From Boundary Data to Bound States

TL;DR

This paper establishes a holographic-like correspondence between boundary gravitational scattering data and bulk bound orbits, enabling the computation of orbital parameters directly from scattering amplitudes across all orders in the Post-Minkowskian expansion.

Contribution

It introduces a novel boundary-to-bulk map linking scattering observables to bound orbit invariants, facilitating analytic calculations of orbital dynamics from scattering data.

Findings

Derived expressions for orbital frequency and periastron advance at various PM orders.

Established a method to compute bound orbit parameters from scattering amplitudes.

Showed the map captures higher order terms beyond the test-particle limit.

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 $\Delta\Phi$) 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, $\Omega(E)$, directly from scattering information. As an example, using the results in Bern et al. [1901.04424, 1908.01493], we readily derive $\Omega(E)$ and $\Delta\Phi(J,E)$ 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.