Total angular momentum in Schwarzschild spacetime

Difficulty:   ★★★☆☆   undergraduate general relativity

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 §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:

    \[u^\mu = \bigg( \cdots, \pm\frac{\sqrt{\ell^2-\ell_z^2\csc^2\theta}}{r^2}, \frac{\ell_z}{r^2\sin^2\theta} \bigg),\]

where the first two components are the same as the previous vector. The expressions are simpler with a lowered index u_\mu.

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