Total angular momentum in Schwarzschild spacetime

Difficulty: ★★★☆☆ undergraduate

In relativity, distances and times are relative to an observer’s velocity. Hence one should be careful when defining an angular momentum. Speaking generally, a natural parametrisation of 4-velocities uses Killing vector fields, if the spacetime has any. In Schwarzschild spacetime, Hartle (2003 book §9.3) defines the Killing energy per mass and Killing angular momentum per mass as:

$e := -\langle\mathbf u,\partial_t\rangle, \qquad \ell_z := \langle\mathbf u,\partial_\phi\rangle.$

The angle brackets are the metric scalar product, $\phi$ has range $[0,2\pi)$ , and we will take $\mathbf u$ to be a 4-velocity. I have relabeled Hartle’s $\ell$ as $\ell_z$ . While $\partial_t$ and $\partial_\phi$ are just coordinate basis vectors for Schwarzschild coordinates, as Killing vector fields (KVFs) they have geometric significance beyond this convenient description. [ $\partial_t$ is the unique KVF which as $r \rightarrow \infty$ in “our universe” (region I), is future-pointing with squared-norm $-1$ . On the other hand $\partial_\phi$ has squared-norm $r^2\sin^2\theta$ , so is partly determined by having maximum squared-norm $r^2$ amongst points at any given $r$ , which implies it is orthogonal to $\partial_t$ , although the specific orientation is not otherwise determined geometrically.]

In fact $\ell_z$ is the portion of angular momentum (per mass) about the $z$ -axis. In Cartesian coordinates $(t,x,y,z)$ , the KVF $\mathbf Z := \partial_\phi$ has components $(0,-y,x,0)$ . Similarly, we can define angular momentum about the $x$ -axis using the KVF $X^\mu := (0,0,z,-y)$ , which in spherical coordinates is $(0,0,\sin\phi,\cot\theta\cos\phi)$ . For the $y$ -axis we use $Y^\mu := (0,-z,0,x)$ , which is $(0,0,-\cos\phi,\cot\theta\sin\phi)$ in the original coordinates. Then:

$\ell_x := \langle\mathbf u,\mathbf X\rangle, \qquad \ell_y := \langle\mathbf u,\mathbf Y\rangle, \qquad \ell_z = \langle\mathbf u,\mathbf Z\rangle.$

Hence we can define the total angular momentum as the Pythagorean relation $\ell^2 := \ell_x^2 + \ell_y^2 + \ell_z^2$ , that is:

$\ell^2 := \langle\mathbf u,\mathbf X\rangle^2 + \langle\mathbf u,\mathbf Y\rangle^2 + \langle\mathbf u,\mathbf Z\rangle^2.$

This is a natural quantity determined from the geometry alone, unlike the individual $\ell_z$ etc. which rely on an arbitrary choice of axes. It is non-negative. I came up with this independently, but do not claim originality, and the general idea could be centuries old. Similarly quantum mechanics uses $J^2$ and $J_z$ , which I first encountered in a 3rd year course, although these are operators on flat space.

One 4-velocity field which conveniently implements the total angular momentum is:

$u^\mu = \bigg( \frac{e}{1-\frac{2M}{r}}, \pm\sqrt{e^2-\Big(1-\frac{2M}{r}\Big)\Big(1+\frac{\ell^2}{r^2}\Big)},\frac{\ell}{r^2},0 \bigg).$

In this case the axial momenta are $\ell_x = \ell\sin\phi$ , $\ell_y = -\ell\cos\phi$ , and $\ell_z = 0$ , for a total Killing angular momentum $\ell$ as claimed. There are restrictions on the parameters, in particular the “ $\pm$ ” must be a minus in the black hole interior. Incidentally this field is geodesic since $\nabla_{\mathbf u}\mathbf u = 0$ . It also has zero vorticity (I wrote a technical post on the kinematic decomposition previously), so we might say it has macroscopic rotation but no microscopic rotation. Another possibility is in terms of $\ell_z$ and $\ell$ :