Hello I'm having a little trouble understanding how two observers in two different inertial frames of reference would view the same simple electromagnetic event. Let's call the first frame K, and we can use cartesian coordinates x, y, z, and t for time in K. There is an electric field E in K that is constant in time, and is everywhere directed in the y-direction with the same magnitude. At t=0, a small positively charged free-moving particle, q, is fired with a gun from the origin in the x-direction with an initial velocity v. I would think that in K, we'd see the motion of this particle describe a parabola starting at the origin, and arcing upward in the x-y plane. That's because the only force on it is q[E] which gives it constant acceleration in the y-direction as it continues to move in the x-direction with velocity v. The second frame is K'. This frame moves with velocity v in the x-direction with respect to K -- the same velocity as the x-component of the charged particle -- and at t=0 and t'=0, the coordinate axes of K and K' all coincide. I'm very confused in trying to understand how an observer in K' would view the motion of this particle. On the one hand , may seem quite obvious: since he's moving along the x-axis of K at the same speed as the particle, all he sees is its acceleration in the y'-direction, and therefore he observes it moving in a straight line in the y'-direction with constant acceleration. But this would seem inconsistent with the electromagnetic field transformation from K to K'. That's because in K', there is a field B' in the negative-z' direction. And so in K', as E' is accelerating the free particle in the y'-direction, I would think B' would be causing it to spiral. And so either I'm wrong about it describing a parabola in K, or I'm wrong about it spiralling in K'. Maybe both. Thanks!