  The Bethe-Salpeter equation is a non-perturbative, relativistic and covariant
description of two-body bound states. We derive the classical Bethe-Salpeter
equation for two massive point particles (with or without spin) in a bound
gravitational system. This is a recursion relation which involves
two-massive-particle-irreducible diagrams in the space of classical amplitudes,
defined by quotienting out by symmetrization over internal graviton exchanges.
In this context, we observe that the leading eikonal approximation to two-body
scattering arises directly from unitarity techniques with a coherent state of
virtual gravitons. More generally, we solve the classical Bethe-Salpeter
equation analytically at all orders by exponentiating the classical kernel in
impact parameter space. We clarify the connection between this classical kernel
and the Hamilton-Jacobi action, making manifest the analytic continuation
between classical bound and scattering observables. Using explicit analytic
resummations of classical (spinless and spinning) amplitudes in momentum space,
we further explore the relation between poles with bound state energies and
residues with bound state wavefunctions. Finally, we discuss a relativistic
analogue of Sommerfeld enhancement which occurs for bound state cross sections.
