I always read that conservation of angular momentum is with respect to an origin of our choice, so if we want to compare the angular momentum of two situations, we have to calculate the angular momentum in these situations with respect to the same origin. However - I've seen in some questions that there's conservation of angular momentum, but the calculation of it in each state is with respect to the center of mass. Now comes the twist - between these two states the center of mass itself has changed it's position in space (like with a rod that spins as it's making translation movement as well - in t=0 the rod can be at the left side of a room and in t=t' it can be at the right side of the room, it's clear that the center of mass of the rod is not at the same position in space). I'd like to know what is the mathmatical justification for this comparison even though the origin between the two states is not at the same location
Isn't is just a moving coordinate system with the origin being the center of mass so that the translational motion of the rod (or whatever) is invariant with respect to the coordinate system. Maybe I don't completely understand completely because this makes sense to me in a classical limit but I'm not sure this holds true for relativistic speeds.