This is about a famous thought experiment, cited by Einstein at the beginning of his first 1905 paper on SR, and discussed in textbooks and on this forum. I've fleshed it out with a specific set up, shown in the thumbnail. Seen in the S frame, a wire aligned in the y direction moves at speed v in the x-direction through a magnetic field B in the z direction. A charge q in the wire experiences a magnetic force qvB in the -y direction, so there is an emf qvBL / q = BvL in the wire. Seen in the S' frame, moving to the right at speed v relative to the S frame, the wire is stationary and the magnet is moving to the left at speed v. Because the wire is stationary there are no magnetic forces on the charge carriers in it. But there must still be an emf: viewing from a different frame can't change what happens. The force on the charge carriers must therefore arise from an electric field, in this new frame. And that's just what we know from Maxwell/Lorentz/Einstein theory: Bz transforms to give an Ey. But the transformation is [tex]E'_y=-\gamma v B_z.[/tex] This means that in the S' frame the force on charge q is [itex]qE'=-q \gamma v B[/itex] in the y-direction. Transverse lengths are Lorentz-invariant, so the emf in the S' frame is presumably [itex]- \gamma Lv B[/itex]. It's that [itex]\gamma[/itex] that worries me. Is it really the case that the emf's are different according to the frame you're observing from? Oddly enough, I have no difficulty accepting that the forces are different!