From Boundary Data to Bound States

TL;DR

The paper builds a gauge-invariant dictionary linking boundary scattering data to adiabatic invariants of bound two-body orbits in General Relativity, valid to all PM orders. Central to the construction is the impetus formula, which expresses the classical radial momentum $\boldsymbol{p}^2(r,E)$ as a boundary Fourier transform of the scattering amplitude, enabling direct extraction of observables such as the periastron advance and orbital frequency from scattering data, without intermediate Hamiltonians. By analytic continuation from hyperbolic to elliptic orbits and employing a radial-action framework, the authors derive closed-form PM expressions for circular-orbit frequencies and two-loop periastron advances, and demonstrate a PM-resummed approach via a no-recoil approximation that reproduces 2PM results and a subset of higher-order terms. The framework offers a novel, gauge-invariant route to the binary problem, with potential extensions to radiation, spin, and non-perturbative regimes, and connections to the classical double copy. Overall, the work provides both exact PM structures and practical resummation schemes that enrich the toolkit for gravitational-wave modeling beyond traditional Hamiltonian methods.

Abstract

We introduce a -- somewhat holographic -- dictionary between gravitational observables for scattering processes (measured at the "boundary") and adiabatic invariants for bound orbits (in the "bulk"), to all orders in the Post-Minkowskian (PM) expansion. Our map relies on remarkable connections between the relative momentum of the two-body problem, the classical limit of the scattering amplitude and the deflection angle in hyperbolic motion. These relationships allow us to compute observables for generic orbits (such as the periastron advance $ΔΦ$) through analytic continuation, via a radial action depending only on boundary data. A simplified (more geometrical) map can be obtained for circular orbits, enabling us to extract the orbital frequency as a function of the (conserved) binding energy, $Ω(E)$, directly from scattering information. As an example, using the results in Bern et al. [1901.04424, 1908.01493], we readily derive $Ω(E)$ and $ΔΦ(J,E)$ to two-loop orders. We also provide closed-form expressions for the orbital frequency and periastron advance at tree-level and one-loop order, respectively, which capture a series of exact terms in the Post-Newtonian expansion. We then perform a partial PM resummation, using a "no-recoil" approximation for the amplitude. This limit is behind the map between the scattering angle for a test-particle and the two-body dynamics to 2PM. We show that it also captures a subset of higher order terms beyond the test-particle limit. While a (rather lengthy) Hamiltonian may be derived as an intermediate step, our map applies directly between gauge invariant quantities. Our findings provide a starting point for an alternative approach to the binary problem. We conclude with future directions and some speculations on the classical double copy.

Figures (3)

• Figure 1: The geometry of the scattering problem in the center of mass frame (gray dot). The motion of the bodies traces two hyperbolas, which are separated by $\tilde{r}_-$ at the point of closest approach.
• Figure 2: Bound elliptic motion in the center of mass frame (gray dot). The black ellipses mark the paths of each individual body. The heavier one lies on the focus of the green dashed ellipse, which describes the worldline of the lighter body in the companion's reference frame. The dotted circle of radius $a$ defines the eccentric anomaly, $u$, which can be used to paramaterize the orbit. See the text for a description of the orbital elements.
• Figure 3: Sample of Feynman diagrams needed for the computation of the scattering angle in the EFT approach to 3PM order. Only the diagrams to one-loop order in (a)-(c) are needed to 2PM.