Double Copy and Classical Two-Body Observables

The double copy can turn gauge-theory building blocks into gravitational amplitudes, and those amplitudes can then be used to calculate impulses, scattering angles, and radiation. The first step is an algebraic relation between quantum field theories; the second is a classical-limit and observable-extraction problem. Conflating them obscures both the successes and the open questions. At present, the method is a powerful generator of specific classical results, not a theorem that every gauge-theory numerator replacement produces every observable of a binary in pure general relativity.

Required background. Events, Intervals, and Minkowski Geometry supplies the on-shell invariants and state spaces; Weak Metric Perturbations and Gauge Freedom distinguishes physical gravitons from gauge and trace modes. Helpful background. The Einstein Equation fixes what “pure general relativity” means; Presymplectic Potentials and Currents motivates comparisons through asymptotic charges; Relativistic Two-Body Dynamics supplies the invariant observables.

Evidence cutoff. Claims and literature on this page were checked through 2026-08-12. Later calculations, corrections, and independent reproductions lie outside this evidence window.

Color–kinematics duality and its output

Write a gauge-theory amplitude in a cubic-graph representation,

$$\mathcal A_n = g^{n-2}\sum_i\frac{c_i n_i}{D_i},$$

where $c_i$ is a color factor, $n_i$ a kinematic numerator, and $D_i$ the product of propagator denominators. Whenever three graphs differ only around one internal edge, their color factors obey a Jacobi identity. A color–kinematics representation makes the corresponding numerators obey the same algebra:

$$c_i+c_j+c_k=0 \quad\Longleftrightarrow\quad n_i+n_j+n_k=0.$$

Replacing color by a second compatible numerator set gives, with a common overall amplitude phase stripped from both sides,

$$\mathcal M_n = \left(\frac{\kappa}{2}\right)^{n-2} \sum_i\frac{n_i\widetilde n_i}{D_i}, \qquad \kappa^2=32\pi G.$$

The gravitational theory is determined by both gauge-theory factors and by the external-state product. Pure Yang–Mills tensored with pure Yang–Mills naturally contains a graviton, a dilaton, and an antisymmetric two-form. It is not automatically pure Einstein gravity.

Numerators are nonunique. A generalized gauge shift $n_i\to n_i+\Delta_i$ may leave the gauge amplitude invariant because

$$\sum_i\frac{c_i\Delta_i}{D_i}=0.$$

The same shift need not leave a naive numerator square invariant. If color–kinematics duality is not manifest, violations of the kinematic Jacobi identities generate additional local contact terms. Bern and collaborators derive these corrections from generalized cuts in Bern et al. 2017, pp. 1–3, especially Eqs. (2)–(10), PDF. Tree-level relations are established for broad classes of theories; at loop level there are many highly nontrivial constructions, but no unrestricted all-theory, all-loop theorem.

This algebra produces an amplitude or integrand. A classical observable still has the schematic form

$$\mathcal O_{\mathrm{cl}} = \lim_{\hbar\to0} \mathfrak F_{\mathrm{obs}} \left[ \mathcal M_{\mathrm{double\ copy}} -\mathcal M_{\mathrm{iterations}}; \text{states},\ i0,\ \text{regulator} \right].$$

Here $\mathfrak F_{\mathrm{obs}}$ denotes the wave-packet, impact-parameter, or radiative transform appropriate to the desired quantity. The subtraction, boundary condition, state projection, and correlated $q^\mu=\hbar\bar q^\mu$ limit are physical parts of the calculation; color replacement alone does not perform them.

A low-order observable check

The cleanest shared test is the spinless 1PM impulse. For conserved point-particle sources, the pure-graviton polarization projector in four dimensions is

$$P^{(h)}_{\mu\nu,\rho\sigma} = \frac12 \left( \eta_{\mu\rho}\eta_{\nu\sigma} +\eta_{\mu\sigma}\eta_{\nu\rho} \right) -\frac12\eta_{\mu\nu}\eta_{\rho\sigma}.$$

With $T_A^{\mu\nu}\propto p_A^\mu p_A^\nu$, $p_A^2=-m_A^2$, and $\sigma=-p_1\cdot p_2/(m_1m_2)$, its contraction gives

$$T_1^{\mu\nu}P^{(h)}_{\mu\nu,\rho\sigma}T_2^{\rho\sigma} \propto m_1^2m_2^2 \left(\sigma^2-\frac12\right).$$

Thus the symmetric-traceless state projection supplies the trace subtraction that turns the squared gauge numerator into the pure-Einstein factor $2\sigma^2-1$. A naive tensor product without this subtraction retains the dilaton contribution. With relativistic state normalization, the long-range amplitude is

$$\mathcal M_{\mathrm{GR}}^{\mathrm{tree}}(\mathbf q) = \frac{16\pi Gm_1^2m_2^2}{\mathbf q^2} \left(2\sigma^2-1\right),$$

and the impact-parameter transform yields

$$\Delta p_{1\perp}^{\mu} = -\frac{2Gm_1m_2(2\sigma^2-1)} {\sqrt{\sigma^2-1}} \frac{b^\mu}{b^2}, \qquad \Delta p_{2\perp}^{\mu}=-\Delta p_{1\perp}^{\mu}.$$

Direct integration of the linearized Einstein equations gives the same invariant impulse and angle in Damour 2016, §V, pp. 6–7, Eqs. (55)–(61), PDF. This comparison tests the coupling normalization, the graviton state projection, the sign of the Fourier transform, and the observable map. At 1PM there is no Born subtraction, spin, or emitted radiation. Consequently the agreement does not validate a proposed loop contact prescription, spin dictionary, or radiation kernel; those require their own comparisons.

Comparison of methods and results

The following statements have different evidential strengths and should not be compressed into the slogan “gravity is gauge theory squared.”

Domain	Demonstrated result by the cutoff	Limitation relevant to two-body physics
Amplitude integrands	Tree amplitudes admit double-copy representations in broad settings; many loop integrands are known, and generalized double copy restores contact terms when Jacobi identities are not manifest.	The object is an integrand in a specified gravitational theory, not yet an impulse, angle, or waveform. Loop-level existence is not established without qualification for arbitrary theories and orders.
Spinless conservative scattering	Double-copy tree building blocks, generalized unitarity, region expansion, and effective-theory matching produced the spinless 3PM Hamiltonian and scattering angle, with independent consistency checks.	Potential-region selection and lower-order iteration subtraction remain essential; success at 3PM is not an all-PM proof.
Worldline impulse and radiation	Worldline quantum field theory gives a color–kinematics form for the eikonal through next-to-leading coupling order and leading radiation, with direct agreement on the dilaton-gravity side.	The demonstrated target is dilaton gravity, and locality information is supplied by an additional biadjoint-scalar kernel. Boundary conditions and radiative cuts remain separate inputs.
Spinning sources	Linear-in-spin radiation and later spinning-worldline eikonal and radiation constructions satisfy nontrivial Ward-identity and direct-computation checks in gravity theories with extra massless fields.	The state product, spin map, contact terms, and unwanted dilaton or two-form modes are model dependent. Generic finite-size and absorption data are not fixed by color–kinematics duality.
Kerr-inspired all-spin scattering	Local all-spin models remove chosen Compton spurious poles and generate aligned-spin 2PM amplitudes and angles by a classical double copy.	The model’s higher-point contacts are a proposal; its author explicitly does not claim that all features of true Kerr interactions are captured without direct GR matching.
Exact classical solutions	Schwarzschild and Kerr have special exact Kerr–Schild or Weyl double copies; a 2025 construction added a type-I Kasner example with a flat-space single copy.	These highly structured solutions do not establish an exact double copy for a dynamical generic binary spacetime.

For the spinless 3PM calculation, the double copy constructs gravitational tree data inside cuts, while the observable arises only after integration and effective-theory matching; see Bern et al. 2019, §§3.3–6, pp. 14–42, and §§9–11, pp. 82–100, PDF. For the worldline result, Shi and Plefka explicitly compare the next-to-leading eikonal and leading radiation with direct dilaton-gravity calculations in Shi and Plefka 2022, §§III–IV, pp. 5–9, PDF.

The spinning evidence is narrower. Goldberger, Li, and Prabhu obtain radiation linear in spin in graviton–dilaton–two-form theory and identify the state-content issue in Goldberger, Li, and Prabhu 2018, §I, pp. 1–2, PDF. Comberiati and Shi verify a spinning worldline double copy for the eikonal through next-to-leading order and radiation at leading order in a specified dilaton-gravity model in Comberiati and Shi 2022, §5, pp. 26–31, PDF. Neither result is a general pure-GR Kerr waveform theorem.

For all-spin Kerr-inspired scattering, Alessio explains the competing Compton prescriptions and the need for direct Teukolsky or GR comparison in Alessio 2024, §1, pp. 1–3, PDF, then describes the constructed theory as a minimal model rather than a complete Kerr theory in Alessio 2024, §6, pp. 19–20, PDF. That self-imposed boundary is essential when quoting its 2PM angle.

What the contrary result does and does not show

An especially informative negative result concerns an off-shell effective action. A proposed rule reproduced dilaton-gravity at lower orders but failed at 3PM, equivalently 2PN, when applied to the gauge-dependent two-body effective action. The discrepancy was checked against the known result, a probe limit, and direct Feynman diagrams. Plefka and collaborators emphasize that the failed object is gauge variant and off shell in Plefka et al. 2019, §I, pp. 1–2, PDF.

This is a concrete counterexample to that off-shell prescription. It is not a counterexample to on-shell BCJ or KLT relations, nor to every observable-level worldline construction. Conversely, later on-shell successes do not repair that particular off-shell rule. The correct conclusion is conditional: a classical double-copy claim must specify its object—amplitude, integrand, field, effective action, eikonal, impulse, or radiation—and its gravitational state content.

Two credible positions

The systematic-generator position emphasizes that on-shell amplitudes already determine asymptotic observables, that double-copy tree data scale efficiently through generalized unitarity, and that low-order conservative and radiative checks have succeeded. On this view, remaining complications are technical dictionaries that can be solved order by order. The 2022 review summarizes the amplitude, observable, and exact-solution achievements while also identifying higher-order integration and state-content problems in Kosower, Monteiro, and O’Connell 2022, §7.4, pp. 40–41, and §8, pp. 41–43, PDF.

The conditional-tool position stresses that generalized gauge freedom, nonunique local contact terms, lower-order iterations, infrared regions, spin supplementary choices, finite-size coefficients, extra states, and retarded boundary conditions are not determined by the slogan alone. It therefore accepts a proposed double copy only after a gravitational observable is reproduced in the same theory and regime.

These positions are compatible at low order and diverge in how strongly they extrapolate. The available evidence supports using the double copy as a systematic candidate generator with stringent observable-level checks. It does not support assuming a universal classical dictionary in advance.

Exact-solution work illustrates the same distinction. Kent and Zimmerman broadened the known examples to an algebraically general Kasner spacetime in Kent and Zimmerman 2025, pp. 1–2, PDF, but explicitly frame a sufficiently general definition and nontrivial dynamical examples as open. A cosmological type-I solution is evidence that algebraic speciality is not an absolute obstruction; it is not evidence that a generic inspiralling binary admits the same metric factorization.

Observable-level acceptance test

A classical two-body double-copy result is compelling only when all of the following are fixed and checked:

1. Theory and states. State both gauge theories and project the product onto the intended graviton, dilaton, and two-form content.
2. Local completion. Verify a spanning set of cuts and include contact terms required by Jacobi violations, locality, and spinning Compton amplitudes.
3. Classical scaling. Restore $\hbar$, take $q=\hbar\bar q$, and show which soft, potential, and radiation regions survive.
4. Iterations and infrared terms. Separate new dynamics from eikonal or Born iterations, and combine conservative and radiative pieces where their scale dependence overlaps.
5. Body dictionary. Declare the spin supplementary condition and the permanent-multipole, tidal, and absorptive coefficients.
6. Invariant comparison. Match an impulse, scattering angle, spin kick, radiated momentum, or waveform directly to a calculation in the target gravitational theory.

Agreement of integrands up to generalized gauge transformations is necessary at the construction stage but insufficient at the last stage. Conversely, different potentials can be acceptable if they give the same invariant after a canonical transformation and the same iteration convention.

Limitations and open questions

No general construction covered by the cutoff simultaneously proves pure-Einstein state projection, arbitrary PM order, generic spin orientation to all multipoles, radiation reaction, absorption, and a direct GR observable comparison. Claims outside the tested sector should therefore be labeled as conjectures or model results.

This assessment should be revisited if an independently reproduced higher-PM spinning or radiative observable resolves a current mapping ambiguity, or if a concrete counterexample invalidates a proposed observable dictionary. Other material triggers would be a general proof of the required loop-level color–kinematics representation for the relevant massive theory, or a generic dynamical spacetime construction that demonstrably matches asymptotic binary observables.

Exercise

A calculation squares Yang–Mills numerators for two spinning worldlines, obtains an integrand with correct factorization poles, and calls the result a 2PM Kerr scattering angle. List the additional information needed before that claim is established.

Solution

First identify the gravitational spectrum. A pure Yang–Mills square generally includes a dilaton and two-form, so a pure-Einstein claim needs an explicit projection or cancellation prescription. Second verify the local four-point completion: correct pole residues do not fix every spin-dependent contact term, and generalized-gauge changes can expose missing Jacobi corrections.

Third restore $\hbar$ and isolate the classical potential region. Subtract the iteration of the 1PM spin-dependent exchange and show that infrared phases cancel or exponentiate in the observable. Fourth translate quantum polarizations into a covariant spin vector with a declared SSC, and specify the Kerr multipole coefficients; otherwise the result may describe a different extended body.

Finally transform the kernel into an invariant angle or impulse and compare it with a direct GR calculation in the same spin configuration and approximation. Passing factorization alone establishes a candidate amplitude, not the asserted Kerr observable.

Return to the chapter overview to place this conditional method beside the classical-limit, loop, cut, and spin dictionaries.

References

• Alessio, Francesco, “Kerr Binary Dynamics from Minimal Coupling and Double Copy”, arXiv:2303.12784v4 [hep-th] (2024); Open PDF.
• Bern, Zvi, Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, and Mao Zeng, “Black Hole Binary Dynamics from the Double Copy and Effective Theory”, Journal of High Energy Physics 2019, 206 (2019); Open PDF.
• Bern, Zvi, John Joseph Carrasco, Wei-Ming Chen, Henrik Johansson, and Radu Roiban, “Gravity Amplitudes as Generalized Double Copies”, Physical Review Letters 118, 181602 (2017); Open PDF.
• Comberiati, Francesco, and Canxin Shi, “Classical Double Copy of Spinning Worldline Quantum Field Theory”, arXiv:2212.13855v2 [hep-th] (2022); Open PDF.
• Damour, Thibault, “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory”, Physical Review D 94, 104015 (2016); Open PDF.
• Goldberger, Walter D., Jingping Li, and Siddharth G. Prabhu, “Spinning Particles, Axion Radiation, and the Classical Double Copy”, Physical Review D 97, 105018 (2018); Open PDF.
• Kent, Brian, and Aaron Zimmerman, “New Framework for Classical Double Copies”, arXiv:2505.03887v3 [hep-th] (2025); Open PDF.
• Kosower, David A., Ricardo Monteiro, and Donal O’Connell, “The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes”, arXiv:2203.13025v2 [hep-th] (2022); Open PDF.
• Plefka, Jan, Canxin Shi, Jan Steinhoff, and Tianheng Wang, “Breakdown of the Classical Double Copy for the Effective Action of Dilaton-Gravity at NNLO”, Physical Review D 100, 086006 (2019); Open PDF.
• Shi, Canxin, and Jan Plefka, “Classical Double Copy of Worldline Quantum Field Theory”, Physical Review D 105, 026007 (2022); Open PDF.