Let's explore why Newtonian linear momentum is so easy, and a parallel argument will apply to the more subtle case of angular momentum. Then, we can look at the linear momentum question in general relativity and give its resolution. Again, parallel arguments will apply to angular momentum, but this time only up to a point. This exploration will reveal something about general relativity as well as something about the problem of defining angular momentum in general relativity. Newtonian physics assumes that the laws of Euclidean geometry apply in space and that the time variable is totally separable from the space variables.
Euclidean space is completely symmetrical under translation.
In an empty Euclidean space there is no absolute origin. Hence, one can think of the underlying space as being, in some sense, independent of the laws of Newtonian mechanics. The laws of Newtonian mechanics do in fact also respect the translational symmetry. This means there is nothing in those laws or in the underlying space that will distinguish one point in space from another. One can recast this qualitative outline in more mathematical terms.