REVIEW
The Hyperfine Einstein-Infeld-Hoffmann Potential
General Relativity and Quantum Cosmology · Submitted 21 Apr 2006 · Published version · 21 references
Pith's verdict: A substantial EFT derivation of the first nonlinear 3PN spin-spin potential, with a real but non-fatal audit gap at the three-graviton vertex; worth refereeing.
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Referee report
The paper
The paper claims
First nonlinear spin-spin potential between spinning bodies computed
The reading
What carries the argument
The machinery is the worldline effective field theory for spinning bodies, with spin treated via a vierbein and the canonical spin gauge, plus a velocity power-counting rule that assigns a definite scaling to every vertex. The calculation is organized by Feynman diagrams: leading and subleading spin-graviton vertices, the $p_0^2/\vec p^4$ correction to the graviton propagator, double-graviton exchange from the $L_{h^2}$ terms, and the three-graviton vertex. The three-graviton vertex is the load-bearing new ingredient: it is the only piece that produces the nonlinear $1/r^4$ term, and the paper does not print it, stating that it is handled with a symbolic manipulation program.
Extended reading notes
Core claim
Using the effective field theory for spinning point particles, the paper derives the complete 3PN spin-spin potential for two compact bodies, Eq. (12). It combines one-graviton exchange with subleading spin couplings, the first retardation correction to the graviton propagator, double-graviton exchange from the quadratic vierbein and connection terms, and the three-graviton vertex with two spin insertions and one mass insertion. The new nonlinear term, $3 G_N^2 (m_1+m_2)/r^4 (\vec S_1\cdot\vec S_2 - 3\,\vec n\cdot\vec S_1\,\vec n\cdot\vec S_2)$, is the first spin-spin effect that would vanish if gravity were linear; the authors identify it as the spin-dependent analogue of the Einstein-Infeld-Hoffmann potential. They also show that the leading finite-size quadrupole operator does not contribute to the spin-spin potential until 3.5PN, so Eq. (12) is complete at 3PN for compact objects regardless of their internal structure.
Load-bearing premise
The derivation relies on the three-graviton vertex contribution, which is not shown in the paper but is stated to have been computed with a symbolic manipulation program; if that program omitted a diagram or mis-assigned a sign, Eq. (12) would not be the complete 3PN spin-spin potential.
Editorial extensions
If this is right
- Gravitational-wave templates for inspiralling spinning binaries must include Eq. (12) at 3PN; omitting it introduces a phase error that accumulates over the long inspiral.
- The nonlinear $1/r^4$ term means that spin-spin dynamics cannot be correctly described by superposing linearized-gravity interactions; gravity's self-interaction enters at this order.
- The equations of motion obtained from Eq. (12) by Hamiltonian procedure give explicit 3PN spin-spin accelerations for use in orbital evolution and waveform models.
- Because finite-size quadrupole effects are shown to be absent from the 3PN spin-spin potential, Eq. (12) applies equally to black holes and neutron stars at this order, with the body's internal structure only entering later.
- The identification with the Einstein-Infeld-Hoffmann potential suggests the same diagram set can generate analogous nonlinear spin terms at higher post-Newtonian orders.
Reading between the lines
- An independent derivation in a different coordinate gauge should reproduce the coefficient $3 G_N^2(m_1+m_2)$ in front of the $1/r^4$ spin-spin term; agreement or disagreement would provide a sharp check of the symbolic three-graviton computation.
- The structure of Eq. (12) suggests a 'spin EIH' effect: the total mass $m_1+m_2$ enters as if the two-body system's center-of-mass energy sources the nonlinear graviton field that couples the two spins, a feature that could be generalized to systems with more than two spins.
- In gravitational-wave data analysis, the $1/r^4$ term will produce a characteristic dephasing that scales with the total mass and spin projections; searches can be designed to look specifically for this signature in low-frequency space-based detectors.
- The vanishing of the 3PN finite-size contribution hints that the worldline effective field theory has a selection rule protecting spin-spin interactions from quadrupole operators at this order; extending the calculation to 4PN would test whether the selection rule persists.
Original abstract
We use recently developed effective field theory techniques to calculate the third order post-Newtonian correction to the spin-spin potential between two spinning objects. This correction represents the first contribution to the spin-spin interaction due to the non-linear nature of general relativity and will play an important role in forthcoming gravity wave experiments.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We use recently developed effective field theory techniques to calculate the third order post-Newtonian correction to the spin-spin potential between two spinning objects. This correction represents the first contribution to the spin-spin interaction due to the non-linear nature of general relativity and will play an important role in forthcoming gravity wave experiments.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We use recently developed effective field theory techniques to calculate the third order post-Newtonian correction to the spin-spin potential between two spinning objects. | contribution 0.88 | 0.95 | 0.20 | asserted 1.00 |
| s1 | This correction represents the first contribution to the spin-spin interaction due to the non-linear nature of general relativity and will play an important role in forthcoming gravity wave experiments. | interpretation 0.99 | 0.39 | 0.51 | asserted 0.89 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:gr-qc/0604099 by the authors.
References
How later papers used it
91 later papers cite it
1 later paper by other authors is listed below: 1 compared its own results with it.
Compares with it 1 of 8 papers with this label
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Stated that its own results are also equivalent to those previously calculated in the EFT approach in this paper, after taking into account the addition appearing in another work.
Next-to-leading order gravitational spin1-spin2 coupling with Kaluza-Klein reduction (2008) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
88 arXiv papers cite this in 164 sentences
Papers citing it · 94 sentences without a confident label
Contests it 2 papers
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Notice that no agreement with the result in [ref] could be found, see [ref] .
Canonical Formulation of Spin in General Relativity · Introduction · contest 0.81
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Though the result in [ref] was derived within the EFT approach, it was not based on the Routhian method but on a direct insertion of a Newton-Wigner SSC.[However, in [ref] this procedure is not justified.
Elimination of the spin supplementary condition in the effective field theory approach to the post-Newtonian approximation · The NLO spin(1)-spin(2) Hamiltonian of Porto and Rothstein · contest 0.71
Builds on it 1 paper
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Ref. [ref] extended the worldline EFT methods of Ref. [ref] in this direction, and since then there have been substantial developments [ref] (see Refs. [ref] for reviews).
Quadratic-in-Spin Hamiltonian at $\mathcal{O}(G^2)$ from Scattering Amplitudes · Introduction · 0.84
Uses its method or tool 10 papers
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The derivation involves several ingredients, which we obtain using the worldline effective field theory (EFT) framework in the PN regime [ref] extended to spinning bodies [ref] .
Gravitational radiation from inspiralling compact objects: Spin effects to fourth Post-Newtonian order · Introduction · 0.92
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The necessary Feynman rules are [ref] L_sg^v^2 = 1/2m_pH_i0,kS^ik, L^v^3_sg = 1/2m_p(H_ij,kS^ikv^j + H_00,kS^0k), L^v^4_sg = 1/2m_p(H_0j,kS^0kv^j + H_i0,0S^i0) L^v^5_sg = 1/2m_p H_ik,0S^k0 v^i , for the terms linear in the metric, whereas to $O(H^2)$ L^v^4_SH^2 = 1/4m^2_pS^ij H^λ_j (H_0λ,i - H_0i,λ) L^v^5_SH^2 = 1/4m^2_p[S^ij H^l_j (H_kl,i - H_ki,l)v^k + S^i0(H_00H_00,i+H^l_iH_00,l)] , where we only kept the terms that lead to non-zero contractions at the desired order.
Next to leading order spin-orbit effects in the motion of inspiralling compact binaries · Spin effects in the Effective Field Theory approach · 0.88
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In this work, we used a generalisation of the NRGR approach [ref] to compute the leading spin-orbit effects that would arise when a compact binary is coupled both conformally and disformally to a light scalar field.
Spin-orbit effects for compact binaries in scalar-tensor gravity · Summary of main results · 0.94
Compares with it 7 papers
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Our results coincide with those of [ref] obtained by the ADM Hamiltonian formalism, and are also equivalent to those previously calculated in the EFT approach in [ref] , after taking into account the addition appearing in [ref] .
Next-to-leading order gravitational spin1-spin2 coupling with Kaluza-Klein reduction · Summary of results · 1.00
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The equations of motion follow from a canonical procedure, e.g. _1 = ∂V^NW_SS/∂_1×_1 .The precession equation is equivalent to the one from [ref] and [ref], after a spin redefinition [ref] .
The Effective Field Theorist's Approach to Gravitational Dynamics · Newton-Wigner SSC · 0.93
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After translating between various conventions, this agrees with the known value in the literature [ref] .
Spin Effects in the Effective Field Theory Approach to Post-Minkowskian Conservative Dynamics · Binding Energy · 0.85
Cites it as background 40 papers
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The leading order (LO) spin effects in the conservative dynamics were derived through the NRGR formalism in [ref] , while the next-to-leading order (NLO) and next-to-next-to leading order (N$^2$LO) spin effects were studied in [ref] and [ref] , respectively.
Next-to-leading order spin-orbit effects in the equations of motion, energy loss and phase evolution of binaries of compact bodies in the effective field theory approach · Introduction · 0.96
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Spin couplings were introduced in the form of an action in [ref] or a Routhian in [ref] .
Spin induced multipole moments for the gravitational wave flux from binary inspirals to third Post-Newtonian order · Spin Couplings · 0.95
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Presently, spin effects in the PN conservative dynamics of binary compact systems are known up to N$^2$LO [ref] , with partial results also at higher orders, e.g. [ref] .[On the other hand, spin-dependent radiation effects are only known to NLO and up to quadratic order in the spins [ref] .
Spin Effects in the Effective Field Theory Approach to Post-Minkowskian Conservative Dynamics · Introduction · 0.93
Discussion
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