How do permanant magnets work (In terms of electric charge and relativity)

Question

I've been reading about magnetism from the point of view of reference frames and relativity.

The concept of a test charge moving next to a wire (and how the choice of reference frame will cause Lorentz contractions of the charges in the wire, resulting in an electric force in the test charge's frame makes perfect sense to me. It explains how what we interpret as a "magnetic field" is really a consequence of combining electrostatics with relativity.

But I haven't been able to find a good explanation of permanent magnets from this point of view. How would someone (and could they) explain these permanent fields using relativity?

Apologies if my question isn't airtight in terms of terminology, just a curious layman. Thank you!

Answer 1

It explains how what we interpret as a "magnetic field" is really a consequence of combining electrostatics with relativity.

This is a very common misconception which arises when people start to look at relativistic electrodynamics.

It true that when moving between reference frames, electric and magnetic fields will generally "mix" in the following way:

$$\mathbf E_{\parallel}' = \mathbf E_\parallel$$ $$\mathbf B_{\parallel}' = \mathbf B_\parallel$$

$$\mathbf E_\perp ' = \gamma(\mathbf E_\perp - \frac{1}{c}\mathbf v \times \mathbf B)$$ $$\mathbf B_\perp ' = \gamma(\mathbf B_\perp + \frac{1}{c}\mathbf v \times \mathbf E)$$

where $\mathbf v$ is the velocity of the new frame with respect to the old one and the primes denote the fields as observed in the new frame. Therefore, the electric and magnetic fields are not relativistic invariants; in one frame you may see only an electric field, while in another you may see an electric field and a magnetic field.

However, this does not mean that magnetic fields are "not real," nor does it mean that all magnetic fields can be regarded as the result of the Lorentz contraction of moving charges. To illustrate this, note that the quantity $\epsilon_0 E^2 - B^2/\mu_0$ is a relativistic invariant - which means that if there is a frame in which the electric field vanishes but the magnetic field doesn't, then there is not a frame in which the reverse is true.