- #1

SiennaTheGr8

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- 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

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?

*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?