Relativistic angular momentum

Remarks

The space components $M^{ij}$ with $i, j \in \set{1, 2, 3}$ are the components of the classical angular momentum $\vec L = \vec x \times \vec p$; for example $M^{12} = x^1 p^2 - x^2 p^1 = L_3$. The mixed components are $M^{0i} = c t \, p^i - x^i E/c$. For a free particle, $x(\tau) = x(0) + \tau p/m$, and the term $\tau p/m$ drops out of $x^\mu p^\nu - x^\nu p^\mu$, so $M$ is constant along the motion. Example: a particle with $\vec p = (p, 0, 0)$ moving on the line $y = b$, $z = 0$ has $L_3 = M^{12} = x^1 p^2 - x^2 p^1 = -bp$ at every time. Non-example: the vector $\vec x \times \vec p$ alone is not a four-dimensional tensor; under a boost it mixes with the components $M^{0i}$. Under a translation of the origin by $a$, $M^{\mu\nu}$ changes to $M^{\mu\nu} - (a^\mu p^\nu - a^\nu p^\mu)$.