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 .