On the multipole expansion of the gravitational field

Luc Blanchet

REVIEW

On the multipole expansion of the gravitational field

Luc Blanchet

General Relativity and Quantum Cosmology · Submitted 29 Jan 1998 · 42 references

Pith's verdict: A solid, foundational methods paper: the all-orders source multipole moments as explicit integrals of the total pseudo-tensor, with the analytic-continuation step as the one load-bearing ingredient that is sketched rather than proved.

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CitationLuc Blanchet. "On the multipole expansion of the gravitational field." arXiv:gr-qc/9801101 (1998). https://pith.science/paper/AU7MQUHF

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This is the paper that gives you the source multipole moments to all post-Newtonian orders as explicit integrals of the total stress-energy pseudo-tensor, rather than order by order. If you work on PN inspiral templates, this is the reference you actually use for I_L and J_L, and it is the paper that made the 3PN program tractable. It holds up. The central result, (3.11)-(3.14), writes the multipole expansion outside a slowly moving source as a finite-part retarded integral of the multipolar source plus a sum of STF moments, each an integral of tau with the |y/r0|^B weight. Section V then decomposes the linearized metric into the six sets of irreducible moments and gives (5.15)-(5.20) in closed form. The W, X, Y, Z moments at general l are new here, and the checks are real: Damour-Iyer [16] in the linearized limit, paper II at 2PN, [37] at Newtonian order. Those are cross-checks against published results, not self-agreement. The derivation is double-covered, Section III by analytic continuation and Appendix A by a compact-support argument. On the soft spots, the reader and the stress-test both land on the right step: the vanishing of the complete radial integral after (3.5), which is what lets the near-zone ball integral be traded for the far-zone one. The stress-test's sharpening is correct and worth knowing: Appendix A is not fully independent, since (A6) to (A7) discards the M(Lambda) subtraction by the same identity. So that one ingredient is shared by both routes. It is still a standard scale-free-integral result of the analytic-continuation type, used consistently across papers I and II, and the sketch given is essentially the proof for power-law integrands. I would call this mild, not a hidden error. The (3.10) to (3.12) cancellation is asserted in the main line, but Appendix A does cover it. The author himself flags the counterintuitive consequence, that the moments depend on the PN expansion integrated formally to spatial infinity, which is honest. Assumption (i) excludes point particles and black holes, acknowledged, with only cited indications of extension. Who should read it: anyone computing PN source moments or building the source-to-radiative map. It deserves a serious referee, though I expect it would come back with only minor requests.

Referee report

Significance

If correct, this is a substantial and useful result: it makes the source multipole moments computable, to arbitrary PN order, from the matter model alone, and combined with the nonlinear functionals of papers I, [27], [34], [35] it yields the radiative moments at infinity to any PN order. This is the general-order completion of a program that had previously been carried out order by order, and it is the natural foundation for the 3PN binary calculations cited as [38]. The paper's strengths are real and worth naming: it gives two independent derivations of the central formula (Section III and Appendix A), it verifies the result in three separate limits against published expressions, and Appendix B reproduces the modified harmonicity algorithm in full so that the reader is not required to consult the 'to appear' reference [35]. The derivation is a computation from the stress-energy pseudo-tensor, not a fit to a target, so there is no circularity concern.

Major objections (2)

[Section II.B, Assumption (i), p. 12; Abstract and Section I] Assumption (i) states explicitly that 'we assume no singularities in the distribution of matter fields (no point particles or black holes)' and that 'the formalism will be a priori valid only for continuous matter distributions.' This is an honest and clearly stated restriction, but it sits in tension with the paper's stated motivation: the abstract and the introduction motivate the work by the LIGO/VIRGO observations of coalescing compact binaries, which are precisely the singular sources excluded by assumption (i). The extension to point particles is asserted by reference to [34,36,38] ('there are indications that the formalism applies also to point particles'), not demonstrated here. I do not ask for that extension to be derived in this paper, but the abstract and the opening of Section I should state the continuous-matter restriction explicitly, so that a reader does not take the headline result to apply directly to binary inspiral without the separate regularization step.
[Section III, paragraph following Eq. (3.5)] The passage from the near-zone integral (3.5) to the far-zone integral (3.6) rests on the claim that the complete radial integral of |y/r0|^B |y|^{2+l+a} (ln|y/r0|)^p from 0 to infinity vanishes identically for all complex B by analytic continuation. The sketch given (split at a constant A, continue each piece, sum; handle the logarithms by (d/dB)^p) is essentially complete for power-law integrands, and I believe the claim is correct. However, this is the one ingredient shared by both derivations: Appendix A discards the M(Λ) subtraction term in passing from (A6) to (A7) by exactly the same vanishing-integral argument. Since the identity is load-bearing for the central result and is the step the author himself flags as counterintuitive in the paragraph after (3.10), it would strengthen the paper to state the argument as a short lemma, with the treatment of the logarithms and of the interchange of the finite part with the (formal) infinite sum over the near-zone series written out explicitly rather than in square brackets.

Minor objections (8)

[Section II, Eqs. (2.23) and (2.25)] The overbar is used for both the near-zone expansion (2.23) and the post-Newtonian expansion (2.25). The author acknowledges this as an abuse of notation, but since the matching equation (2.28) equates the two barred objects, a reader can easily lose track of which expansion is meant at a given step. A footnote at (2.28) restating the convention would help.
[Section III, sentence following Eq. (3.12)] The r0-independence of the multipole expansion M(h^{mu nu}) is asserted ('It can be shown that ... the r0's cancel out between the two terms of (3.11)') without demonstration. Since r0 is an arbitrary length scale introduced by the analytic-continuation regulator, a one-line indication of how the cancellation occurs would be reassuring.
[Section III, passage from (3.10) to (3.12)] The cancellation of the second term in (3.10) is disposed of with 'It is not difficult to check.' Appendix A does cover this step by an independent route, but a forward cross-reference at this point in the main text would save the reader a search.
[Section V, Eqs. (5.6a)-(5.6c)] The special cases l = 0, 1 for I_L and l = 1 for J_L are stated separately from the generic expressions (5.7). It would be helpful to say explicitly whether (5.6) follows from (5.7) by taking the limit, or whether the l = 0, 1 cases require separate treatment because of the terms involving time antiderivatives.
[Section V, paragraph before Eq. (5.6)] The author notes that 'our definitions differ from paper I by some constant factors.' Since the moments I_L, J_L, W_L, ..., Z_L are the paper's main deliverable and will be compared against paper I by users, a short table of the conversion factors would be a useful addition.
[Section III, paragraph following Eq. (3.15)] The suggestion that (3.11)-(3.15) 'may have in fact a domain of validity larger than the one of MPM approximate solutions' is explicitly flagged as speculation, which is appropriate, but it should be marked as a conjecture in the text rather than left as an open suggestion, so that it is not mistaken for a derived result.
[Section II.B, p. 12] Typographical: 'saﬁsﬁes' should read 'satisfies'; in Section V, 'We subsitute the decompositions' should read 'substitute'; in Eq. (3.5), 'given explicitely' should read 'explicitly'.
[References] Reference [35] is listed as 'to appear in Class. Quantum Gravity'; Appendix B reproduces the relevant algorithm in full, so the paper is self-contained on this point, which is appreciated. Reference [38] is listed as 'in preparation' and is cited only for the forward-looking 3PN program, which is acceptable, but the citation should be flagged as unpublished where it appears in Section VI.

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The paper

The paper claims

Blanchet derives explicit integral formulas for the gravitational multipole moments of a slowly moving source, valid to all post-Newtonian orders, by integrating the total stress-energy pseudo-tensor with an analytic-continuation weight.

The reading

The paper asks a concrete question: for a slowly moving, isolated gravitating body, what exactly are the multipole moments that seed the exterior gravitational field, and can they be written as integrals over the matter and the gravitational field itself? In linearized gravity the answer was known — the moments are integrals of the matter stress-energy over a compact source — but at higher post-Newtonian order the gravitational field carries energy far outside the matter, the integrals diverge, and the standard formulas stop working. Blanchet's answer is to define each moment as a finite part, at $B=0$, of an integral of the total stress-energy pseudo-tensor weighted by $|y/r_0|^B$, with the weight analytically continued. The payoff is a single closed prescription that reproduces the known linearized and second-post-Newtonian results as limits, and that plugs directly into the existing post-Minkowskian machinery for propagating the field outward to the radiation zone. If the construction holds, the waveform from a binary inspiral can in principle be computed to any post-Newtonian order from the matter model alone.

What carries the argument

The load-bearing device is the finite part at $B=0$ of the $B$-dependent retarded integral, $FP_{B=0} Box_R^{-1}[|y/r_0|^B ...]$, borrowed from the earlier post-Minkowskian construction and now applied to the source's own multipole expansion. Its role is to make sense of integrals whose integrand is not compactly supported, by analytic continuation in the exponent $B$. The pivotal step is the observation that the complete radial integral $int_0^infty d|y| |y/r_0|^B |y|^{2+l+a} (ln|y/r_0|)^p$ vanishes identically for all complex $B$ by analytic continuation, which lets the author move the integration region from an infinitesimal neighbourhood of the origin out to spatial infinity. That jump is what allows the moments to be written as integrals of the post-Newtonian expansion of $tau$ over all space, and it is also why the author flags the result as counterintuitive: the moments formally depend on a post-Newtonian expansion that is only supposed to be valid near the source.

Extended reading notes

Core claim

The central result is equations (3.11)–(3.14): the multipole expansion of the gravitational field outside a slowly moving isolated source equals a finite-part retarded integral of the multipolar gravitational source term plus a sum over multipoles, $M(h^{ij}) = FP_{B=0}[-4G/c^4] + ...$, where each symmetric trace-free moment is an explicit integral of the total stress-energy pseudo-tensor $F^{ij}_L(u) = FP_{B=0} int d^3y |y/r_0|^B y_L int_{-1}^{1} dz delta_l(z) tau^{ij}(y, u + z|y|/c)$. Section V then decomposes the linearized metric into irreducible pieces and extracts six sets of source moments — the mass-type $I_L$, the current-type $J_L$, and four gauge moments $W_L, X_L, Y_L, Z_L$ — as closed integrals over combinations of $tau^{00}, tau^{0i}, tau^{ij}$ (equations 5.15–5.20). The author verifies that in the linearized limit these reduce exactly to the Damour–Iyer expressions, and that at second post-Newtonian order they agree with the earlier matching calculation. The moments so defined parametrize the 'particular' linearized metric that seeds the post-Minkowskian iteration, so the radiation-zone moments follow as nonlinear functionals of them.

Load-bearing premise

The derivation rests on the claim that the full radial integral of the weighted integrand from zero to infinity vanishes identically for every complex value of the continuation parameter, which is what allows the integration region to be moved from the source out to spatial infinity.

Editorial extensions

If this is right

Every source multipole moment becomes computable from a chosen matter model, so the chain from stress-energy to exterior field to radiated waveform is closed in principle at any post-Newtonian order.
The linearized-gravity moments of Damour and Iyer, and the second-post-Newtonian moments obtained earlier by asymptotic matching, both fall out as limits, giving a single formula that subsumes the earlier separate derivations.
Because the four gauge moments $W_L, X_L, Y_L, Z_L$ are also given in closed form, the corrections that first enter at 2.5 post-Newtonian order can be computed rather than inferred.
The result slots directly into the existing post-Minkowskian iteration, so the tail and other nonlinear corrections to the radiative moments can be carried through without redoing the near-zone analysis.
The author states the construction should extend to point-particle models of compact objects via regularization, which is the case of practical interest for binary inspiral waveforms.

Reading between the lines

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

The construction makes a sharp prediction that can be checked independently: the moments must be independent of the arbitrary length scale $r_0$ introduced by the weight, since it cancels between the two terms of (3.11). A computation that showed residual $r_0$ dependence would indicate the finite-part prescription is incomplete.
Because the moments are defined by an analytic continuation rather than by a boundary condition at the source, two sources with the same post-Newtonian expansion coefficients but different far-field behaviour would be assigned identical moments; whether that is physically correct is a question the paper does not settle.
The framework suggests a natural consistency test at third post-Newtonian order: compute $I_L$ and $J_L$ both from these integrals and from an independent near-zone matching, and compare. The author notes third-order work was in preparation, so this comparison is the near-term place where the prescription either holds or does not.
The reliance on analytic continuation to move an integral from the origin to infinity echoes regularization techniques used elsewhere in classical field theory; if the same identity can be justified by a contour or dimensional-regularization argument, the derivation would no longer rest on a sketched lemma.

Original abstract

This paper constructs the multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source. We introduce some definitions for the source multipole moments, valid to all orders in a post-Newtonian expansion, and depending in a well-defined way on the total stress-energy pseudo-tensor of the material and gravitational fields. Previously obtained expressions of the source multipole moments are recovered in the appropriate limits. The source moments parametrize the linearized approximation of the gravitational field exterior to the source, as computed by means of a specific post-Minkowskian algorithm defined in a previous work. Since the radiative multipole moments parametrizing the radiation field far from the source can be obtained as non-linear functionals of the source moments, the present paper permits relating the radiation field far from a slowly-moving source to the stress-energy pseudo-tensor of the source. This should be useful when comparing to theory the future observations of gravitational radiation by the LIGO and VIRGO experiments.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.88 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

This paper constructs the multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source. We introduce some definitions for the source multipole moments, valid to all orders in a post-Newtonian expansion, and depending in a well-defined way on the total stress-energy pseudo-tensor of the material and gravitational fields. Previously obtained expressions of the source multipole moments are recovered in the appropriate limits. The source moments parametrize the linearized approximation of the gravitational field exterior to the source, as computed by means of a specific post-Minkowskian algorithm defined in a previous work. Since the radiative multipole moments parametrizing the radiation field far from the source can be obtained as non-linear functionals of the source moments, the present paper permits relating the radiation field far from a slowly-moving source to the stress-energy pseudo-tensor of the source. This should be useful when comparing to theory the future observations of gravitational radiation by the LIGO and VIRGO experiments.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This paper constructs the multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source.	contribution 1.00	0.94	0.16	asserted 1.00
s1	We introduce some definitions for the source multipole moments, valid to all orders in a post-Newtonian expansion, and depending in a well-defined way on the total stress-energy pseudo-tensor of the material and gravitational fields.	contribution 0.86	0.94	0.08	asserted 1.00
s2	Previously obtained expressions of the source multipole moments are recovered in the appropriate limits.	result 0.78	0.63	0.05	asserted 1.00
s3	The source moments parametrize the linearized approximation of the gravitational field exterior to the source, as computed by means of a specific post-Minkowskian algorithm defined in a previous work.	method 0.37	0.44	0.10	asserted 1.00
s4	Since the radiative multipole moments parametrizing the radiation field far from the source can be obtained as non-linear functionals of the source moments, the present paper permits relating the radiation field far from a slowly-moving source to the stress-energy pseudo-tensor of the source.	interpretation 0.76	0.58	0.53	asserted 0.84
s5	This should be useful when comparing to theory the future observations of gravitational radiation by the LIGO and VIRGO experiments.	interpretation 0.58	0.08	0.08	hedged possibility 0.82

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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Our conventions and notation are the following : signat ure − + ++ ; greek indices =0,1,2,3; latin indices =1,2,3; r = |x| = ( x21 + x22 + x23)1/2; ni = ni = xi/r; ∂i = ∂/∂x i; xL = xL = xi1xi2 . . . xil and ∂L = ∂i1∂i2 . . . ∂il, where L = i1i2 . . . il is a multi-index with l indices; xL−1 = xi1...xil−1, xaL−1 = xaxi1 . . . xil−1, etc. . . ; ˆxL and ˆ∂L...

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hep-th 2024-02 accept novelty 6.0 of 10

The post-Newtonian expansion of the next-to-leading post-Minkowskian gravitational waveform agrees with the classical multipolar post-Minkowskian result up to 2.5 post-Newtonian order once the BMS frame and infrared s...
Radiated momentum and radiation-reaction in gravitational two-body scattering including time-asymmetric effects
gr-qc 2022-10 conditional novelty 6.0 of 10

Including time-asymmetric tail and radiation-reaction effects is necessary for the post-Minkowskian expansion of the radiated four-momentum to be polynomial in the two masses, and that polynomiality then fixes the 4PM...
Gravitational waves in scalar-tensor theory to one-and-a-half post-Newtonian order
gr-qc 2022-01 conditional novelty 6.0 of 10

The authors derive the tensor and scalar gravitational waveforms and energy fluxes of inspiralling compact binaries in massless scalar-tensor theory through 1.5 post-Newtonian order, including dipolar tail and memory ...
The current-type quadrupole moment and gravitational-wave mode $(\ell,m)=(2,1)$ of compact binary systems at the third post-Newtonian order
gr-qc 2021-05 conditional novelty 6.0 of 10

The current-type quadrupole moment of a non-spinning compact binary is computed to 3PN order, yielding the (2,1) gravitational-wave mode at 3.5PN accuracy in a closed-form expression in the orbital frequency.
Gravitational scattering at the seventh order in $G$: nonlocal contribution at the sixth post-Newtonian accuracy
gr-qc 2020-12 accept novelty 6.0 of 10

All integrals entering the nonlocal-in-time gravitational scattering angle at 6PN order and O(G^7) are computed analytically, fixing the previously unknown coefficient D to the rational value -12607/108.
Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates
gr-qc 2020-11 accept novelty 6.0 of 10

The authors derive the explicit coordinate transformation and resulting Newman-Unti metric, through quadratic order, that maps the multipolar post-Minkowskian gravitational field of an isolated source into Bondi-like ...
Hamiltonian for tidal interactions in compact binary systems to next-to-next-to-leading post-Newtonian order
gr-qc 2020-09 accept novelty 6.0 of 10

The authors compute the NNL/7PN tidal Hamiltonian in isotropic, ADM-like, and Delaunay coordinates, matching independent scattering-amplitude results in the overlap and adding new G^4 terms.
Tidal effects in the gravitational-wave phase evolution of compact binary systems to next-to-next-to-leading post-Newtonian order
gr-qc 2020-05 accept novelty 6.0 of 10

The tidal contribution to the gravitational-wave energy flux and phase of a non-spinning compact binary is computed through next-to-next-to-leading post-Newtonian order, formally 7.5PN including tail effects.
The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order
gr-qc 2020-03 conditional novelty 6.0 of 10

The 4PN mass quadrupole moment of a non-spinning compact binary is derived in the MPM-PN formalism, with UV divergences removed by dimensional regularization and the 4PN equation-of-motion shifts, and the result quote...
Gravitational multipole moments from Noether charges
hep-th 2017-11 conditional novelty 6.0 of 10

Mass and current multipole moments of Einstein gravity are shown to equal Noether surface charges of specific residual gauge symmetries, and the same charges are proposed as a definition for arbitrary gravity theories.
Quadratic-in-spin effects in the orbital dynamics and gravitational-wave energy flux of compact binaries at the 3PN order
gr-qc 2015-01 accept novelty 6.0 of 10

The authors derive the previously unknown 3PN quadratic-in-spin contributions to the gravitational-wave energy flux of spinning compact binaries, and confirm existing 3PN spin-spin equations of motion.
Cubic order spin effects in the dynamics and gravitational wave energy flux of compact object binaries
gr-qc 2014-11 conditional novelty 6.0 of 10

A Lagrangian octupolar formalism yields the leading 3.5PN cubic-in-spin contributions to binary dynamics and gravitational-wave energy flux, checked against Kerr and black-hole perturbation results in limiting cases.
Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order
gr-qc 2014-09 conditional novelty 6.0 of 10

The authors compute the mass-type octupole moment of non-spinning compact binaries to 3PN order and the resulting (3,3) and (3,1) gravitational-wave modes to 3.5PN, together with all non-linear multipole couplings nee...
Next-to-leading tail-induced spin-orbit effects in the gravitational radiation flux of compact binaries
gr-qc 2013-07 conditional novelty 6.0 of 10

The authors derive the 4PN tail-induced spin-orbit terms in the energy flux and phasing of quasi-circular spinning binaries, finding agreement with Kerr perturbation theory in the test-mass limit.
Next-to-next-to-leading order spin-orbit effects in the gravitational wave flux and orbital phasing of compact binaries
gr-qc 2013-03 accept novelty 6.0 of 10

The authors derive the 3.5 post-Newtonian spin-orbit contributions to the gravitational-wave flux and orbital phasing of spinning compact binaries, checked against the Kerr test-mass limit and the boosted-Kerr one-body limit.
The third and a half post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries
gr-qc 2012-04 accept novelty 6.0 of 10

The (2,2) quadrupole mode of the gravitational waveform from non-spinning compact binaries on quasi-circular orbits is derived through 3.5 post-Newtonian order.
Multipole expansion at the level of the action
gr-qc 2012-02 conditional novelty 6.0 of 10

The paper gives closed-form, all-orders expressions for the multipole moments of a linear source action in scalar, electromagnetic, and linearized-gravity theories, organized into symmetric trace-free tensors.
Third post-Newtonian angular momentum flux and the secular evolution of orbital elements for inspiralling compact binaries in quasi-elliptical orbits
gr-qc 2009-08 accept novelty 6.0 of 10

The 3PN angular momentum flux and the resulting secular evolution of orbital elements are derived for inspiralling compact binaries in quasi-elliptical orbits, completing the eccentric-orbit phasing input at 3PN.
The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits
gr-qc 2008-02 accept novelty 6.0 of 10

The authors derive the complete third post-Newtonian gravitational waveform and its spin-weighted spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits.
Tail effects in the third post-Newtonian gravitational wave energy flux of compact binaries in quasi-elliptical orbits
gr-qc 2007-11 conditional novelty 6.0 of 10

The hereditary tail contribution to the 3PN gravitational-wave energy flux of eccentric compact binaries is computed as a set of eccentricity-dependent enhancement functions multiplying the known circular-orbit coefficients.
Higher-order spin effects in the dynamics of compact binaries II. Radiation field
gr-qc 2006-05 conditional novelty 6.0 of 10

Spin-orbit corrections to the mass and current quadrupole moments, the gravitational-wave energy flux, and the orbital phasing of a spinning compact binary are derived through 2.5 post-Newtonian order.
Spin induced multipole moments for the gravitational wave amplitude from binary inspirals to 2.5 Post-Newtonian order
gr-qc 2012-03 conditional novelty 5.0 of 10

The authors derive the remaining spin-induced source multipole moments, including the next-to-leading current octupole for the first time, completing the spin contribution to the binary inspiral waveform at 2.5 post-N...
The Overlap of Numerical Relativity, Perturbation Theory and Post-Newtonian Theory in the Binary Black Hole Problem
gr-qc 2014-08 accept novelty 3.0 of 10

Coordinate-invariant comparisons across post-Newtonian theory, black hole perturbation theory, and numerical relativity show that perturbation theory, rescaled by the symmetric mass ratio, tracks comparable-mass binar...

How the literature uses it

38 arXiv papers cite this in 89 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 57 sentences without a confident label

Builds on it 1 paper

The ensuing formulas will generalize the 3-dimensional Eqs. (5.1)-(5.2) in [ref] and the $d$-dimensional ones (3.32)-(3.33) in [ref] .

The current-type quadrupole moment and gravitational-wave mode $(\ell,m)=(2,1)$ of compact binary systems at the third post-Newtonian order · The irreducible decomposition of the multipole expansion · 1.00

Uses its method or tool 15 papers

In the present paper, we computed the matter stress-energy tensor of the compact binary from the same effective action, and inserted it into a GW generation formalism based on MPM approximations for the external field [ref] , which are matched to the PN expansion of the inner field [ref] .

Tidal effects in the gravitational-wave phase evolution of compact binary systems to next-to-next-to-leading post-Newtonian order · Summary and conclusions · 1.00
Following the multipolar-post-Minkowskian (MPM) formalism [ref] , we shall express the radiative quadrupole moment $U_ij$ in terms successively of canonical moments ${M_L, S_L}$ and then source moments ${I_L, J_L, W_L, X_L, Y_L, Z_L}$ for general sources at 3.5PN order.

The third and a half post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries · Expression of the dominant quadrupole mode · 0.97
In line with Papers I & II, we use the multipolar post-Newtonian approach to gravitational radiation, which combines a multipolar-post-Minkowskian expansion for the vacuum field in the exterior of the matter source [ref] , together with a matching to the post-Newtonian field inside the source [ref] (see Ref. [ref] for a review).

Next-to-leading tail-induced spin-orbit effects in the gravitational radiation flux of compact binaries · Introduction · 0.99

Compares with it 3 papers

This formalism has exactly the same scope as the one of Refs. [ref] , i.e., it applies to any isolated post-Newtonian sources, but it differs in the definition of the source multipole moments and in many technical details when properly implemented [ref] .

Post-Newtonian Theory for Gravitational Waves · Analytic approximations and wave generation formalism · 0.91
We contrast and compare these results to the analogous classical derivation of the corrections to the Einstein quadrupole formula using the well-established multipolar post-Minkowskian formalism [ref] .

Scattering Amplitude Techniques in Classical Gauge Theories and Gravity · Introduction · 1.00
We contrast and compare these results to the analogous classical derivation of the corrections to the Einstein quadrupole formula using the well-established multipolar post-Minkowskian formalism [ref] .

Post-Newtonian Waveforms from Spinning Scattering Amplitudes · Introduction · 0.99

Cites it as background 7 papers

The symmetric-trace-free (STF) multipole moments of isolated PN radiative sources are known from a matching between the inner PN expansion in the system near zone and the outer MPM expansion in the far zone [ref] .

Tidal effects in the gravitational-wave phase evolution of compact binary systems to next-to-next-to-leading post-Newtonian order · Computation of source multipole moments · 0.88
These are based on the PN solution of the Einstein equations in the near zone, followed by a matching with the multipole expansion of the gravitational field in the exterior zone and its propagation to the far zone, where it determines the radiative multipoles [ref] (see [ref] for a comprehensive review).

Post-Newtonian Multipoles from the Next-to-Leading Post-Minkowskian Gravitational Waveform · Introduction. · 0.92
In the multipolar post-Minkowskian (MPM) formalism [ref] , this uniqueness results from the fact that the metric in harmonic coordinates is uniquely determined as the solution of the matching between the MPM metric outside the source and the metric in the near zone, the latter being in the form of a PN expansion [ref] .

Gravitational radiation reaction for compact binary systems at the fourth-and-a-half post-Newtonian order in harmonic coordinates · Introduction · 0.82

Discussion

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