Angular momentum in general relativity

In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant.citation needed

In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element:

${\displaystyle \mathbf{L}=\mathbf{r}\wedge <!--\; \wedge \; -->\mathbf{p},}$

in which the exterior product ∧ replaces the cross product × (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined from the x and p vectors, and the expression is true in any number of dimensions (two or higher). In Cartesian coordinates:

or more compactly in index notation:

${\displaystyle L_{ij}=x_i{p}_j-<!--\; -\; -->x_j{p}_i.}$

The angular velocity can also be defined as an antisymmetric second order tensor, with components ω_ij. The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor:

${\displaystyle L_{ij}=I_{ijk\mathrm{\ell <!--\; \ell \; -->}}{\mathrm{\omega <!--\; \omega \; -->}}_{k\mathrm{\ell <!--\; \ell \; -->}}.}$

Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them.

In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an antisymmetric tensor of second order:

${\displaystyle M_{\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{\beta <!--\; \beta \; -->}}=X_{\mathrm{\alpha <!--\; \alpha \; -->}}{P}_{\mathrm{\beta <!--\; \beta \; -->}}-<!--\; -\; -->X_{\mathrm{\beta <!--\; \beta \; -->}}{P}_{\mathrm{\alpha <!--\; \alpha \; -->}}}$

in the language of four-vectors, namely the four position X and the four momentum P, and absorbs the above L together with the motion of the centre of mass of the particle.

In each of the above cases, for a system of particles, the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system.