 [gr-qc]

    2013 arXiv

  47. [55]

    Foffa, P

    S. Foffa, P. Mastrolia, R. Sturani, and C. Sturm, Effective field theory approach to the gravitational two-body dynamics at fourth post-Newtonian order and quintic in the newton constant, Phys. Rev. D95, 104009 (2017), arXiv:1612.00482 [gr-qc]

    2017 arXiv

  48. [56]

    R. A. Porto and I. Z. Rothstein, Apparent ambiguities in the post-newtonian expansion for binary systems, Physical Review D96, 024062 (2017), arXiv:1703.06463 [gr-qc]

    2017 arXiv

  49. [57]

    Foffa and R

    S. Foffa and R. Sturani, Conservative dynamics of binary systems to fourth post-Newtonian order in the EFT approach i: Regularized Lagrangian, Phys. Rev. D100, 024047 (2019), arXiv:1903.05113 [gr-qc]

    2019 arXiv

  50. [58]

    Foffa, R

    S. Foffa, R. Porto, I. Rothstein, and R. Sturani, Conservative dynamics of binary systems to fourth post-Newtonian order in the EFT approach ii: Renormalized Lagrangian, Phys. Rev. D100, 024048 (2019), arXiv:1903.05118 [gr-qc]

    2019 arXiv

  51. [59]

    Bl¨ umlein, A

    J. Bl¨ umlein, A. Maier, P. Marquard, and G. Sch¨ afer, Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach, Nuclear Physics B955, 115041 (2020), arXiv:2003.01692 [gr-qc]

    2020 arXiv

  52. [60]

    Foffa and R

    S. Foffa and R. Sturani, Tail terms in gravitational radiation reaction via effective field theory, Phys. Rev. D87, 044056 (2012), arXiv:1111.5488 [gr-qc]

    2012 arXiv

  53. [61]

    C. R. Galley, A. K. Leibovich, R. A. Porto, and A. Ross, Tail effect in gravitational radiation reaction: Time nonlocality and renormalization group evolution, Phys. Rev. D93, 124010 (2016), arXiv:arXiv:1511.07379 [gr-qc] [gr-qc]

    2016 arXiv

  54. [62]

    Trestini, Schott term in the binding energy for compact binaries on circular orbits at fourth post-Newtonian order, Phys

    D. Trestini, Schott term in the binding energy for compact binaries on circular orbits at fourth post-Newtonian order, Phys. Rev. D112, 024076 (2025), arXiv:2504.13245 [gr-qc]

    2025 arXiv

  55. [63]

    Blanchet, G

    L. Blanchet, G. Faye, Q. Henry, F. Larrouturou, and D. Trestini, Gravitational-Wave Phasing of Quasicircular Compact Binary Systems to the Fourth-and-a-Half Post-Newtonian Order, Phys. Rev. Lett.131, 121402 (2023), arXiv:2304.11185 [gr-qc]

    2023 arXiv

  56. [64]

    Blanchet, G

    L. Blanchet, G. Faye, Q. Henry, F. Larrouturou, and D. Trestini, Gravitational-wave flux and quadrupole modes from quasicircular nonspinning compact binaries to the fourth post-Newtonian order, Phys. Rev. D108, 064041 (2023), arXiv:2304.11186 [gr-qc]

    2023 arXiv

  57. [65]

    Sachs and P

    R. Sachs and P. Bergmann, Structure of particles in linearized gravitational theory, Phys. Rev.112, 674 (1958)

    1958

  58. [66]

    Pirani, Introduction to gravitational radiation theory, inLectures on General Relativity, Brandeis Summer Institute in Theoretical Physics, Vol

    F. Pirani, Introduction to gravitational radiation theory, inLectures on General Relativity, Brandeis Summer Institute in Theoretical Physics, Vol. 1, edited by A. Trautman, F. Pirani, and H. Bondi (Prentice-Hall, Englewood Cliffs, 1964) pp. 249–373

    1964

  59. [67]

    Thorne, Multipole expansions of gravitational radiation, Rev

    K. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys.52, 299 (1980)

    1980

  60. [68]

    Blanchet, T

    L. Blanchet, T. Damour, G. Esposito-Far` ese, and B. R. Iyer, Dimensional regularization of the third post-Newtonian gravitational wave generation of two point masses, Phys. Rev. D71, 124004 (2005), gr-qc/0503044

    2005 arXiv

  61. [69]

    Henry, G

    Q. Henry, G. Faye, and L. Blanchet, The current-type quadrupole moment and gravitational-wave mode (ℓ, m) = (2, 1) of compact binary systems at the third post-Newtonian order, Class. Quant. Grav.38, 185004 (2021), arXiv:2105.10876 [gr-qc]

    2021 arXiv

  62. [70]

    Born and E

    M. Born and E. Wolf,Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013)

    2013

  63. [71]

    Blanchet, G

    L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha, The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits, Class. Quant. Grav.25, 165003 (2008), arXiv:0802.1249 [gr-qc]

    2008 arXiv

  64. [72]

    Blanchet, G

    L. Blanchet, G. Faye, and F. Larrouturou, The quadrupole moment of compact binaries to the fourth post-Newtonian order: from source to canonical moment, Classical and Quantum Gravity39, 195003 (2022)

    2022

  65. [73]

    Trestini, F

    D. Trestini, F. Larrouturou, and L. Blanchet, The quadrupole moment of compact binaries to the fourth post-newtonian order: relating the harmonic and radiative metrics, Classical and Quantum Gravity40, 055006 (2023)

    2023

  66. [74]

    Blanchet and T

    L. Blanchet and T. Damour, Post-Newtonian generation of gravitational waves, Annales Inst. H. Poincar´ e Phys. Th´ eor. 50, 377 (1989)

    1989

  67. [75]

    Marchand, Q

    T. Marchand, Q. Henry, F. Larrouturou, S. Marsat, G. Faye, and L. Blanchet, The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order, Class. Quant. Grav.37, 215006 (2020), arXiv:2003.13672 [gr-qc]. 46

    2020 arXiv

  68. [76]

    Larrouturou, Q

    F. Larrouturou, Q. Henry, L. Blanchet, and G. Faye, The quadrupole moment of compact binaries to the fourth post-Newtonian order: I. Non-locality in time and infra-red divergencies, Class. Quant. Grav.39, 115007 (2022), arXiv:2110.02240 [gr-qc]

    2022 arXiv

  69. [77]

    Larrouturou, L

    F. Larrouturou, L. Blanchet, Q. Henry, and G. Faye, The quadrupole moment of compact binaries to the fourth post-Newtonian order: II. Dimensional regularization and renormalization, Class. Quant. Grav.39, 115008 (2022), arXiv:2110.02243 [gr-qc]

    2022 arXiv

  70. [78]

    Blanchet, T

    L. Blanchet, T. Damour, and G. Esposito-Far` ese, Dimensional regularization of the third post-Newtonian dynamics of point particles in harmonic coordinates, Phys. Rev. D69, 124007 (2004), gr-qc/0311052

    2004 arXiv

  71. [79]

    Blanchet and G

    L. Blanchet and G. Faye, Hadamard regularization, J. Math. Phys.41, 7675 (2000), gr-qc/0004008

    2000 arXiv

  72. [80]

    Schwartz,Th´ eorie des distributions(Hermann, Paris, 1978)

    L. Schwartz,Th´ eorie des distributions(Hermann, Paris, 1978)

    1978

  73. [81]

    I. M. Gel’fand and G. E. Shilov,Generalized functions(Academic Press, New York, 1964)

    1964

  74. [82]

    Note that this file is optimally read usingMathematica, but it can be straightforwardly parsed by any text editor

    The ancillary fileSupplementalMaterial.wlcontains the acceleration and related quantities in machine-readable format. Note that this file is optimally read usingMathematica, but it can be straightforwardly parsed by any text editor

  75. [83]

    Bonetti, F

    M. Bonetti, F. Haardt, A. Sesana, and E. Barausse, Post-Newtonian evolution of massive black hole triplets in galactic nuclei – I. Numerical implementation and tests, Mon. Not. Roy. Astron. Soc.461, 4419 (2016), arXiv:1604.08770 [astro- ph.GA]

    2016 arXiv

  76. [84]

    Blanchet and G

    L. Blanchet and G. Faye, Flux-balance equations for linear momentum and center-of-mass position of self-gravitating post-Newtonian systems, Classical and Quantum Gravity36, 085003 (2019)

    2019

  77. [85]

    Kozameh, J

    C. Kozameh, J. Nieva, and G. Quirega, Spin and center of mass comparison between the PN approach and the asymptotic formulation, Phys. Rev. D98, 064032 (2018), arXiv:1711.11375 [gr-qc]

    2018 arXiv

  78. [86]

    Comp` ere, R

    G. Comp` ere, R. Oliveri, and A. Seraj, The Poincar´ e and BMS flux-balance laws with application to binary systems, JHEP 10, 116, [Erratum: JHEP 06, 045 (2024)], arXiv:1912.03164 [gr-qc]

    2024 arXiv

  79. [87]

    Blanchet and G

    L. Blanchet and G. Faye, Lorentzian regularization and the problem of point-like particles in general relativity, J. Math. Phys.42, 4391 (2001), gr-qc/0006100

    2001 arXiv

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