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3rd post-Newtonian higher order Hamilton dynamics for two-body point-mass systems
General Relativity and Quantum Cosmology · Submitted 17 Dec 1997
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order.
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The abstract, sentence by sentence
The paper presents the conservative dynamics of two-body point-mass systems up to the third post-Newtonian order ($1/c^6$). The two-body dynamics is given in terms of a higher order ADM Hamilton function which results from a third post-Newtonian Routh functional for the total field-plus-matter system. The applied regularization procedures, together with making use of distributional differentiation of homogeneous functions, give unique results for the terms in the Hamilton function apart from the coefficient of the term $(\nu p_{i}{\pa_{i}})^2r^{-1}$. The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The paper presents the conservative dynamics of two-body point-mass systems up to the third post-Newtonian order ($1/c^6$). | contribution 0.99 | 0.91 | 0.09 | asserted 1.00 |
| s1 | The two-body dynamics is given in terms of a higher order ADM Hamilton function which results from a third post-Newtonian Routh functional for the total field-plus-matter system. | method 0.62 | 0.88 | 0.05 | asserted 1.00 |
| s2 | The applied regularization procedures, together with making use of distributional differentiation of homogeneous functions, give unique results for the terms in the Hamilton function apart from the coefficient of the term $(\nu p_{i}{\pa_{i}})^2r^{-1}$. | method 0.56 | 0.89 | 0.06 | asserted 1.00 |
| s3 | The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order. | interpretation 0.99 | 0.73 | 0.75 | hedged possibility 1.00 |
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The paper
Original abstract
The paper presents the conservative dynamics of two-body point-mass systems up to the third post-Newtonian order ($1/c^6$). The two-body dynamics is given in terms of a higher order ADM Hamilton function which results from a third post-Newtonian Routh functional for the total field-plus-matter system. The applied regularization procedures, together with making use of distributional differentiation of homogeneous functions, give unique results for the terms in the Hamilton function apart from the coefficient of the term $(\nu p_{i}{\pa_{i}})^2r^{-1}$. The result suggests an invalidation of the binary point-mass model at the third post-Newtonian order.
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References
How later papers used it
69 later papers cite it
2 later papers by other authors are listed below: 1 used its method or tool and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.
Uses its method or tool 2 of 9 papers with this label · 1 by the same authors
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Used a result from the theory of distributions given in this paper to define the distributional derivative of a positive homogeneous function.
Next-to-next-to-leading order post-Newtonian linear-in-spin binary Hamiltonians (2013) · 0.88 · paraphrased: the licence does not allow quoting
By the same authors
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Adopted suitable regularization procedures of integrals involved, citing the appendix in this paper, to determine the Hamiltonian after expanding the field equations in powers of G.
Reduced Hamiltonian for next-to-leading order Spin-Squared Dynamics of General Compact Binaries (2010) (same authors) · 0.93 · paraphrased: the licence does not allow quoting
Compares with it 1 of 5 papers with this label
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Checked that in the limit of large mass ratio its own result agrees with this paper's potential for a test body orbiting a Schwarzschild black hole to O(G^2) and all orders in velocity.
From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion (2018) · 0.98 · paraphrased: the licence does not allow quoting
How the literature uses it
62 arXiv papers cite this in 106 sentences
Papers citing it · 57 sentences without a confident label
Contests it 2 papers
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We therefore suggest that Bernard et al. [ref] should (similarly to what happened at the 3PN level when using only Hadamard-type regularization [ref] ) acknowledge the presence of further ambiguities in their result, parametrized by new ambiguity parameters.
Conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity · Suggestion for adding more IR ambiguity parameters in Ref. CITEMARK173Z · contest 0.77
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Unfortunately we know that the HR, even in the extended variant, is not entirely satisfying and cannot give a completely consistent picture at the 3PN order [ref] .
Post-Newtonian and Numerical Calculations of the Gravitational Self-Force for Circular Orbits in the Schwarzschild Geometry · Relation to Hadamard regularization · contest 0.76
Builds on it 1 paper
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As illustrated in table [ref], the analysis of this sector builds on the N$^2$LO spin-orbit sector obtained in [ref] , and in [ref] via the EFT of spinning gravitating objects, and on the non-spinning N$^3$LO sector (at the $3$PN order) [ref] .
N$^3$LO gravitational spin-orbit coupling at order $G^4$ · Diagrammatic expansion · 0.99
Uses its method or tool 9 papers
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From the Hamiltonian obtained by integrating the density from above one can go to a Routhian (a Hamiltonian in matter degrees of freedom and a Lagrangian in the propagating field degrees of freedom) as suggested in [ref] and can eliminate the propagating degrees of freedom by inserting their approximate solutions in terms of the matter variables [ref] .
Next-to-next-to-leading order post-Newtonian spin-orbit Hamiltonian for self-gravitating binaries · Outline of the calculation · 0.97
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Using the ordinary derivative it would give [equation] But as we already know from the constraint equations the second derivative of $2$ should be [equation] Fortunately there is a result from the theory of distributions [ref] which defines the so-called distributional derivative [equation] Here $f$ is a positive homogeneous function of degree $λ$ (i.e. $f(ax) = a^λf(x)$ for $a≥0$) and $k := -λ+ 1 -$ is a non-negative integer.
Next-to-next-to-leading order post-Newtonian linear-in-spin binary Hamiltonians · Distributional Contributions · 0.88
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At the level at which we are working, this regularization is not ambiguous and can, for instance, be simply performed by using Hadamard's “partie finie” regularization (as explained, e.g., in Appendix B of Ref. [ref] ).
Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling · Derivation of the spin-orbit Hamiltonian in ADM coordinates · 0.97
Uses its data or values 1 paper
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The explicit expressions for the metric functions entering Eqs. [ref] can be found e.g. in Appendix A of Ref. [ref] (where the functions $ϕ_(2)$, $ϕ_(4)$, $π̃_(3)^ij$, and $h^TT_(4)ij$ can be found) and in Ref. [ref] (where the functions $α_(2)$, $α_(4)$ and $β_(3)i$, $β_(5)i$ are given).
Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling · Derivation of the spin-orbit Hamiltonian in ADM coordinates · 0.83
Compares with it 3 papers
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The comparison of the result with the work [ref] , on the basis of the computation of the invariant energy of compact binaries moving on circular orbits, revealed [ref] that
Post-Newtonian Theory for Gravitational Waves · Hadamard regularization ambiguities · 0.99
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We checked intermediate results against [ref] .
Next-to-next-to-leading order post-Newtonian spin-orbit Hamiltonian for self-gravitating binaries · Outline of the calculation · 1.00
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We have computed the effective Lagrangian reproducing known results [ref] and paving the way for the yet uncomputed 4PN dynamics.
Effective field theory calculation of conservative binary dynamics at third post-Newtonian order · Conclusions · 0.92
Cites it as background 25 papers
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While the Post-Minkowskian (PM) and Post-Newtonian (PN) expansion schemes have been highly successful [ref] , a framework that seamlessly interpolates between weak and strong coupling remains a central theoretical goal.
Classical Renormalization Group Equations for General Relativity · Introduction · 0.92
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Besides the well-known Newtonian and 1PN Hamiltonians, previous results to this order for point-masses are the 2PN [ref] , 2.5PN [ref] , 3PN [ref] , and 3.5PN [ref] Hamiltonians.
Next-to-next-to-leading order post-Newtonian spin-orbit Hamiltonian for self-gravitating binaries · Introduction · 0.81
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Many studies have contributed to deriving the 4PN conservative dynamics, for nonspinning binaries [ref] , at SO level [ref] , SS [ref] , and at higher orders in the spins [ref] .
Theoretical groundwork supporting the precessing-spin two-body dynamics of the effective-one-body waveform models SEOBNRv5 · Hamiltonian · 0.85
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