My question concerns the situation where two positive ions initially move in the same direction and are side by side in a vacuum. An observer also co-moves with the ions at their speed of v. I will refer to the co-moving observer as C and an observer at rest as R.

1. C considers his moving frame to be at rest, so his clock is unaffected and the ions (which initially appear to be at rest) just have a rest mass of m_{0}. C observes each ion accelerates at a_{C}. From the equation F=ma we can say a_{C}= F_{E}/m_{0}where F_{E}is the electrostatic repulsion.

2. For R at rest there is a magnetic force of attraction which varies linearly with the ions’ speed of v. This can be expressed as kv, so for R the repulsion is reduced to F_{E}– kv. The mass of the moving ions is increased by the Lorentz factor of γ or (1 - v^{2}/c^{2})to give a further reduction in acceleration. Hence R should see a lesser acceleration of a^{-0.5}_{R}= (F_{E}– kv)/γm_{0.}

C and R see each other’s clock is slow by a factor of 1/ γ. So each can say the other’s measurement of speed is increased by a factor of γ and acceleration by γ^{2}. But I don’t see how non-linear factors can reconcile acceleration differences which basically vary with v. A single acceleration event seems to produce fundamentally different accelerations which cannot be reconciled by Lorentz transformations. Electric and magnetic fields are said to be the same when viewed from different frames, but the same fields would have the same effects. Where am I going wrong?