Fundamentals of Time and Relativity

Angular Momentum

Poincaré invariance is the fundamental symmetry in relativity; see Poincaré Group. This includes, apart from the invariance under spacetime translations considered on the previous page, the invariance under infinitesimal homogeneous Lorentz transformations $δ x^{μ} = δ a^{μ}_{ν} x^{ν}$ . The infinitesimal parameters $δ a^{μ}_{ν}$ form an anti-symmetric tensor $δ a^{μ}_{ν} = - δ a_{ν}^{μ}$ , because the length of any four-vector is conserved under Lorentz transformations, that is

10.14

(x^{μ} + δ a^{μ}_{ν} x^{ν}) (x_{μ} + δ a_{μ}^{ν} x_{ν}) = x^{μ} x_{μ}

Since the action is a Lorentz invariant, one gets, following the reasoning of Noether’s theorem, the conservation law

10.15

δ S_{free} [\bar{x} (τ)] = - m \frac{d {\bar{x}}^{μ} (τ)}{d τ} {δ a_{μ ν} {\bar{x}}^{ν} (τ) |}_{1}^{2} = \bar{x} (τ) \cdot {δ a \cdot \bar{p} (τ) |}_{1}^{2} = 0

with $\bar{x} (τ) = {\bar{x}}^{μ} (τ) e_{μ}$ a solution of the equation of motion.

The anti-symmetric tensor product of the coordinate and momentum four-vectors in (10.15) defines the relativistic angular momentum tensor $l^{μ ν} : = x^{μ} p^{ν} - x^{ν} p^{μ}$ , or in STA notation, the angular momentum bivector

10.16

l : = \frac{1}{2} (x p - p x) = x \land p

The connection between the angular momentum tensor components and bivector is the same as the one between the electromagnetic field tensor and bivector (9.16):

10.17

l^{μ ν} = e^{μ} \cdot l \cdot e^{ν} = (e^{ν} \land e^{μ}) \cdot l

The spatial components of the angular momentum tensor coincide with the 3-d angular momentum vector $l = x \times p$ . For the mixed components the conservation of angular momentum leads to the constant (nameless) vector $x^{0} p - p^{0} x$ . When energy and momentum are also conserved, this implies $x - v t = const$ with $v$ the constant velocity of the particle.

The split of the covariant angular momentum into the angular momentum 3d-vector $l = x \times p$ and the velocity $v$ determined by the mixed components, corresponds exactly to the split of the electromagnetic field tensor in (9.3).
The conservation of the 3-d angular momentum is associated with the invariance of the action under spatial rotations.