  A technique for translating the classical scattering function of two
gravitationally interacting bodies into a corresponding (effective one-body)
Hamiltonian description has been recently introduced [Phys.\ Rev.\ D {\bf 94},
104015 (2016)]. Using this technique, we derive, for the first time, to
second-order in Newton's constant (i.e. one classical loop) the Hamiltonian of
two point masses having an arbitrary (possibly relativistic) relative velocity.
The resulting (second post-Minkowskian) Hamiltonian is found to have a tame
high-energy structure which we relate both to gravitational self-force studies
of large mass-ratio binary systems, and to the ultra high-energy quantum
scattering results of Amati, Ciafaloni and Veneziano. We derive several
consequences of our second post-Minkowskian Hamiltonian: (i) the need to use
special phase-space gauges to get a tame high-energy limit; and (ii)
predictions about a (rest-mass independent) linear Regge trajectory behavior of
high-angular-momenta, high-energy circular orbits. Ways of testing these
predictions by dedicated numerical simulations are indicated. We finally
indicate a way to connect our classical results to the quantum gravitational
scattering amplitude of two particles, and we urge amplitude experts to use
their novel techniques to compute the 2-loop scattering amplitude of scalar
masses, from which one could deduce the third post-Minkowskian effective
one-body Hamiltonian.
