I have been upon occasion asked to discuss the relationship between relativity and permanent magnets. I do so here only qualitatively, since the mathematics necessary to "do it right" is a bit challenging. The post here should, I hope, convince the reader of the veracity of the effect and the skeptical reader can have fun trying to debunk it mathematically if they want. I have another post in the experts threadthat goes into some of the important math and physics in more detail, but this post is intended to show the relevant effects.

First we must consider the effect in the magnetic realm. We must first recall that permanent magnets are caused by the motion of electrons around atoms. The electron orbits are aligned, causing a net magnetic moment. We can for simplicity treat these orbits as little current loops. While an electron is negatively charged, I choose to use a positive moving charge in the loop, so as to use the usual convention of current. If you know enough physics to object, you know enough physics to correct on the fly.

We start by recalling that the magnetic field along the axis of a current loop is perpendicular to the loop. If a current loop has a counter-clockwise current, the magnetic field is out of the screen.

We now take a positively-charged particle with a velocity and cause it to move across the axis of the current loop.

From the right-hand rule, we deduce that the particle will feel a magnetic force to the right.

We now consider the situation in the rest frame of the positively charged particle flying over the loop. For simplicity we look when the particle is coincident with the loop's axis. Since this particle has no velocity in this rest frame, it can have no magnetic force. Thus the only thing that can push it to the right is electrostatic forces. Without Lorentzian contractions, the current loop is symmetric and the forces will balance. Thus we must consider the relativistic effects.

First we need to realize that the velocity of the current in the current loop will be different on the right and left-hand side. First take a small differential current element from the right and left hand side. We see the relative velocity in the figure:

Because we see in the figure that there are different velocities, the two current elements will experience different Lorentz contractions, as shown in this figure:

Recall that the charged particle is not moving in this frame and therefore cannot be feeling any magnetic force. However we see that the current element (equivalently charge element) on the left is contracted more than the one on the right. Thus while both current elements will repel the charged particle along the loop's axis, the fact that the charge is more dense on the left-hand side will induce a larger electrostatic repulsion than that on the right. Thus the charged particle will feel a net force to the right.

Thus we see here that a force that is purely magnetic in one reference frame is entirely electrostatic in another frame. This effect is only possible due to relativistic contraction. There is no Newtonian explanation.