Meaning of the invariants built from the angular momentum tensor

[SiennaTheGr8]

TL;DR Summary

What is the significance of the Lorentz-invariants you can construct from the angular momentum rank-2 tensor?

In special relativity, there's an antisymmetric rank-2 angular-momentum tensor that's "structurally" very similar to the electromagnetic field tensor. Much like you can extract from the latter (and its Hodge dual) a pair of invariants through double contractions (##\vec E \cdot \vec B## and ##E^2 - B^2##), you can extract from the former a pair of Lorentz invariants: ##\vec L \cdot \vec N## and ##L^2 - N^2##, where ##\vec L = \vec r \times \vec p## is the angular-momentum pseudovector and ##\vec N = E \vec r - t \vec p## (of course, ##\vec r## is three-position, ##\vec p## is three-momentum, ##E## is energy, and ##t## is coordinate time). The first scalar (##\vec L \cdot \vec N##) is trivially zero (which I suppose makes it Poincaré-invariant, too). The second (##L^2 - N^2##) is not, but reduces to ##m^2 r^2## in the center-of-momentum frame.

I'm wondering whether the Lorentz-invariance of ##L^2 - N^2## has a straightforward physical interpretation. In the center-of-momentum frame, I guess ##L^2 - N^2## means ##\sum_{i = 1}^n m_i \vec r_i \cdot m_i \vec r_i## (for a system of ##n## particles), which is (maybe) notable because it's related to the numerator of the Newtonian center-of-mass, ##\frac{\sum_{i = 1}^n m_i \vec r_i}{\sum_{i = 1}^n m_i}##. That's all I've got. Am I missing something obvious here? Does the Lorentz-invariance of ##L^2 - N^2## have a simple physical meaning?

 

[renormalize]

That's all I've got. Am I missing something obvious here? Does the Lorentz-invariance of ##L^2 - N^2## have a simple physical meaning?

You correctly infer that ##\vec{N}## relates to the so-called relativistic "center-of-energy" or "center-of-inertia" (see e.g. Landau & Lifshitz, Classical Theory of Fields, pg. 41). Note that for an isolated system, conservation requires that the ten quantities ##E,\vec{p},\vec{L},\vec{N}## all be constant. So in particular, ##\text{const.}=\frac{\vec{N}}{E}=\vec{r}-\left(\frac{\vec{p}}{E}\right)t\equiv\vec{r}_{0}-\vec{v}_{0}t##, where ##\vec{r}_{0},\vec{v}_{0}## are the position and velocity of the system's center-of-energy. But for this reason, Weinberg (Gravitation and Cosmology, pg. 47) says about ##\vec{N}##: "These components have no clear physical significance, and in fact can be made to vanish if we fix the origin of coordinates to coincide with the "center of energy" at ##t=0##, that is, if at ##t=0## the moment ##\int x^{i}T^{00}d^{3}x## vanishes." He then points out that this is due to the fact that the angular-momentum tensor ##J^{\alpha\beta}## is not invariant under 4-translations since orbital angular momentum is always defined with respect to some center of rotation. Instead, to characterize the "internal" portion of the angular momentum, one must use the so-called Pauli-Lubanski spin vector ##S_{\alpha}\equiv\frac{1}{2}\varepsilon_{\alpha\beta\gamma\delta}\,\frac{J^{\beta\gamma}P^{\delta}}{\sqrt{P^{2}}}##, which is sensibly invariant under translations and reduces in the rest frame to the ordinary 3D total angular momentum.