September 2021 – CMacLaurin.com

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 book §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 book §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 book §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 research paper (arXiv) §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…

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 book §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 $u_\mu$ .

Month: September 2021

Coordinates adapted to observer 4-velocity field

Total angular momentum in Schwarzschild spacetime