The following simple problem seems to lead to a contradiction when analyzed in different frames.
Consider an infinitely long stationary wire with positive line charge density $\lambda_0$ and negative charge density $-\lambda_0$. In its rest frame, the wire is electrically neutral.
Now consider driving a current so that the negative charges move to the right with uniform speed $v$ while the positive charges stay fixed. At a distance $y$ from the wire, a positive charge $q$ is moving to the right with the same speed $v$ parallel to the wire.
In the lab frame, the wire now is negatively charged, because the motion of the negative charges leads to a Lorentz contraction, making net charge density $(1-\gamma)\lambda_0$. In addition, the moving charges also create a current, with $I=-\gamma \lambda_0 v$, which creates a magnetic field. Therefore, by the right hand rule, the magnetic force on the positive charge points away from the wire and has magnitude $F_B=qvB= \frac{\mu_0 q\gamma\lambda_0 v^2}{2\pi y}$. In addition, the contribution from the electric field points in the oppositive direction, and has magnitude $F_E=qE=\frac{q(\gamma-1)\lambda_0}{2\pi\epsilon_0 y}$. The total force, analyzed in the lab frame, is thus $F=\frac{\mu_0 q\gamma\lambda_0 v^2}{2\pi y}-\frac{q(\gamma-1)\lambda_0}{2\pi\epsilon_0 y}=\frac{\lambda_0 q}{2\pi\epsilon_0 y}(\epsilon_0\mu_0v^2 \gamma -\gamma+1) =\frac{\lambda_0 q}{2\pi\epsilon_0 y}(1-\frac{1}{\gamma})$, using $\epsilon_0\mu_0=1/c^2$ and $1/\gamma=\sqrt{1-v^2/c^2}$. Since $1-\frac{1}{\gamma} >0$, the charge will be repelled from the wire.
However, in the frame of the moving charge, there is no magnetic force since the charge itself is stationary in its own frame. As for the electric field, in this frame the negative charges in the wire are stationary (because they move at the same speed $v$ in the lab frame), whereas the positive charges moves at speed $-v$. This means that viewed in this frame, the wire has a net positive charge $\lambda=(\gamma-1)\lambda_0$, which results in a force $F'=\frac{(\gamma-1)\lambda_0 q}{2\pi\epsilon_0 y}$ on the charge.
Clearly, the expression for $F$ does not agree with $F'$ ($F'=\gamma F$), which leads to different physical pictures. What is wrong here? (I'm guessing that F should also transform under a change of frame?)