# Did the Goudsmit-Uhlenbeck analysis of spin consider relativity?

It's frequently mentioned in introductory quantum mechanics texts that Goudsmit and Uhlenbeck conjectured that the magnetic moment of an electron was due to angular momentum arising from the electron rotating around its own axis. But then when they tried to calculate how fast it would have to be spinning, assuming that the electron is a rigid sphere with radius equal to the classical electron radius, they found that a point on the equator would be moving with a speed greater than the speed of light, so they were embarrassed for publishing their work.

My question is, did they do this calculation using Newtonian mechanics or special relativity? If we do take relativity into account, and consider a (Born-) rigid sphere with radius equal to the classical electron radius, and then we tried to find out what speed the sphere would need to rotate at in order to have an angular momentum that produces the magnetic moment of an electron, would we still get a speed faster than light? Momentum goes to infinity as speed approaches c, but what happens to angular momentum? I'm aware that angular momentum becomes really complicated in special relativity, with tensors and bivectors and the like, but is there a simple (or even approximate) expression that can give us some idea of what would happen in this case?

This is of course just a curiosity, because there are other problems with the classical theory of spin, like the fact that a rotation of 720 degrees is required (for an electron) rather than a rotation of 360 to get you back to your initial state, due to the double cover property of SU(2).

Any help would be greatly appreciated.

It's true that angular momentum can increase without bound as the linear speed approaches $c$, but the magnetic moment cannot, because it's proportional to the current density $\mathbf{J}$, which is in turn proportional to the ordinary velocity $\mathbf{v}$. So, indeed, there is a simple linear relationship between the rotational speed and the magnetic moment, even at relativistic speeds, and it remains true that you can't possibly explain the magnetic dipole moment of the electron by modelling it as a rotating sphere of charge. (Unless, of course, you assume some areas of the electron are positively charged...)