## Coordinates adapted to observer 4-velocity field

Suppose you have a 4-velocity field , which might be interpreted physically as observers or a fluid. It may be useful to derive a time coordinate which both coincides with proper time for the observers, and synchronises them in the usual way. Here we consider only the geodesic and vorticity-free case. Define:

The “flat” symbol is just a fancy way to denote lowering the index, so the RHS is just . On the LHS, is the gradient of a scalar, which may be expressed using the familiar chain rule:

where is a coordinate basis. Technically is a covector, with components in the cobasis . Similarly , so we must match the components: . For our purposes we do not need to integrate explicitly, it is sufficient to know the original equation is well-defined. (No such time coordinate exists if there is acceleration or vorticity, which is a corollary of the Frobenius theorem, see Ellis+ 2012 §4.6.2.)

The new coordinate is timelike, since . One can show its change with proper time is . Further, the hypersurfaces are orthogonal to , since the normal vector is parallel to . This orthogonality means that at each point, the hypersurface agrees with the usual simultaneity defined locally by the observer at that point. (Orthogonality corresponds to the Poincaré-Einstein convention, so named by H. Brown 2005 §4.6).

We want to replace the -coordinate by , and keep the others. What are the resulting metric components for this new coordinate? (Of course it’s the same metric, just a different expression of this tensor.) Notice the original components of the inverse metric satisfy . Similarly one new component is . Also , where . The are the same by symmetry, and the remaining components are unchanged. Hence the new components in terms of original components are:

The matrix inverse gives the new metric components . The 4-velocity components are: by the original equation. Also , and the are unchanged. Hence .

Anecdote: I used to write out , rearrange for , and substitute it into the original line element. This works but is clunky. My original inspiration was Taylor & Wheeler 2000 §B4, and I was thrilled to discover their derivation of Gullstrand-Painlevé coordinates from Schwarzschild coordinates plus certain radial velocities. (I give more references in MacLaurin 2019  §3.) I imagine that if a textbook presented the material above — given limited space and more formality — it may seem as if the more elegant approach were obvious. However I only (re?)-discovered it today by accident, using a specific 4-velocity from the previous post, and noticing the inverse metric components looked simple and familiar…

## 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 .