  As recent work continues to demonstrate, the study of relativistic scattering
processes leads to valuable insights and computational tools applicable to the
relativistic bound-orbit two-body problem. This is particularly relevant in the
post-Minkowskian approach to the gravitational two-body problem, where the
field has only recently reached a full description of certain physical
observables for scattering orbits, including radiative effects, at the third
post-Minkowskian (3PM) order. As an historically instructive simpler example,
we consider here the analogous problem in electromagnetism in flat spacetime.
We compute for the first time the changes in linear momentum of each particle
and the total radiated linear momentum, in the relativistic classical
scattering of two point-charges, at sixth order in the charges (analogous to
3PM order in gravity). We accomplish this here via direct iteration of the
classical equations of motion, while making comparisons where possible to
results from quantum scattering amplitudes, with the aim of contributing to the
elucidation of conceptual issues and scalability on both sides. We also discuss
further extensions to radiative quantities of recently established relations
which analytically continue certain observables from the scattering regime to
the regime of bound orbits, applicable for both the electromagnetic and
gravitational cases.
