REVIEW
Dimensional regularization of the gravitational interaction of point masses
General Relativity and Quantum Cosmology · Submitted 11 May 2001 · 28 references
Pith's verdict: The pole cancellation and ω_k = 41/24 are real, checkable results; the ω_s = 0 headline rests on a step the letter asserts and defers to a paper in preparation.
T0 · AI review
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Referee report
The paper
The paper claims
Dimensional regularization, applied to the canonical ADM description of two point masses at third post-Newtonian order, yields a finite 3PN Hamiltonian with no free parameter: the static ambiguity is fixed at $ω_s=0$.
The reading
What carries the argument
The argument is carried by dimensional regularization inside the Arnowitt-Deser-Misner canonical formalism in the transverse-traceless gauge. Each field is solved in $d$ spatial dimensions; the solution of $Δu_a=-δ_a$ is $u_a=k r_a^{2-d}$, and the potentially dangerous terms near a particle are isolated as ten local integrands $T_A$ whose coefficients $c_A(d)$ are tabulated. The pole cancellation is the identity $∑_{A=1}^{10} c_A(3)=0$. The finite difference from the earlier Riesz-Hadamard Hamiltonian is controlled by the derivative coefficients, $∑_A c'_A(3)=41/1536$, which yields $ω_k=41/24$; because this difference is quadratic in momenta, the static parameter is left at $ω_s=0$.
Extended reading notes
Core claim
The central claim is that dimensional continuation of the 3PN ADM Hamiltonian yields a finite, unambiguous Hamiltonian with no pole part as $d→3$, and that this Hamiltonian uniquely determines the previously free static parameter to be $ω_s=0$. The authors also report a direct calculation of the kinetic parameter $ω_k$, obtained by summing about fifty dimensionally continued contributions, which gives $ω_k=41/24$—the unique value compatible with global Poincaré invariance. The paper presents this agreement as evidence that dimensional regularization preserves the perturbative gauge consistency of general relativity, unlike the Riesz-Hadamard variants used in earlier 3PN work.
Load-bearing premise
The load-bearing premise is that the momentum-independent pieces $h_1$ and $h_2$ can be rearranged by allowed integration by parts into forms with no dangerous $r_1^{-3}$-type terms as $d→3$; the paper states that this was checked and that $ΔH_1=ΔH_2=0$, but it does not display the check.
Editorial extensions
If this is right
- The conservative 3PN two-body Hamiltonian for non-spinning compact bodies is fixed with no adjustable regularization parameter, so its predictions depend only on $m_1$ and $m_2$.
- The last stable orbit and binding energy of binary black holes can be estimated at 3PN without the earlier $ω_s$ ambiguity; the paper quotes a 1.67% $(m_1+m_2)c^2$ binding energy for equal-mass non-spinning black holes using a Padé-resummed effective-one-body approach.
- Dimensional regularization should be used for the 3PN gravitational-wave flux, where the earlier regularization could introduce spurious ambiguities.
- The conformally flat truncation used in some numerical-relativity initial data is inconsistent at 3PN because it spoils the pole cancellation and global Poincaré invariance.
- The result supplies a benchmark for independent harmonic-coordinate and post-test-mass calculations of $ω_s$.
Reading between the lines
- If $ω_s=0$ survives independent checks, any remaining discrepancy between 3PN-resummed last-stable-orbit predictions and numerical-relativity initial data is more likely to come from resummation or from missing radiation or initial-data effects than from an unresolved regularization parameter.
- The paper's contact-term “threading” property—dimensional regularization evaluating $φ^4$ at a delta-function source as the fourth power of the smooth field there—suggests a practical rule for mapping 3PN results between regularization schemes; checking that map term by term would be a useful cross-check.
- Because the calculation relies on ADMTT gauge to suppress stronger ultraviolet divergences, a harmonic-gauge repetition is not merely a consistency check: it would test whether the finite value of $ω_s$ is gauge-independent, as the paper's gauge-symmetry argument requires.
- If the conformally flat truncation is indeed inconsistent at 3PN, numerical-relativity initial-data pipelines could test this by adding transverse-traceless gravitational-wave content and checking whether the measured initial-data parameters move toward the 3PN predictions.
Original abstract
We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous ``static'' parameter: namely, $\omega_s=0$. Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the ``kinetic'' parameter $\omega_k$, giving the unique answer compatible with global Poincar\'e invariance ($\omega_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous ``static'' parameter: namely, $\omega_s=0$. Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the ``kinetic'' parameter $\omega_k$, giving the unique answer compatible with global Poincar\'e invariance ($\omega_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We show how to use dimensional regularization to determine, within the Arnowitt-Deser-Misner canonical formalism, the reduced Hamiltonian describing the dynamics of two gravitationally interacting point masses. | contribution 0.86 | 0.93 | 0.15 | asserted 1.00 |
| s1 | Implementing, at the third post-Newtonian (3PN) accuracy, our procedure we find that dimensional continuation yields a finite, unambiguous (no pole part) 3PN Hamiltonian which uniquely determines the heretofore ambiguous ``static'' parameter: namely, $\omega_s=0$. | result 1.00 | 0.97 | 0.59 | asserted 1.00 |
| s2 | Our work also provides a remarkable check of the perturbative consistency (compatibility with gauge symmetry) of dimensional continuation through a direct calculation of the ``kinetic'' parameter $\omega_k$, giving the unique answer compatible with global Poincar\'e invariance ($\omega_k={41/24}$) by summing $\sim50$ different dimensionally continued contributions. | result 0.82 | 0.96 | 0.14 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:gr-qc/0105038 by the authors.
References
How later papers used it
98 later papers cite it
1 later paper by other authors is listed below: 1 built on it. 1 more paper by the same authors is listed: 1 compared its own results with it.
Builds on it 1 of 3 papers with this label
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Extended the 3PN-accurate conservative orbital dynamics presented in this paper.
Generalized quasi-Keplerian solution for eccentric, non-spinning compact binaries at 4PN order and the associated IMR waveform (2021) · 0.98 · paraphrased: the licence does not allow quoting
Compares with it 1 of 2 papers with this label · 1 by the same authors
By the same authors
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Changed all occurrences of a parameter to -2 to gain a result comparable to this paper.
Next-to-next-to-leading order post-Newtonian linear-in-spin binary Hamiltonians (2013) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
92 arXiv papers cite this in 143 sentences
Papers citing it · 91 sentences without a confident label
Builds on it 3 papers
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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
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The resulting compact binary dynamics extends the 3PN-accurate conservative orbital dynamics, presented in Ref. [ref] ,
Generalized quasi-Keplerian solution for eccentric, non-spinning compact binaries at 4PN order and the associated IMR waveform · Incorporating local-in-time 4PN Hamiltonian: generalized quasi-Keplerian solution · 0.98
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* We “flex” the currently known 3 PN-accurate EOB Hamiltonian [ref] by introducing an (effective) 4 PN Hamiltonian parameter $a_5$, parametrizing an additional contribution $+ a_5 ν/r^5$ in the main EOB radial function $A(r)$.
Faithful Effective-One-Body waveforms of equal-mass coalescing black-hole binaries · Effective-One-Body (EOB) method and waveform · 0.86
Uses its method or tool 11 papers
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These ambiguities can only be ruled out or corrected by doing the UV-analysis explained in [ref] , which relies on the complete $$-dependence of the integrands.
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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We performed parameter estimation runs using parallel bilby [ref] for all sources in the population with four different models, which were TaylorF2 (TF2) [ref] , IMRPhenomD_NRTidal (PhenDNRT) [ref] , IMRPhenomD_NRTidalv2 (PhenDNRTv2), and SEOBNRv4_ROM_NRTidalv2 (SEOBNRTv2) [ref] resulting in a total of 152 simulations.
Impact of gravitational waveform model systematics on the measurement of the Hubble constant · Population of synthetic BNS sources · 0.95
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The technique developed by [ref] to control local (or UV) divergences boils down to the computation of the difference lim_d→3 H^loc(d) - H^RH loc(3), where $H^RH loc(3)$ is the “local part” of the Hamiltonian obtained by means of the three-dimensional RH regularization [it is the sum of all integrals of the type $I^RH(3;ϵ_1,ϵ_2)$ introduced in Eq. [ref]], $H^loc(d)$ is its $d$-dimensional counterpart.
Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries · Local and asymptotic dimensional regularization · 0.99
Uses its data or values 3 papers
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Combining the GSF result [ref] and the 3PN-EOB results [ref] , we determined in Eq ([ref]) the value of a linear combination of $a_5 , a_6 , a_7$, etc$…$ If the contributions of $a_7 , a_8$, etc$…$ are relatively small (which can be argued for) this leads to determining a linear combination of $a_5$ and $a_6$, see Eqs. ([ref]) or ([ref]).
Gravitational Self Force in a Schwarzschild Background and the Effective One Body Formalism · Conclusions · 0.90
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In terms of the variables $ε$ and $j$, the ADM mass, orbital angular momentum and radial action are given by [ref]
First Law of Mechanics for Compact Binaries on Eccentric Orbits · Verification of the first law · 0.83
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The basic inputs for all families are the PN expressions for the conserved 3PN energy (per unit total mass) [ref] $E_3(v)$ and 3.5PN energy flux [ref] $F_3.5(v)$, [equation] where $γ=0.577216…$ is the Euler constant.
Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors · The PN Approximants · 0.80
Compares with it 2 papers
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Performing the mentioned integration by parts leads to the following formal 3PN ADM Hamiltonian which can also be compared to [ref] , [equation] We changed all occurrences of $$ to $= -2$ to gain a result which is comparable to [ref] .
Next-to-next-to-leading order post-Newtonian linear-in-spin binary Hamiltonians · The formal 3PN ADM Hamiltonian · 1.00
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To check our code we recalculated parts of the 3PN point-mass Hamiltonian given in [ref] .
Next-to-next-to-leading order post-Newtonian spin-orbit Hamiltonian for self-gravitating binaries · Result · 0.90
Cites it as background 29 papers
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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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They commute with the spin vectors $S_a$, where “$a$” denotes the particle label, $a=1,2$. $PP$ is the conservative point-particle ADM Hamiltonian known up to 3, see, e.g., [ref] and [ref] .
Motion and gravitational wave forms of eccentric compact binaries with orbital-angular-momentum-aligned spins under next-to-leading order in spin-orbit and leading order in spin(1)-spin(2) and spin-squared couplings · Spin and orbital dynamics · 0.89
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The first complete 1PN computation was preformed by Einstein, Infeld and Hoffmann in [ref] ; since then, the evaluation of the higher-order terms has been a formidable effort, whose current state of the art, after the calculation of the 2PN [ref] and 3PN [ref] contributions, is represented by the determination of the energy at 4PN order, which was achieved for the first time in [ref] and later confirmed in [ref] and in [ref] .
Static two-body potential at fifth post-Newtonian order · Introduction · 0.97
Discussion
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