The subtlety is due to the fact that total 4-angular momentum needs that the individual 4-angular momenta, to be summed over, are evaluated

simultaneously, i.e. at the same coordinate time t, as I mentioned in my solution to the exercise referring to total

N, that is the spatial part of total 4 angular momentum.

For simplicity, let analyse a system made of just two particles, P1 and P2.

A first observer, in coordinate frame F, evaluates the total 4-angular momentum

M summing the 4-angular momentum of first particle,

M1, measured at space-time point (x1, y1, z1, t1) and of second particle,

M2, measured at space-time point (x2, y2, z2, t2) ; this has to be done simultaneously, i.e. at time t1=t2=t

For a second observer point of view, in coordinate frame F', the first observer summed up the 4-angular momentum of first particle measured at space-time point (x1', y1', z1', t1') and of second particle at space-time point (x2', y2', z2', t2'); but now t1' is different from t2'; therefore the result is not, for the second observer, a total 4-angular momentum, because the momenta of the two particle are measured at different times.

This means that total 4-angular momentum is not (in general) Lorentz-invariant, i.e. is not a tensor, i.e. simply makes no sense.

However, the definition makes sense if interactions between the particles occurs only when the two particles are at the same space-time point (and we may call these local interaction collisions).

When there is no interaction, evaluating individual momenta at different times is not a problem, because they don't change with time, by angular momentum conservation.

When there is an interaction, occurring at the same space-time point, the two momenta can be measured just after collision simoultaneously in all frames.

Then with this restriction that total 4-angular momentum is Lorentz invariant.

You wrote that this restriction is needed to have total angular momentum conservation; actually it is needed also to have total angular momentum Lorentz invariance.

You may find more details at

http://panda.unm.edu/Courses/Finley/P49 ... angmom.pdf I found this result quite amazing: if you just assume that:

a) special relativity is correct, that is obviously very reasonable

b) total 4-angular momentum exists (i.e. you can define it as a 4-tensor), that is also very reasonable, taking into account the relation between angular momentum and spin (see chapter 22 of RTR)

then you may deduce that all interactions that may exist in nature occur via collisions among particles.

This is very elegant, because would imply that all interactions are due to a single and simple physical mechanism: collision , i.e. interaction occurring when the particles have the same spatial position at same time, and that have instantaneous duration.

In particular, it would be possible to think that any field (e.g. electromagnetic field) is made by particles, that colliding with other particles causes the interaction. But this is the basic principle of quantum field theory (see chapter 26 of RTR)!