*A basic question about magnetism*describes magnetic and electric fields as being the same in each reference frame. That is to say there is no separate perpendicular magnetic field; motion just modifies the electric field. So this 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 m0. C observes each ion accelerates at aC. From the equation F=ma we can say aC = FE/m0 where FE 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 FE – kv. The mass of the moving ions is increased by the Lorentz factor of γ or (1 - v2/c2)-0.5 to give a further reduction in acceleration. Hence R should see a lesser acceleration of aR = (FE – kv)/γm0.

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 γ2. But I don’t see how non-linear factors can reconcile acceleration differences which basically vary linearly with v. A single acceleration event seems to produce fundamentally different accelerations which cannot be reconciled by Lorentz transformations. Conventionally there are said to be separate electric and magnetic fields that are the same when viewed from different frames. In which case these same fields should have the same effects for all values of v.

Where am I going wrong?