  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.
