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…