Undergrad Meaning of the invariants built from the angular momentum tensor

[SiennaTheGr8]

TL;DR

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.