Suppose you are floating in outer space and there is a massive sphere of mass $1 \text{kg}$ next to you and radius $1$ meter and an angular velocity of $1$ revolutions a second. Using the formula for the moment of inertia of a solid sphere $I = \frac{2}{5}MR^2$ and the formula for angular momentum $L = I\omega$ we conclude that the sphere has an angular momentum of $\frac{2}{5}$ $\frac{\text{kg} \cdot \text{m}^2 }{\text{s}}$
Now a mysterious force suddenly sends the sphere flying away from you at $0.99c$ according to your frame of reference (maybe a bomb goes off on one side of the sphere). Naturally now that the sphere is moving very fast you proceed to calculate the rate of rotation of the sphere and conclude that it is rotating at $\sqrt{1 - \frac{(0.99c)^2}{c^2} } $ revolutions per second or about $0.14$ revolutions per second.
You thus conclude the total angular momentum in this universe is now $\frac{2}{5} * 0.14$
Of course this feels a bit strange considering that angular momentum is normally a conserved quantity so where has the angular momentum GONE in this system?
The "bomb" going off is a suspicious culprit here but that could easily be replaced with say a bunch of lasers pushing on the sphere and again it would appear that lasers can just make angular momentum disappear in SR.
Perhaps a more sophisticated way to ask this question is, if not angular momentum itself, what is the correct generalization of angular momentum that is preserved in special relativity?
