I see, that makes a lot of sense. So in the primed frame, ##\gamma = 1## since the particle is initially at rest. Thus ##A^\prime = (0,0,\alpha,0)##, and transforming this gives ##A=(0,0,\alpha,0)##, which means that you can read off the three-acceleration as ##a_y = \alpha/\gamma^2##. (Obvious...