Coordinates adapted to observer 4-velocity field

Difficulty:   ★★★☆☆   undergraduate

Suppose you have a 4-velocity field \mathbf u, which might be interpreted physically as observers or a fluid. It may be useful to derive a time coordinate T 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:

    \[dT := -\mathbf u^\flat.\]

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

    \[dT = \frac{\partial T}{\partial x^0}dx^0 + \frac{\partial T}{\partial x^1}dx^1 + \cdots,\]

where x^\mu is a coordinate basis. Technically dT is a covector, with components (dT)_\mu = \partial T/\partial x^\mu in the cobasis dx^\mu. Similarly -\mathbf u^\flat = -u_0dx^0 -u_1dx^1 -\cdots, so we must match the components: \partial T/\partial x^\mu = -u_\mu. 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 \langle dT,dT\rangle = \langle -\mathbf u^\flat,-\mathbf u^\flat\rangle = -1. One can show its change with proper time is dT/d\tau = \langle dT,\mathbf u\rangle = 1. Further, the T = \textrm{const} hypersurfaces are orthogonal to \mathbf u, since the normal vector (dT)^\sharp is parallel to \mathbf u. 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 x^0-coordinate by T, 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 g^{\mu\nu} = \langle dx^\mu,dx^\nu\rangle. Similarly one new component is g'^{TT} = \langle dT,dT\rangle = -1. Also g'^{Ti} = \langle dT,dx^i\rangle = -u^i, where i = 1,2,3. The g'^{iT} are the same by symmetry, and the remaining components are unchanged. Hence the new components in terms of original components are:

    \[g'^{\mu\nu} = \begin{pmatrix} -1 & -u^1 & -u^2 & -u^3 \\ -u^1 & g^{11} & g^{12} & g^{13} \\ -u^2 & g^{21} & g^{22} & g^{23} \\ -u^3 & g^{31} & g^{32} & g^{33} \end{pmatrix}.\]

The matrix inverse gives the new metric components g'_{\mu\nu}. The 4-velocity components are: u'_\mu = (-1,0,0,0) by the original equation. Also u'^T = \langle dT,\mathbf u\rangle = 1, and the u'^i = \langle dx^i,\mathbf u\rangle = u^i are unchanged. Hence u'^\mu = (1,u^1,u^2,u^3).

Anecdote: I used to write out dT = -u_0dx^0 - u_1dx^1 - \cdots, rearrange for dx^0, 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

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 §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.