An axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved spacetime

Robert M. Wald; Theodore C. Quinn

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An axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved spacetime

Theodore C. Quinn, Robert M. Wald

General Relativity and Quantum Cosmology · Submitted 24 Oct 1996 · 18 references

Pith's verdict: The method is the contribution, not the equations — and the gravitational cross-check against Mino et al. is weaker than it's usually given credit for.

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CitationRobert M. Wald, Theodore C. Quinn. "An axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved spacetime." arXiv:gr-qc/9610053 (1996). https://pith.science/paper/ANKNBGYY

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Editor's note

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The real contribution here isn't the equations. Eq. (25) is DeWitt-Brehme as corrected by Hobbs, and eq. (51) is Mino-Sasaki-Tanaka. What's new is the method: a comparison axiom that gets the self-force by differencing two particles of equal acceleration magnitude, which turns a singular stress-energy computation into a page of Hadamard expansion plus spherical averaging. That's the idea people ended up using, and it's worth reading for that alone. The electromagnetic half is solid — short derivation, checkable algebra, endpoint matching a published result. The extended-body motivation in II.A, where the volume integral difference is argued to vanish in the point-particle limit while the surface term survives, is the honest core of the paper and it's laid out well. The reduction-of-order step in the gravitational case, dropping the Abraham-Lorentz terms because you can't keep m times gamma_in cross terms when you're only linear in both, is correct and worth noting as such. Two things to keep in mind. First, the axioms are postulates. The authors say so plainly in the Conclusions, so it isn't a hidden gap, but the paper is a derivation conditional on two assumptions justified by plausibility rather than proof. Fine as far as it goes. Second, if you're treating the gravitational result as independently confirmed twice, don't. Section III says outright that (45) and (46) disagree with Mino et al.'s (2.27) and (2.33): a sign in the acceleration term, a batch of non-curvature terms missing from theirs, several curvature signs, and they didn't track down the non-curvature discrepancies. Then eq. (51) is said to agree with [7]. The contraction-and-projection argument that would carry you from 'our (47) differs from theirs' to 'our (51) matches theirs' is shown only for the advanced-versus-retarded difference, not for the full discrepancy set. And [7] was a preprint at the time, not a published result. So the second check is weaker than it looks. The MathTensor verification of (47) is the real support there, and that's a check on algebra, not on physics. None of this breaks eq. (51) as far as I can tell. For readers: anyone working on self-force or EMRIs, and anyone who wants a template for extracting equations of motion without computing the singular stress tensor. Deserves a serious referee.

Referee report

Significance

If the axioms hold, the paper supplies a short and conceptually clean route to two results that otherwise require considerably heavier stress-energy calculations: the DeWitt-Brehme-Hobbs electromagnetic self-force and the leading gravitational self-force. The electromagnetic leg is a genuine independent rederivation of a published result, and the gravitational leg isolates the tail integral as the sole source of deviation from geodesic motion at linear order in the mass, which is a sharp and falsifiable structural statement. The methodological framing, that point-particle equations of motion encode limits of extended bodies rather than fundamental objects, is a useful contribution in its own right. The Hadamard expansions (45)-(47) are stated to have been checked with MathTensor, and I verified that the projection of the local term in Eq. (49) onto the subspace orthogonal to u^a reproduces the coefficient and sign structure of Eq. (50), so the internal algebra of the reduction is self-consistent.

Major objections (3)

[Section III, Eqs. (45)-(47), and abstract sentence s4] The abstract and Section III claim agreement of Eq. (51) with Mino et al. [7], but the manuscript itself documents unresolved discrepancies between its Hadamard expansions and those of [7]: a sign discrepancy in the ±4κ^{-2}a_a u_b term of (45) versus [7] Eq. (2.27), many non-curvature terms of (46) absent from [7] Eq. (2.33), and several curvature sign discrepancies, of which the authors state they 'have not attempted to trace down the origin of the other discrepancies.' The step from 'our (47) disagrees with [7] (2.33)' to 'our (51) agrees with [7]'s final equation' is not shown. Since (51) follows from (47) by contraction with u^b u^c, spherical averaging, and projection transverse to u^a, agreement requires that each discrepant term either be pure gauge or contract and average to something parallel to u^a. The manuscript demonstrates this only for the advanced-versus-retarded difference, not for the full set of discrepancies with [7]. Either supply a term-by-term demonstration that the discrepant terms drop out under contraction, averaging, and projection, or soften the abstract so that the agreement with [7] is presented as expected rather than established.
[Section II.A, Eq. (16), and Section II.B, Axiom 1] The comparison axiom rests on the assertion, made after Eq. (23), that the radial derivatives of the finite, direction-dependent terms in the Hadamard expansion are bounded, so that the volume integral in Eq. (16) vanishes in the point-particle limit while the surface term survives. The manuscript says only that this 'seems plausible.' Because the axiom is the load-bearing postulate of the entire construction, a scaling or bounding argument for the volume term would materially strengthen it: for instance, an explicit estimate of the difference of the two fields and of its radial derivative in Riemann normal coordinates, showing that the integrand of the volume term is integrable with vanishing measure as R goes to zero. Relatedly, the axiom matches only the magnitude of the four-acceleration, not its derivative; the ȧ-dependent terms in (23) are finite, are not matched between the two bodies, and do contribute to the final force (they produce the (2/3)e^2 ȧ^a term in Eq. (25)). The statement of the axiom should make explicit that the field difference is taken with each trajectory carrying its own ȧ^a, so that the reader does not read the axiom as requiring ȧ^a to be identified as well.
[Section III, paragraph following Eq. (48)] The manuscript states that, unlike the electromagnetic case, the quantity −C^abc u_b u_c is not gauge invariant and differs between the advanced and retarded solutions even as r goes to zero, but that 'it can be verified that this difference between the forces for the advanced and retarded solutions is parallel to the four-velocity of the particle.' This is asserted without a computation, yet it is precisely what makes Gravitational Axiom 2 consistent with the projected equation of motion and what licenses the use of the half-advanced, half-retarded field in the axiom. A short explicit calculation of the difference, showing that it is proportional to u^a, should be included, or the force Eq. (49) should be stated to be defined only up to a term parallel to u^a, with the physical content residing in the projected Eq. (50).

Minor objections (7)

[Eqs. (24) and (48)] Both displayed axioms omit the overall factor of charge or mass: Eq. (24) should carry a factor e and Eq. (48) a factor m on the right side, as is clear from the corresponding Eqs. (25) and (49).
[Abstract, sentence s4] The statement that the deviation from geodesic motion 'arises entirely from a tail term' holds only at linear order in m and in the incoming perturbation γ^in, as Section III correctly explains in the paragraph preceding Eq. (51). The abstract should carry that qualification, since a reader encountering only the abstract will take the statement as unconditional.
[Section II.C, final paragraph] The prescription for non-globally-hyperbolic spacetimes, choosing a sufficiently small globally hyperbolic neighborhood and defining F^in and the tail term 'relative to that neighborhood,' would benefit from one more sentence explaining why the resulting force is independent of the neighborhood chosen, given that the split F = F^in + F^- and the lower limit of the tail integral both depend on that choice.
[Section IV, Eq. (57)] The intermediate step displaying the term (1/3)v_k v_k δ_ij is superfluous, since it is contracted against the traceless d^5Q_ij/dt^5 and vanishes identically; the reduction from 6/45 to 2/15 could also be shown explicitly.
[Reference [3] and the Introduction] The text attributes the regularization approach to Penrose [3], but reference [3] bundles Penrose and Rindler's Spinors and Space-time with Unruh's 1976 paper. Please indicate which of these is the source of the approach being compared against.
[Section II.A, Eq. (16)] The step from the surface integral to e⟨E^α N⟩_R, and the notation ⟨⟩_R for the spherical average, would be clearer if the averaging convention were defined at the point of first use rather than only in words.
[Throughout] There are several typographical errors that should be corrected: 'bve' in the Conclusions ('It should bve emphasized'), 'Not that eq. (51)' for 'Note that eq. (51)', 'we we worked' in the paragraph before Eq. (51), 'there still remain a several notable differences' in Section III, the missing verb in 'Given eq. (2), we that a^a ≡ ...' in Section II.A, and the broken title 'gravitationa l'.

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The paper

The paper claims

Two axioms are enough to pin down the radiation-reaction force on a small charged or massive body in curved spacetime, reproducing the known electromagnetic result and yielding a gravitational equation of motion whose only departure from a

The reading

The paper asks a narrow but consequential question: what is the leading self-force — the back-reaction of a body's own field — on a small, nearly spherical body moving in a curved background? Direct point-particle limits fail, because holding charge or mass fixed while shrinking the body makes the self-field and its derivatives blow up. The authors replace the limit with two postulates. The first, a comparison axiom, says that two particles of equal charge whose accelerations have the same magnitude differ in self-force exactly by the ordinary Lorentz force of the difference of their fields, once neighbourhoods are identified by Riemann normal coordinates with velocities and accelerations aligned. The second fixes the answer for a uniformly accelerating charge in flat spacetime with the half-advanced, half-retarded field to be zero. From these, plus the standard Hadamard expansion of the retarded field near the world line, the authors recover the DeWitt-Brehme expression as corrected by Hobbs, and then run the same machinery in the gravitational case. The gravitational payoff is that, after reduction of order, the local acceleration and jerk terms drop out at linear order in the mass, leaving a nonlocal tail integral as the entire deviation from geodesic motion — the same conclusion reached by Mino, Sasaki and Tanaka.

What carries the argument

The load-bearing device is the comparison axiom. It converts an ill-posed point-particle limit into a statement about differences: if two bodies have accelerations of equal magnitude, the singular parts of their self-fields cancel under a Riemann-normal-coordinate identification that aligns $u^a$ and $a^a$, so the residual force is the ordinary Lorentz force (or its gravitational analogue $-C_{abc}u^bu^c$) of the field difference, averaged over a sphere of geodesic radius $r$ and taken as $r → 0$. The averaging sphere is what makes the answer finite and unambiguous. Feeding this with the Hadamard expansion of the retarded field near the world line — whose divergent terms depend only on $u^a$ and $a^a$, not on curvature or $ȧ^a$ — produces the explicit force formulas without any computation of the electromagnetic or gravitational stress-energy near the world line. Reduction of order, treating $e^2/m$ (or $m$) as a small parameter and substituting back the differentiated equation, then removes the runaway solutions.

Extended reading notes

Core claim

The paper's central claim is that the leading self-force on a small body in curved spacetime is fixed by a comparison axiom rather than by a detailed model of the body. In the electromagnetic case the axiom plus the flat-spacetime vanishing condition yields $f^a_{EM} = e(F^{in})^{ab}u_b + (2/3)e^2(ȧ^a - a^2u^a) + (1/3)e^2(R^a{}_bu^b + u^aR_{bc}u^bu^c) + e^2u^b∫^{τ^-}_{-∞}∇_{[b}(G^-)^{a]}{}_{c'}u^{c'}(τ')dτ'$, which agrees with DeWitt and Brehme as corrected by Hobbs. In the gravitational case the analogous axioms give a force containing a term parallel to the four-velocity (which only renormalises how $u^a$ is normalised) and, after projection and reduction of order, the equation of motion $a^a = (½∇^aγ^{in}_{bc} - ∇_b(γ^{in})_c{}^a - ½u^au^d∇_dγ^{in}_{bc})u^bu^c + mu^bu^c∫^{τ^-}_{-∞}(½∇^aG^-{}_{bc}{}^{a'b'} - ∇_b(G^-)_c{}^{aa'b'} - ½u^au^d∇_dG^-{}_{bc}{}^{a'b'})u_{a'}u_{b'}dτ'$. Because the paper works only to linear order in both $m$ and the incoming perturbation $γ^{in}_{ab}$, the Abraham-Lorentz-type terms $(ȧ^a - a^2u^a)$ cannot be retained consistently, and the tail integral survives as the sole self-force contribution.

Load-bearing premise

The comparison axiom itself: that when two particles have accelerations of equal magnitude, the difference in their self-forces is exactly the averaged ordinary force of the difference of their fields, because the volume integral of the field difference vanishes as both bodies shrink. The authors argue this only by plausibility, from the structure of the extended-body force integral, and state in the Conclusions that they did not prove the axioms follow from Maxwell's equations or from Einstein's equation.

Editorial extensions

If this is right

The electromagnetic self-force formula of DeWitt and Brehme, as corrected by Hobbs, follows from two postulates plus the Hadamard expansion, without computing the singular stress-energy tensor near the world line.
For a small body in vacuum, the leading correction to geodesic motion at order $m$ is a nonlocal tail integral over the past world line, so the equation of motion is integro-differential rather than local.
The local acceleration and jerk terms that appear at intermediate stages do not survive the reduction of order at linear order in $m$, so any local 'Abraham-Lorentz' contribution in the gravitational case is an artefact of truncation.
The result is stated to agree with the independent calculation of Mino, Sasaki and Tanaka, giving two routes to the same equation of motion.
In the slow-motion, weak-field limit the tail term is expected to become effectively local, mimicking the standard Abraham-Lorentz (electromagnetic) and quadrupole-derived (gravitational) damping forces.

Reading between the lines

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

If the comparison axiom is correct, it should generalise beyond Maxwell and linearised gravity: any linear field equation with a distributional point source whose Hadamard expansion has velocity-and-acceleration-only divergences would admit the same difference-of-forces construction.
The paper's own framing suggests a clean numerical test: compute the self-force for two small extended bodies of equal acceleration magnitude but different internal structure, and check whether their force difference converges to the averaged field-difference Lorentz force as the bodies shrink.
Because the tail integral is the sole surviving gravitational term at linear order, the practical content of the result for astrophysical inspirals is a statement about how much of the past orbit must be retained; the size of the effective memory of the tail is the natural quantity to quantify next.
The authors note that they did not derive the axioms from the field equations; a derivation would likely proceed by controlling the volume integral in their equation (16) uniformly over a one-parameter family of extended bodies, which is exactly where they say the main obstacle lies.

Original abstract

The problem of determining the electromagnetic and gravitational ``self-force'' on a particle in a curved spacetime is investigated using an axiomatic approach. In the electromagnetic case, our key postulate is a ``comparison axiom'', which states that whenever two particles of the same charge $e$ have the same magnitude of acceleration, the difference in their self-force is given by the ordinary Lorentz force of the difference in their (suitably compared) electromagnetic fields. We thereby derive an expression for the electromagnetic self-force which agrees with that of DeWitt and Brehme as corrected by Hobbs. Despite several important differences, our analysis of the gravitational self-force proceeds in close parallel with the electromagnetic case. In the gravitational case, our final expression for the (reduced order) equations of motion shows that the deviation from geodesic motion arises entirely from a ``tail term'', in agreement with recent results of Mino et al. Throughout the paper, we take the view that ``point particles'' do not make sense as fundamental objects, but that ``point particle equations of motion'' do make sense as means of encoding information about the motion of an extended body in the limit where not only the size but also the charge and mass of the body go to zero at a suitable rate. Plausibility arguments for the validity of our comparison axiom are given by considering the limiting behavior of the self-force on extended bodies.

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

The problem of determining the electromagnetic and gravitational ``self-force'' on a particle in a curved spacetime is investigated using an axiomatic approach. In the electromagnetic case, our key postulate is a ``comparison axiom'', which states that whenever two particles of the same charge $e$ have the same magnitude of acceleration, the difference in their self-force is given by the ordinary Lorentz force of the difference in their (suitably compared) electromagnetic fields. We thereby derive an expression for the electromagnetic self-force which agrees with that of DeWitt and Brehme as corrected by Hobbs. Despite several important differences, our analysis of the gravitational self-force proceeds in close parallel with the electromagnetic case. In the gravitational case, our final expression for the (reduced order) equations of motion shows that the deviation from geodesic motion arises entirely from a ``tail term'', in agreement with recent results of Mino et al. Throughout the paper, we take the view that ``point particles'' do not make sense as fundamental objects, but that ``point particle equations of motion'' do make sense as means of encoding information about the motion of an extended body in the limit where not only the size but also the charge and mass of the body go to zero at a suitable rate. Plausibility arguments for the validity of our comparison axiom are given by considering the limiting behavior of the self-force on extended bodies.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	The problem of determining the electromagnetic and gravitational ``self-force'' on a particle in a curved spacetime is investigated using an axiomatic approach.	contribution 0.78	0.84	0.07	asserted 0.92
s1	In the electromagnetic case, our key postulate is a ``comparison axiom'', which states that whenever two particles of the same charge $e$ have the same magnitude of acceleration, the difference in their self-force is given by the ordinary Lorentz force of the difference in their (suitably compared) electromagnetic fields.	method 0.83	0.68	0.08	asserted 0.99
s2	We thereby derive an expression for the electromagnetic self-force which agrees with that of DeWitt and Brehme as corrected by Hobbs.	result 1.00	0.92	0.16	asserted 1.00
s3	Despite several important differences, our analysis of the gravitational self-force proceeds in close parallel with the electromagnetic case.	method 0.77	0.84	0.05	asserted 1.00
s4	In the gravitational case, our final expression for the (reduced order) equations of motion shows that the deviation from geodesic motion arises entirely from a ``tail term'', in agreement with recent results of Mino et al. Throughout the paper, we take the view that ``point particles'' do not make sense as fundamental objects, but that ``point particle equations of motion'' do make sense as means of encoding information about the motion of an extended body in the limit where not only the size but also the charge and mass of the body go to zero at a suitable rate.	result 0.91	0.87	0.40	asserted 0.99
s5	Plausibility arguments for the validity of our comparison axiom are given by considering the limiting behavior of the self-force on extended bodies.	method 0.57	0.79	0.05	hedged possibility 0.72

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 0.99.

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sources read

arXiv listing abstract v1

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References

Works cited (18)

[7]

Y. Mino, M. Sasaki, and T. Tanaka, preprint gr-qc/960601 8

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[1]

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R. Penrose and W. Rindler, Spinors and Space-time, vol. I (Cambridge University Press, Cambridge, 1984). W.G. Unruh, Proc. Roy. Soc. Lond. A 348, 447 (1976)

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[4]

W. G. Dixon, Trans. Roy. Soc. Lond. A. 277, 59 (1974)

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[5]

B. S. DeWitt and R. W. Brehme, Annals of Physics 9, 220 (1960)

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[6]

J. M. Hobbs, Ann. Phys. 47, 141 (1968)

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[8]

W. Beiglb¨ ock, Commun. Math. Phys. 5, 106 (1967)

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[9]

W. G. Dixon, Nuovo Cim. 34, 319 (1964)

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L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators IV (Springer, Berlin, 1985)

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C.W. Misner, K.S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1970). R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984)

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[12]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975)

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D. Boulware, Ann. Phys. 124, 169 (1980)

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J. B. Hartle and K. S. Thorne, Phys. Rev. D 31, 1815 (1985)

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A. Einstein, L. Infeld, and B. Hoﬀmann, Ann. Math. 39, 65 (1938)

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[16]

E. E. Flanagan and R. M. Wald, Phys Rev. D (in press), prep rint gr-qc/9602052

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[17]

P. Havas, Phys. Rev. 108, 1351 (1957). (A crucial sign error in eq. (10) of the referen ce was corrected in P. Havas and J. N. Goldberg, Phys. Rev. 128, 398 (1962).) 36

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[18]

B. S. DeWitt and C. M. DeWitt, Physics 1, 3 (1964). 37

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Multiple Hamiltonian definitions of the conservative second-order self-force are identified in a nonlinear scalar toy model, restricted to unbound scattering trajectories.
Post-adiabatic self-force waveforms: slowly spinning primary and precessing secondary
gr-qc 2025-10 unverdicted novelty 6.0 of 10

Extended 1PA self-force waveforms for slowly spinning primary and precessing secondary, with re-summed 1PAT1R variant showing improved accuracy against NR for q ≳ 5 and |χ1| ≲ 0.1.
Radiation and Reaction at One Loop
hep-th 2023-03 conditional novelty 6.0 of 10

The imaginary part of a one-loop five-point amplitude computes the radiation emitted under a particle's own self-field (radiation reaction), verified explicitly in QED and proposed for Yang-Mills and gravity.
Energetics and scattering of gravitational two-body systems at fourth post-Minkowskian order
gr-qc 2022-04 accept novelty 6.0 of 10

Fourth post-Minkowskian conservative two-body dynamics, embedded in effective-one-body Hamiltonians, agrees with numerical relativity to a few percent for equal-mass binaries near merger and for scattering at relative...
Black hole perturbation theory and gravitational self-force
gr-qc 2021-01 conditional novelty 6.0 of 10

The paper provides a unified review and a new multiscale framework for generating post-adiabatic gravitational waveforms from generic nonresonant orbits around Kerr black holes.
Signatures of the Quantization of Gravity at Gravitational Wave Detectors
hep-th 2020-10 conditional novelty 6.0 of 10

Coupling a gravitational wave detector to a quantized gravitational field produces state-dependent stochastic noise in the arm length, described by a Langevin equation with a calculable power spectrum.
Frequency-domain algorithm for the Lorenz-gauge gravitational self-force
gr-qc 2013-08 conditional novelty 6.0 of 10

A frequency-domain solver with a generalized extended-homogeneous-solutions scheme computes the Lorenz-gauge gravitational self-force on eccentric Schwarzschild orbits 10 to 1000 times faster than time-domain codes at...
EMRI corrections to the angular velocity and redshift factor of a mass in circular orbit about a Kerr black hole
gr-qc 2012-07 accept novelty 6.0 of 10

A first radiation-gauge, mode-sum computation of the gauge-invariant self-force quantity H for circular equatorial orbits in Kerr, yielding order-mass corrections to redshift and angular velocity.
High-order expansions of the Detweiler-Whiting singular field in Schwarzschild spacetime
gr-qc 2012-04 accept novelty 6.0 of 10

Two independent expansion methods agree on the singular self-field of a point charge in Schwarzschild spacetime to fourth order, yielding new mode-sum regularization parameters and a benchmark scalar self-force value.
The complete non-spinning effective-one-body metric at linear order in the mass ratio
gr-qc 2011-11 conditional novelty 6.0 of 10

The effective-one-body metric potentials A and D-bar are determined exactly at linear order in the mass ratio by mapping gravitational self-force data for the redshift and periastron advance.
Conservative, gravitational self-force for a particle in circular orbit around a Schwarzschild black hole in a Radiation Gauge
gr-qc 2010-09 accept novelty 6.0 of 10

The conservative gravitational self-force for a particle in circular orbit around a Schwarzschild black hole is computed in a radiation gauge and agrees with independent Lorenz-gauge and Regge-Wheeler-gauge results to...
The self-consistent gravitational self-force
gr-qc 2009-07 conditional novelty 6.0 of 10

A fixed-worldline singular expansion, matched between inner and outer zones, yields a self-consistent derivation of the gravitational self-force and recovers the known MiSaTaQuWa equation.
Dynamics of Relativistic Binaries in Structured and Stochastic Environments: A Lagrange-Fourier-Hansen Framework
gr-qc 2026-06 unverdicted novelty 5.0 of 10

A new framework projects perturbations onto resonant frequencies via Hansen coefficients to produce efficient coupled ODEs for orbital elements in GW-driven relativistic binaries, demonstrated on tidal fields and accr...
Scalar QED as a toy model for higher-order effects in classical gravitational scattering
hep-th 2021-12 conditional novelty 5.0 of 10

Three amplitude-based routes (eikonal phase, KMOC impulse, and a new soft radial action) reproduce the known two-loop scalar QED scattering angle, and the radial-action shortcut is the genuinely new piece.
The Radial Action from Probe Amplitudes to All Orders
hep-th 2021-09 conditional novelty 5.0 of 10

Solving the probe wave equation and taking the classical limit yields the exact all-order scattering angle for a probe mass in pure NUT space, given by a complete elliptic integral.
Gravitational Self-force in a Radiation Gauge
gr-qc 2010-04 conditional novelty 5.0 of 10

Reconstructing the metric perturbation from the perturbed Weyl scalar in a radiation gauge, plus mode-sum renormalization, gives a workable route to the gravitational self-force for orbits around Schwarzschild and Ker...
Gravitational radiative corrections from effective field theory
gr-qc 2009-12 accept novelty 5.0 of 10

The NRGR effective field theory reproduces known post-Newtonian gravitational-wave flux corrections through 3PN and yields a renormalization-group equation for the quadrupole moment that controls the logarithms.
Gravitational self force in extreme mass-ratio inspirals
gr-qc 2009-08 accept novelty 4.0 of 10

The review organizes gravitational self-force methods for extreme mass-ratio inspirals and gives the first detailed derivation of the Kerr mode-sum regularization parameters.
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...
The gravitational eikonal: from particle, string and brane collisions to black-hole encounters
hep-th 2023-06 accept novelty 2.0 of 10

The gravitational eikonal exponentiation provides a practical amplitude-based route to classical gravitational observables, and this review consolidates its techniques across particles, strings, branes, and black-hole...
Snowmass White Paper: the Double Copy and its Applications
hep-th 2022-04 accept novelty 2.0 of 10

The double copy, which constructs gravity amplitudes from squares of gauge-theory amplitudes, is surveyed across its three tree-level formulations, its applications to black hole and gravitational-wave physics, and it...
Snowmass White Paper: Gravitational Waves and Scattering Amplitudes
hep-th 2022-04 conditional novelty 2.0 of 10

A review arguing that on-shell amplitude and effective-field-theory methods have already produced new analytic results in general relativity and can be pushed to the precision future gravitational-wave detectors require.

How later papers used it

146 later papers cite it

Evidence: substantial. 146 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 206 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 7 papers with this label

Used the Quinn-Wald prescription, subtracting the singular field from the retarded field near the particle, averaging over a surrounding sphere, and taking the limit as its radius shrinks to zero.

Gravitational Self-force in a Radiation Gauge (2010) · 0.91 · paraphrased: the licence does not allow quoting

How the literature uses it

120 arXiv papers cite this in 166 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 · 75 sentences without a confident label

Uses its method or tool 6 papers

These formulas were recently developed in [ref] , building on earlier related discussions found in [ref] .

Circular orbits and polarized images of charged particles orbiting Kerr black hole with a weak magnetic field · Introduction · 0.93
Adiabatic waveforms from inspiral trajectories.—We now turn to adiabatic waveforms from orbits that slowly inspiral due to first-order [$O(η)$] GSF effects [ref] , specifically following the two-timescale framework developed in Refs. [ref] .

Adiabatic waveforms from extreme-mass-ratio inspirals: an analytical approach · Introduction · 0.92
Following [ref] , they suggest the finite subtraction be carried out via [equation] where $$ indicates a different spacetime, $r=(σ_b u^b)_τ_+$ and the angular brackets are introduced to average out any directional dependence.

Regularization of a scalar charged particle for generic orbits in Kerr spacetime · The Quinn-Wald regularization procedure · 0.97

Cites it as background 75 papers

The solution for a massive point-particle moving in a strong field and for any velocity was indeed derived in 1997 by Mino, Sasaki and Tanaka [ref] , Quinn and Wald [ref] , in the de Donder [ref] (harmonic) gauge[The harmonic gauge was also proposed by Lanczos [ref] ], around an expansion of the mass ratio, $m/M$, $m$ being the small mass and $M$ the large one.

Self-force driven motion in curved spacetimes · The two-body problem and the perturbation method · 0.90
The first correct expression for a regularised self-force in the case of a scalar charge was produced by Quinn [ref] , which used the same regularization procedure developed by Quinn and Wald [ref] , often referred to as the axiomatic approach.

Regularization of a scalar charged particle for generic orbits in Kerr spacetime · The Quinn-Wald regularization procedure · 0.86
A formal expression for the gravitational self-force was first derived in 1996 by Mino, Sasaki, and Tanaka [ref] and Quinn and Wald [ref] ; the resulting equation of motion is now known as the MiSaTaQuWa equation.

The self-consistent gravitational self-force · Introduction · 0.96

Discussion

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