Total angular momentum in Schwarzschild spacetime

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:

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

In fact is the portion of angular momentum (per mass) about the -axis. In Cartesian coordinates , the KVF has components . Similarly, we can define angular momentum about the -axis using the KVF , which in spherical coordinates is . For the -axis we use , which is in the original coordinates. Then:

Hence we can define the total angular momentum as the Pythagorean relation , that is:

This is a natural quantity determined from the geometry alone, unlike the individual 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 and , 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:

In this case the axial momenta are , , and , for a total Killing angular momentum as claimed. There are restrictions on the parameters, in particular the “” must be a minus in the black hole interior. Incidentally this field is geodesic since . 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 and :

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