Here's a situation similar to the hydrogenoid atom, but simpler as it involves no quantum mechanics.
Two particles collide head-on and rebound elastically, for instance a proton and a positron with speed relativistic but not enough to make new particles. Their common centre of mass is immobile
as is a first observer who sees them reverse their speed.
A second observer has a constant speed u, say perpendicular to the particles' path. He sees both particles having a speed component -u before, during and after the rebound. The punched screens make it more dramatic: the immobile observer sees the particles come back through the holes, hence the moving observer too, and the screens have kept their speed -u.
The particles' momentum along u is u times their relativistic mass before and after the collision. We wish this momentum component to be constant over the collision, so the moving observer must attribute to the particles a mass that is constant over the collision - even when the transverse speed gets smaller or zero as the kinetic energy converts into electrostatic energy. That's consistent with the mass of heavy nuclei, where an observer external to the nucleus weighs the protons' electrostatic repulsion. By the way, this increase doesn't depend on the speed u, which can be small.
Though, the immobile observer computes the collision with a particle mass depending on the speed only, not on the electrostatic energy. Worse, the moving observer too computes the transverse speeds during the rebound using no mass contribution from the electrostatic interaction.
So the mass of the electrostatic interaction depends on the observer, or worse, on his purpose
, yuk. Possibly like: the repulsion energy makes particles heavier except for the acceleration that results from this interaction. O good.
Spread the electrostatic contribution to the mass in the vacuum where the fields of both particles interfere, rather than on the particles? But why wouldn't that mass slow the particles' acceleration due to the repulsion? The interference of the fields moves with the particles.
Uncomfortable too: the lighter particle, which carries the biggest increase of relativistic mass since the centre of mass is immobile, also carries the biggest increase of electrostatic mass, despite both particles experience the same electrostatic potential, including the slope and curvature. So the electrostatic contribution doesn't depend on local field quantities, but on the particle's history through the field, or possibly on the rest mass.
I didn't consider the magnetic induction here, despite charges move. Nor the radiation, despite charges accelerate (but identical particles would radiate little).
Did I botch something? Would someone kindly shed light on this mess?
Marc Schaefer, aka Enthalpy