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Update: I previously didn’t consider that the forces themselves also seem to be acting with a speed of ±c respective to the emitter of the force, otherwise, it would not account for the speed limit of c on accelerators that pull particles instead of pushing them. So, the previous formula for the Doppler effect should only affect the field density, but not the effectiveness of the force applied to a particle.
A force moving with speed v acts upon an object moving with speed w with effectiveness e=(vw)/v = (1w/v). This means for faster velocities w>v, it has a negative effectiveness and slows down the object towards v. Thus, the new formula for the force applied to a particle should now be:
F = F'·f·e = F'·1v/c·(1v/c) = F'·±(1v/c)²
where f=1v/c is the perceived frequency/density of the field waves, and F’ is the original force/intensity of the force field, and e=(1v/c) is the effectiveness of the applied force at the particle’s current relative velocity. The frequency is now an absolute value, since negative frequencies aren’t really possible, and the reversed force effect is now sufficiently described by the new term e.
The implication of this new formula would be that particles always are accelerated/decelerated towards ±c with respect to the force emitter.