Compact binary systems in Einstein-{\AE}ther gravity. II. Radiation reaction to 2.5 post-Newtonian order

Clifford M. Will; Fatemeh Taherasghari

REVIEW 4 major objections 3 minor 65 references

Compact binary systems in Einstein-{\AE}ther gravity. II. Radiation reaction to 2.5 post-Newtonian order

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Compact binaries in Einstein-Æther gravity lose energy through a 1.5PN dipole radiation-reaction term whose coefficient depends only on the transverse aether speed, contradicting published far-zone flux results.

desk verdict A serious but contested 2.5PN radiation-reaction calculation whose central dipole result, independent of vL, contradicts published flux derivations and needs expert adjudication. read the letter →

arxiv 2506.03843 v1 pith:GWQTXPTI submitted 2025-06-04 gr-qc

Fatemeh Taherasghari , Clifford M. Will This is my paper

classification gr-qc

keywords Einstein-Aethertheorypost-NewtonianexpansionradiationreactiondipolegravitationalcompactbinaryenergylossraterelaxedfieldequationsAethersensitivities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to establish the complete radiation-reaction equations of motion for a compact binary in Einstein-Æther gravity through 2.5 post-Newtonian order, using near-zone field solutions rather than far-zone flux. Its central result is a 1.5PN dipole radiation-reaction acceleration and a $-1$PN orbital energy-loss rate whose coefficient depends only on the transverse Æther speed $v_T$. The authors argue that this contradicts published results based on a Noether-current flux construction, which include a $v_L$-dependent term, and that the disagreement at the quadrupole (0PN) order is even stronger. A sympathetic reader would care because the theory's observable predictions for binary-pulsar orbital decay depend on which coefficient is correct.

What carries the argument

The central objects are the relaxed Einstein-Æther field equations in the redefined variables of Paper I, where the metric potentials obey flat-space wave equations with speed 1 and the transverse Æther field obeys one with speed $v_T$, and no longitudinal $v_L$ wave equation appears. The equations of motion are generated by a modified geodesic equation for compact bodies whose masses depend on the Æther field through sensitivities $s_A$, $s'_A$, $s''_A$. Near-zone potentials are obtained by the Direct Integration of the Relaxed Equations method, then specialized to two bodies; the 1.5PN dipole moment $I^i_{ae}$ must be computed to 1PN order, and the 1.5PN correction to the velocity transformation enters at 2.5PN order. These pieces together produce the radiation-reaction accelerations and the energy-loss rate.

What would settle it

One concrete check is to compute the far-zone energy flux for a compact binary using the full nonlinear relaxed equations (keeping all quadratic field contributions) rather than the linearized Noether construction. If the resulting $-1$PN flux reproduces $\frac{2}{c_{14}v_L^3} + \frac{2(2-c_{14})}{c_1 v_T}$ rather than $1+\frac{3(2-c_{14})}{c_1 v_T}$, the radiation-reaction derivation fails at the energy-balance or decoupling step; if it reproduces the paper's coefficient, the previously published constraints are wrong. A numerical-relativity simulation of an Einstein-Æther binary, where feasible, would provide an independent check on the orbital-decay rate.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the radiation-reaction sector of the two-body equations of motion in Einstein-Æther theory, built from the near-zone solutions of Paper I. The 1.5PN dipole acceleration is $a^i_{1.5PN} = -\frac{1}{3(1-s)} \left(1 + \frac{3(2-c_{14})}{c_1 v_T}\right) G \ddot{I}^i_{ae}$, leading to the $-1$PN energy loss $$\frac{dE}{dt}_{-1PN} = -\frac{$G^{3}$ $m_1^{2}$ $m_2^{2}$}{3 $r^{4}$} \left(\frac{s_1-s_2}{(1-s_1)(1-s_2)}\right)^2 \left(1 + \frac{3(2-c_{14})}{c_1 v_T}\right).$$ The authors emphasize that this coefficient contains only $v_T$, whereas the published Noether-current flux results contain $\frac{2}{c_{14} v_L^3} + \frac{2(2-c_{14})}{c_1 v_T}$, and that their 0PN loss expression bears no resemblance to those earlier results. If the near-zone derivation is right, the standard predictions for orbital decay in this theory are wrong.

Load-bearing premise

The load-bearing premise is that the Paper I field redefinition completely decouples the near-zone equations, leaving no physical longitudinal $v_L$ mode in the 1.5PN dipole reaction, and that the near-zone energy loss equals the far-zone flux; if either fails, Eq. (4.7) is incomplete.

Editorial extensions

If this is right

The 1.5PN dipole acceleration and the 2.5PN terms provide, with the Newtonian and 1PN accelerations, the first complete 2.5PN relative equations of motion for two compact bodies in this theory.
The $-1$PN energy-loss rate shows that dipole radiation vanishes for bodies with equal sensitivities and otherwise proceeds at a rate set by $v_T$, with no direct $v_L$ dependence.
The 0PN energy-loss expression reduces to the general-relativistic quadrupole result in the appropriate limit and otherwise contains $v_L$-dependent coefficients, so the quadrupole sector is not simply shape-independent.
Because the published flux-based coefficients are not reproduced, any previous pulsar-timing bound on Einstein-Æther parameters that used those coefficients must be recomputed with the new equations.
The paper's energy-balance caveat means the relation between near-zone radiation reaction and far-zone flux in this theory is itself an open problem, so confirming this result requires an independent far-zone calculation.

Reading between the lines

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

If the paper is right, a far-zone flux calculation that keeps all nonlinearities should reproduce Eq. (4.7); matching the old coefficient instead would locate the error in the field redefinition or in the energy-balance assumption.
The restriction $c_1+c_3=0$ may be reshaping the radiation sector more strongly than previously thought, with the longitudinal wave $v_L$ moved entirely out of the dipole sector and into 0PN and higher terms.
A practical discriminator is binary-pulsar timing: if the sensitivities $s_A$ can be pinned down independently, the predicted ratio of dipole to quadrupole contributions to $\dot P/P$ differs between the two coefficients by a factor that could be tested as more pulsar data accumulate.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: this is a serious, algebra-heavy derivation of compact-binary equations of motion to 2.5PN order in Einstein-Aether, with a new 1.5PN dipole radiation-reaction term and a -1PN energy-loss rate that depends only on vT, not vL. That directly contradicts the published far-zone flux results. The authors are upfront that they can't resolve the disagreement. The paper deserves a serious referee, but the referee needs to focus on one fragile step.

What's genuinely good: the Newtonian and 1PN equations match earlier work, the GR limit c14=0 is recovered, and the paper includes long appendices with explicit multipole moments and energy expressions. The authors don't hide the conflict; Section V is candid about the mismatch and about the energy-balance assumption. The calculation is a real attempt to push the DIRE method into a new theory.

The soft spots are proportional to how central they are. The load-bearing step is the Paper I field redefinition that converts the relaxed Einstein-Aether equations into wave equations with no vL speed. From that, the 1.5PN dipole acceleration (2.25c) and hence Eq. (4.7) are vL-independent. But the paper never shows that the longitudinal mode is pure gauge or that the transformation is non-singular. vL still appears in the transformation and in many potentials, so its absence in the leading radiative-order term is a fragile cancellation. That's not a manufactured objection; it's the same reason the authors themselves can't locate their error — or the prior error. The energy-balance issue is secondary but real: a near-zone radiation reaction should match a far-zone flux that includes all physical modes, and the paper explicitly flags that this is unproven in Einstein-Aether. The 0PN result bearing 'no resemblance' to prior work is a red flag that something systematic is off, even if the prior work is indeed flawed.

I can't verify the long coefficients, and the authors provide no code or machine-checked algebra. That makes independent validation hard, but it doesn't make the paper wrong.

Verdict: send it to peer review. The calculation is important, the conflict with the existing literature is a genuine puzzle, and an expert referee — ideally someone who knows both PN theory and Einstein-Aether — should be able to test the field redefinition and the energy-balance logic. If the vL issue is resolved, this is a major correction. If it isn't, the paper still documents a serious disagreement worth publishing with a clear caveat.

Referee Report

4 major / 3 minor

Summary. The paper derives compact-binary equations of motion in Einstein-Æther gravity through 2.5PN order, using near-zone field solutions from the authors' Paper I and a modified geodesic equation that encodes body sensitivities to the Æther field. The central new results are the 1.5PN dipole radiation-reaction acceleration, Eq. (2.25c), and the resulting −1PN orbital energy-loss rate, Eq. (4.7), which depends on the transverse Æther speed vT but not on the longitudinal speed vL. The paper also reports a 0PN energy-loss rate, Eq. (4.14), and explicitly states that both results disagree with published far-zone flux calculations based on the Noether current, with the coefficient discrepancy spelled out in Eq. (5.3). The authors acknowledge in Sec. V.A that they have not resolved the disagreement.

Significance. If the headline results are correct, the paper would overturn the standard flux-based predictions for binary energy loss in Einstein-Æther theory and would have substantial impact on tests of Lorentz violation with binary pulsars and gravitational-wave observations. The manuscript has clear strengths: the Newtonian and 1PN equations of motion agree with earlier work, the GR limit c14=0 is recovered, the calculations are explicit with detailed appendices, and the authors candidly flag the unresolved conflict with Refs. [11,13,14]. However, the significance hinges entirely on the correctness of the dipole result and the reliability of the field redefinition that removes the vL mode from the wave equations. Because the disagreement with three independent derivations is left unresolved, the paper currently presents an open problem rather than a settled result.

major comments (4)

[Sec. V.A, Eq. (4.7) and Eq. (5.3)] The central −1PN energy-loss result, Eq. (4.7), disagrees with the flux-based coefficient (5.3) of Refs. [11,13,14], and the manuscript states in Sec. V.A that the authors have been unable to find a simple resolution. Since this disagreement concerns the paper's main physical claim, the manuscript currently leaves the correctness of its headline result undetermined. A revision should either pinpoint a concrete error in the earlier flux derivations or provide an independent far-zone flux calculation that reproduces Eq. (4.7).
[Sec. II.A, Eqs. (2.3), (2.5), (2.25c)] The absence of the longitudinal speed vL in the dipole reaction is traced to the variable redefinition of Paper I Eq. (4.7), which converts the relaxed equations into the wave equations (2.3). However, Eq. (2.5) shows that vL is a characteristic speed of the linearized vacuum theory, and a local field redefinition cannot eliminate a physical propagating degree of freedom unless the longitudinal combination is pure gauge or the transformation is singular; neither is demonstrated. Because vL-dependent quantities such as WL in Eq. (2.19) survive in the potentials and source terms, the disappearance of vL from Eq. (2.25c) is a delicate cancellation that needs a proof or a verification against a direct calculation of the far-zone flux including all modes.
[Sec. V.A] The energy-balance assumption is load-bearing for interpreting dE/dt as the binary's energy-loss rate. The manuscript itself notes that there is limited concrete evidence for energy balance in GR and questions whether it holds in Einstein-Æther theory already at dipole order. If energy balance fails, the near-zone radiation-reaction calculation does not determine the orbital decay that would be compared with observations; the paper should either justify energy balance at the orders used here or explicitly limit its claim to radiation reaction.
[Sec. V.B, Eq. (4.14)] The 0PN result, Eq. (4.14), is stated to bear no resemblance to published results, but the explanation in Sec. V.B is qualitative (linearized approximation, PN corrections to the dipole moment). Since this is another central result, the authors should state which specific terms in their derivation differ from the linearized flux calculation and demonstrate, rather than assert, that the linearized approximation fails at this order; otherwise the 0PN energy-loss rate is not established.

minor comments (3)

[Eq. (4.9)] The expression for A4 contains the term "3η(2 − s − 1 − s2)" with an undefined quantity s; it should presumably be s1 or s2, or the notation should be clarified.
[Sec. II.A] The variable redefinition of Paper I Eq. (4.7) is the load-bearing step of the derivation, but it is only referenced and not reproduced; including the transformation, or at least a self-contained summary of its properties, would make the manuscript more assessable.
[Sec. V.A] The sentence "This criticism was somewhat ingenuous" appears to mean "disingenuous"; the wording should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the radiation-reaction terms and energy-loss rates are derived analytically from the relaxed field equations and a covariant compact-body action; the disagreement with prior flux calculations is an unresolved physics question, not a circular reduction.

full rationale

The derivation chain is self-contained and does not reduce to its own inputs. The 1.5PN dipole acceleration (2.25c) and the -1PN energy loss (4.7) follow from the relaxed wave equations (2.3), the near-zone potentials from Paper I, and the modified geodesic equation (2.12) derived from a generally covariant matter action. No parameter is fitted to a target result; the sensitivities s_A, as_A enter as independent strong-field parameters, and the GR limit (c14 -> 0) plus agreement with prior 1PN equations is checked. The vL-independence of Eq. (4.7) is a substantive consequence of the field redefinition of Paper I, which is itself a derivation from the field equations with stated assumptions (c1 + c3 = 0), not a restatement of the target result. The authors' explicit statement that they cannot resolve the disagreement with the Noether-current flux results (Sec. V.A) is a correctness concern about the validity of that redefinition or of energy balance, not evidence of circularity. Self-citations to Paper I and to the authors' earlier modified-geodesic work are load-bearing, but they provide independent derivations rather than assumptions that define the conclusion. The energy-balance assumption is flagged as uncertain, which is a limitation, not a circular step. The reviewer's concern about the longitudinal mode is a physics objection to the field redefinition, not a circularity.

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

The derivation is analytical with no data fitting. Free parameters are the compact-body sensitivities, which are inputs from the Eardley phenomenological scheme rather than fitted numbers. Theory parameters c1, c2, c14 and speeds vT, vL enter as inputs from earlier literature and are not counted here. The main burden is the field-redefinition decoupling and the energy-balance assumption.

free parameters (1)

body sensitivities s_A, s'_A, as_A (and bs_A)
Eqs. (2.14)-(2.16): phenomenological derivatives of m_A(gamma) at gamma=1. They parameterize the unknown strong-field response of each compact body to the ambient Aether field and enter all radiation-reaction terms.

assumptions (5)

domain assumption Theory restricted to c1+c3=0 (with c14=c1+c4) to enforce GW speed cT=1.
Motivated by GW170817/GRB170817 speed constraint (Sec. I); used throughout the field equations and in interpreting vT and vL.
domain assumption Compact bodies are modeled by the Eardley-type point-particle action S_m = -sum integral m_A(gamma) dtau_A, with m_A depending on gamma = -K_ae^mu u_mu.
Eq. (2.9); assumes the Aether field couples to bodies only through internal strong-field effects, not directly to matter. This yields the modified geodesic equation (2.12) and the sensitivity parameters.
ad hoc to paper The variable transformation of Paper I Eq. (4.7) decouples the relaxed field equations into flat-spacetime wave equations (2.3) with propagation speeds 1 and vT only, with no vL mode.
This redefinition is the basis for concluding that the 1.5PN dipole radiation reaction is independent of vL, which is the main point of disagreement with prior work.
domain assumption Energy balance: the orbital energy-loss rate computed from near-zone equations of motion equals the far-zone energy flux.
Required to compare dE/dt (Sec. IV) with the Noether-current flux results of references [11,13,14]. The authors note this may fail in Einstein-Aether theory (Sec. V.A).
standard math Surface terms in integration by parts during near-zone multipole and radiation calculations can be discarded.
Used in deriving N1.5, B1.5, and K_ae^j 1.5 in Sec. V.A; standard in weakly isolated systems but not independently verified here for Aether theory.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Compact binary systems in Einstein-{\AE}ther gravity. II. Radiation reaction to 2.5 post-Newtonian order." pith.science (2026). https://pith.science/paper/GWQTXPTI

@misc{pith2026250603843,
  author       = {Pith},
  title        = {Pith review of: Compact binary systems in Einstein-\AEther gravity. II. Radiation reaction to 2.5 post-Newtonian order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWQTXPTI}},
  note         = {Machine review of arXiv:2506.03843}
}

abstract

We obtain the equations of motion for compact binary systems (black holes or neutron stars) in an alternative theory of gravity known as Einstein-AEther theory, which supplements the standard spacetime metric with a timelike four-vector (the AEther field) that is constrained to have unit norm. The equations make use of solutions obtained in Paper I for the gravitational and AEther field potentials within the near zone of the system, evaluated to 2.5 post-Newtonian (PN) order ($O(v/c)^5$ beyond Newtonian gravity), sufficient to obtain the effects of gravitational radiation reaction to the same order as the quadrupole approximation of general relativity. Those potentials were derived by applying the post-Minkowskian method to the field equations of the theory. Using a modified geodesic equation that is a consequence of the effects of the interaction between the AEther field and the internal strong-gravity fields of the compact bodies, we obtain explicit equations of motion in terms of the positions and velocities of the bodies, focussing on the radiation-reaction terms that contribute at 1.5PN and 2.5PN orders. We obtain the rate of energy loss by the system, including the effects of dipole gravitational radiation (conventionally denoted $-1$PN order) and the analogue of quadrupole radiation (denoted $0$PN order). We find significant disagreements with published results, based on calculating the energy flux in the far zone using a ``Noether current'' construction.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

65 extracted references · 39 canonical work pages

[1]

baryon density

≡ mA and asA ≡ s2 A − sA + s′ A 2, b sA ≡ a′ sA + (sA − 2)asA. (2.16) The “prime” on a′ sA denotes a derivative with respect to γ. C. Conversion of potentials to the baryon density We must now convert all potentials from integrals over σ, σi, σij, and σj ae to integrals over a mass density ρ∗ deﬁned by the constant masses mA of each body, namely ρ∗ ≡ ∑ A ...

work page
[2]

5P N . . . , (2.24) Through 1.5PN order, the results are ai N = 1 1 − s GU ,i , (2.25a) ai P N = 3 2 1 1 − s GΦ ,i 1 − 1 1 − s G2Φ ,i 2 − 4 1 − s G2U ,i U − 3G ˙U vi + 4 ( 1 − c14 2 ) G ˙V i + ((1 − s)(2 − 3s) + as) 2(1 − s)2 GU ,i v2 − ((1 − s)(4 − 3s) − as) (1 − s)2 GU ,j vivj + 8 ( 1 − c14 2 ) GV [i,j ]vj − (2 − c14)s (1 − s)c1 G ˙V i ae − 2(2 − c14)s ...

work page
[3]

5P N + ǫ2ai 2P N + ǫ5/ 2ai

work page
[4]

Eling and T

C. Eling and T. Jacobson, Phys. Rev. D 69, 064005 (2004), gr-qc/0310044

arXiv 2004
[5]

We will defer discussion of the 2PN terms to a future publication

5P N = − 1 3(1 − s) ( 1 + 3(2 − c14)s c1vT ) G¨I i ae , (2.25c) where vi, s, and as refer to the chosen body and the po- tentials are to be evaluated at the location of the chosen body, and where G = 2 G0/(2 − c14) is the gravitational constant. We will defer discussion of the 2PN terms to a future publication. The 2.5PN terms are very lengthy, so we brea...

work page
[6]

(2.26) Terms proportional to v: ai

5P N(v2) = − 1 3 G¨I j ae ( v2δij − 4vivj) − (as − s(1 − s)) 6(1 − s)2 ( 1 + 3(2 − c14) c1vT ) G¨I j ae ( v2δij + 2vivj) . (2.26) Terms proportional to v: ai

work page
[7]

I (ij) ae ) vj + G ( xj

5P N(v) = (2 − c14)G ( (4) I ij +2 ... I (ij) ae ) vj + G ( xj ... I j ae − (1 − 2c14) ... I jj ae + c14 (4) I jj ) vi + 2(2 − c14)s 3(1 − s)c1v3 T G ( x[i ... I j] ae + ... I [ij] ae ) vj − 2as(2 − c14) (1 − s)2c1vT G2U , (j ˙I k) ae ( viδjk + vkδij) 7 − 2(2 − c14)2(2c1 + (1 − 2c1)s) (1 − s)c2 1vT G2U , [i ae ˙I j] aevj − 2(2 − c14)(2c14 + (1 − 2c14)s) (...

work page
[8]

(2.28) Terms with no v factor, proportional to G2 : ai(2)

5P N(v0) = 3 5(1 − s) ( 1 − 5 9 (c14 + (2 − c14)s) ) G (5) I ⟨ij⟩ xj − 2 15(1 − s) ( 1 − 5 6 (c14 + (2 − c14)s) ) G (5) I ijj − 2 3(1 − s) ( 1 − 1 3 (c14 + (2 − c14)s) ) Gǫjik (4) J jk − (c14 + (2 − c14)s) 9(1 − s)v2 L G (5) I jj xi − 1 30(1 − s) { 1 + (2 − c14)sWT c1vT − (c14 + (2 − c14)s) WL (2 − c14) ( 1 + (2 − c14) vT )} × ( δjkδim + 2δijδkm) G (4) I ...

work page

Show all 65 references

[9]

5P N(v0) = G2 1 − s {( (2 − c14)U ,j δik + 1 3 (4 + 3c14)U ,i δjk − 1 2 (2 − c14)X ,ijk) ... I jk + 2(2 − c14)U ,j ¨I (ij) ae + (2 − c14)2 c2 1vT (2c1 + (1 − 2c1)s) Uae ¨I i ae + 8 3 U ,i ¨I j aexj + (1 + 2c14)U ,i ¨I jj ae + 8 3 ( 1 + 3(2 − c14) 4c1vT (c14 + (2 − c14)s) ) U ¨...
[10]

5P N = − G2m1m2 3r3 (s1 − s2)2 (1 − s1)2(1 − s2)2 × [ 1 + 3(2 − c14) c1vT ] ( vi − 3ninv ) , (3.5) while the 2.5PN contribution is given by ai
[11]

Newto- nian

5P N = G2m1m2 r3 × [ ninv ( Q1v2 + Q2nv2 + Q3Gm0 r(1 − s1)(1 − s2) ) +vi ( Q4v2 + Q5nv2 + Q6Gm0 r(1 − s1)(1 − s2) )] , (3.6) where the Qn are complicated functions of the Einstein- Æther parameters, the sensitivities and the functions de- ﬁned in Eqs. (3.4). IV. ORBITAL ENERGY...
[12]

5P N with ηm0v, giving ˙Edirect 0P N = G2(m1m2)2 m0r3 (1 − s1)(1 − s2) [ Q4v4 + Q2 ˙r4 + (Q1 + Q5)v2 ˙r2 + Gm0 (1 − s1)(1 − s2)r (Q6v2 + Q3 ˙r2) ] . (4.11) We use the identity (4.6) with ( s, p, q) = (0 , 3, 2) to sub- stitute ˙r4 = (3 /5) ˙r2(v2 + x · aN ) and then use it wit...
[13]

5P N = 1 1 − s [ N ,j
[14]

transverse Æther speed

5 −s ( ˙K j ae1. 5 + 2K [i,j ] ae1. 5vj )] , (5.1) where N1. 5, B1. 5 and K j ae1. 5 are the 1.5PN contributions listed in Eq. (5.5) of Paper 1, N1. 5 = − 2 3 G (3) I kk − 4 3 Gxk ¨I k ae , B1. 5 = −2G ( 1 − 1 2 c14 ) [ (3) I kk +2¨I kk ae ] , K j ae1. 5 = 2G c1vT ( 1 − 1 2 c1...

2017
[15]

+ (D1 − E1)(nv1 + nv2)nj ] + 1 3 [ ¯m + 3(2 − c14) c1vT (m1s1 + m2s2) ] G ˙I j ae , (B4) where ¯m = m1 + m2. Setting P j = 0, and deﬁning the relative velocity vj ≡ vj 1 − vj 2, we can obtain the transformation between the individual velocities and vj, given by vj 1 = m2(1 − s...
[16]

5P N , vj 2 = − m1(1 − s1) m0 vj + δv j P N + δv j
[17]

5P N , (B5) where δv j P N = 1 2 η(∆ − Cas)v2vj + Gm1m2 m0r [( 3(1 − s1)(1 − s2) (m1 − m2) m0 + ∆( D1 + E1) ) vj + ∆( D1 − E1) ˙rnj ] , (B6) and δv j
[18]

5P N = − 1 3m0 [ (m1 + m2) + 3(2 − c14) c1vT (m1s1 + m2s2) ] G ˙Iae , (B7) where Cas ≡ 1 m2 0 [ m2 1as2 (1 − s1)2 (1 − s2) − m2 2as1 (1 − s2)2 (1 − s1) ] . (B8) The 1PN correction δv j P N will be needed when we treat the 2PN equations of motion in a forthcoming publica- tion;...
[19]

Jacobson and D

T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001), gr-qc/0007031

2001 arXiv
[20]

Mattingly and T

D. Mattingly and T. Jacobson, in CPT and Lorentz Sym- metry, edited by V. A. Kosteleck´ y (World Scientiﬁc, Sin- gapore, 2002), pp. 331–335, gr-qc/0112012

2002 arXiv
[21]

Jacobson and D

T. Jacobson and D. Mattingly, Phys. Rev. D 70, 024003 (2004), gr-qc/0402005

2004 arXiv
[22]

Schumacher, S

K. Schumacher, S. E. Perkins, A. Shaw, K. Yagi, and N. Yunes, Phys. Rev. D 108, 104053 (2023), 2304.06801

2023 arXiv
[23]

B. Z. Foster and T. Jacobson, Phys. Rev. D 73, 064015 (2006), gr-qc/0509083

2006 arXiv
[24]

J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004), astro- ph/0403694

2004
[25]

Hoˇ rava, Phys

P. Hoˇ rava, Phys. Rev. D 79, 084008 (2009), 0901.3775

2009 arXiv
[26]

J. W. Moﬀat, J. Cosmol. Astropart. Phys. 3, 004 (2006), gr-qc/0506021

2006 arXiv
[27]

Skordis, Phys

C. Skordis, Phys. Rev. D 77, 123502 (2008), 0801.1985

2008 arXiv
[28]

Skordis, Class

C. Skordis, Class. Quantum Grav. 26, 143001 (2009), 0903.3602

2009 arXiv
[29]

B. Z. Foster, Phys. Rev. D 73, 104012 (2006), gr- qc/0602004

2006
[30]

B. Z. Foster, Phys. Rev. D 75, 129904 (2007)

2007
[31]

sensitivity parameters

(hereafter referred to as Paper I). In light of the extremely tight constraint on the diﬀerence in speed of gravitational waves compared to electromagnetic waves obtained from observations of the gravitational-waves and gamma ray bursts from a binary neutron star merger in 201...

2017
[32]

B. Z. Foster, Phys. Rev. D 76, 084033 (2007), 0706.0704

2007 arXiv
[33]

K. Yagi, D. Blas, E. Barausse, and N. Yunes, Phys. Rev. D 89, 084067 (2014), 1311.7144

2014 arXiv
[34]

Y. Gong, S. Hou, D. Liang, and E. Papantonopoulos, Phys. Rev. D 97, 084040 (2018), 1801.03382

2018 arXiv
[35]

J. Oost, S. Mukohyama, and A. Wang, Phys. Rev. D 97, 124023 (2018), 1802.04303

2018 arXiv
[36]

Hou and Y

S. Hou and Y. Gong, Universe 4, 84 (2018), 1806.02564

2018 arXiv
[37]

Saﬀer and N

A. Saﬀer and N. Yunes, Phys. Rev. D 98, 124015 (2018), 1807.08049

2018 arXiv
[38]

K. Lin, X. Zhao, C. Zhang, T. Liu, B. Wang, S. Zhang, X. Zhang, W. Zhao, T. Zhu, and A. Wang, Phys. Rev. D 99, 023010 (2019)

2019
[39]

Zhang, X

C. Zhang, X. Zhao, A. Wang, B. Wang, K. Yagi, N. Yunes, W. Zhao, and T. Zhu, Phys. Rev. D 101, 044002 (2020), 1911.10278

2020 arXiv
[40]

Gupta, M

T. Gupta, M. Herrero-Valea, D. Blas, E. Barausse, N. Cornish, K. Yagi, and N. Yunes, Classical and Quan- tum Gravity 38, 195003 (2021), 2104.04596

2021 arXiv
[41]

Xie and T.-Y

Y. Xie and T.-Y. Huang, Phys. Rev. D 77, 124049 (2008), 0805.4067

2008 arXiv
[42]

Blanchet, Living Reviews in Relativity 27, 4 (2024)

L. Blanchet, Living Reviews in Relativity 27, 4 (2024)

2024
[43]

Mirshekari and C

S. Mirshekari and C. M. Will, Phys. Rev. D 87, 084070 (2013), 1301.4680

2013 arXiv
[44]

R. N. Lang, Phys. Rev. D 89, 084014 (2014), 1310.3320

2014 arXiv
[45]

R. N. Lang, Phys. Rev. D 91, 084027 (2015), 1411.3073

2015 arXiv
[46]

Bernard, Phys

L. Bernard, Phys. Rev. D 98, 044004 (2018), 1802.10201

2018 arXiv
[47]

Bernard, L

L. Bernard, L. Blanchet, and D. Trestini, J. Cosmol. As- tropart. Phys. 2022, 008 (2022), 2201.10924

2022 arXiv
[49]

Taherasghari and C

F. Taherasghari and C. M. Will, Phys. Rev. D 108, 124026 (2023), 2308.13243

2023 arXiv
[50]

B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Ad- hikari, V. B. Adya, et al., Phys. Rev. Lett. 119, 161101 (2017), 1710.05832

2017 arXiv
[51]

B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Ad- hikari, V. B. Adya, et al., Astrophys. J. Lett. 848, L13 (2017), 1710.05834

2017 arXiv
[52]

C. M. Will and A. G. Wiseman, Phys. Rev. D 54, 4813 (1996), gr-qc/9608012

1996 arXiv
[53]

M. E. Pati and C. M. Will, Phys. Rev. D 62, 124015 (2000), gr-qc/0007087

2000 arXiv
[54]

D. M. Eardley, Astrophys. J. Lett. 196, L59 (1975)

1975
[55]

C. M. Will, Astrophys. J. 214, 826 (1977)

1977
[56]

Taherasghari and C

F. Taherasghari and C. M. Will, Phys. Rev. D 106, 064021 (2022), 2207.13454

2022 arXiv
[57]

K. S. Thorne, D. L. Lee, and A. P. Lightman, Phys. Rev. D 7, 3563 (1973)

1973
[58]

S. E. Gralla, Phys. Rev. D 81, 084060 (2010), 1002.5045

2010 arXiv
[59]

S. E. Gralla, Phys. Rev. D 87, 104020 (2013), 1303.0269

2013 arXiv
[60]

M. E. Pati and C. M. Will, Phys. Rev. D 65, 104008 (2002), gr-qc/0201001

2002 arXiv
[61]

Poisson and C

E. Poisson and C. M. Will, Gravity: Newtonian, Post- Newtonian, Relativistic (Cambridge University Press, Cambridge, 2014)

2014
[62]

B. R. Iyer and C. M. Will, Phys. Rev. D 52, 6882 (1995)

1995
[63]

Walker and C

M. Walker and C. M. Will, Phys. Rev. Lett. 45, 1741 (1980)

1980
[64]

Blanchet, T

L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wiseman, Phys. Rev. Lett. 74, 3515 (1995), gr- qc/9501027

1995
[65]

Ehlers, A

J. Ehlers, A. Rosenblum, J. N. Goldberg, and P. Havas, Astrophys. J. Lett. 208, L77 (1976)

1976
[66]

Walker and C

M. Walker and C. M. Will, Astrophys. J. Lett. 242, L129 (1980)

1980

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

baryon density

≡ mA and asA ≡ s2 A − sA + s′ A 2, b sA ≡ a′ sA + (sA − 2)asA. (2.16) The “prime” on a′ sA denotes a derivative with respect to γ. C. Conversion of potentials to the baryon density We must now convert all potentials from integrals over σ, σi, σij, and σj ae to integrals over a mass density ρ∗ deﬁned by the constant masses mA of each body, namely ρ∗ ≡ ∑ A ...

work page

[2] [2]

5P N . . . , (2.24) Through 1.5PN order, the results are ai N = 1 1 − s GU ,i , (2.25a) ai P N = 3 2 1 1 − s GΦ ,i 1 − 1 1 − s G2Φ ,i 2 − 4 1 − s G2U ,i U − 3G ˙U vi + 4 ( 1 − c14 2 ) G ˙V i + ((1 − s)(2 − 3s) + as) 2(1 − s)2 GU ,i v2 − ((1 − s)(4 − 3s) − as) (1 − s)2 GU ,j vivj + 8 ( 1 − c14 2 ) GV [i,j ]vj − (2 − c14)s (1 − s)c1 G ˙V i ae − 2(2 − c14)s ...

work page

[3] [3]

5P N + ǫ2ai 2P N + ǫ5/ 2ai

work page

[4] [4]

Eling and T

C. Eling and T. Jacobson, Phys. Rev. D 69, 064005 (2004), gr-qc/0310044

arXiv 2004

[5] [5]

We will defer discussion of the 2PN terms to a future publication

5P N = − 1 3(1 − s) ( 1 + 3(2 − c14)s c1vT ) G¨I i ae , (2.25c) where vi, s, and as refer to the chosen body and the po- tentials are to be evaluated at the location of the chosen body, and where G = 2 G0/(2 − c14) is the gravitational constant. We will defer discussion of the 2PN terms to a future publication. The 2.5PN terms are very lengthy, so we brea...

work page

[6] [6]

(2.26) Terms proportional to v: ai

5P N(v2) = − 1 3 G¨I j ae ( v2δij − 4vivj) − (as − s(1 − s)) 6(1 − s)2 ( 1 + 3(2 − c14) c1vT ) G¨I j ae ( v2δij + 2vivj) . (2.26) Terms proportional to v: ai

work page

[7] [7]

I (ij) ae ) vj + G ( xj

5P N(v) = (2 − c14)G ( (4) I ij +2 ... I (ij) ae ) vj + G ( xj ... I j ae − (1 − 2c14) ... I jj ae + c14 (4) I jj ) vi + 2(2 − c14)s 3(1 − s)c1v3 T G ( x[i ... I j] ae + ... I [ij] ae ) vj − 2as(2 − c14) (1 − s)2c1vT G2U , (j ˙I k) ae ( viδjk + vkδij) 7 − 2(2 − c14)2(2c1 + (1 − 2c1)s) (1 − s)c2 1vT G2U , [i ae ˙I j] aevj − 2(2 − c14)(2c14 + (1 − 2c14)s) (...

work page

[8] [8]

(2.28) Terms with no v factor, proportional to G2 : ai(2)

5P N(v0) = 3 5(1 − s) ( 1 − 5 9 (c14 + (2 − c14)s) ) G (5) I ⟨ij⟩ xj − 2 15(1 − s) ( 1 − 5 6 (c14 + (2 − c14)s) ) G (5) I ijj − 2 3(1 − s) ( 1 − 1 3 (c14 + (2 − c14)s) ) Gǫjik (4) J jk − (c14 + (2 − c14)s) 9(1 − s)v2 L G (5) I jj xi − 1 30(1 − s) { 1 + (2 − c14)sWT c1vT − (c14 + (2 − c14)s) WL (2 − c14) ( 1 + (2 − c14) vT )} × ( δjkδim + 2δijδkm) G (4) I ...

work page