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Frederic Schuller says that velocity has units in Hertz in The WE-Heraeus International Winter School on Gravity and Light. He says:

\begin{align} [v^a]&=\frac 1 T \\ [g_{ab}]&=L^2 \\ \Big[\sqrt{g_{ab \left(v^av^b\right)}}\Big]&=\frac L T \end{align}

with $g_{ab}$ corresponding to the metric tensor on a Riemannian manifold.

Velocity of a parametrized curve on a point of a manifold is just the differential operators at that point forming the tangent space, and evaluated by a test function on a chart to Euclidean space.

The test function doesn't need to have any connections to time; hence, can we conclude that the only way the velocity components will have units of 1/time is if the curve is parametrized by time?

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  • $\begingroup$ There is no simple, universally accepted answer to questions like this. This person's way of talking about it is unusual, but not necessarily wrong. For a detailed treatment of this kind of thing, see section 5.11 of my GR book, which is free online lightandmatter.com/genrel $\endgroup$ – Ben Crowell Sep 30 '19 at 16:05
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Schuller's argument is based on the assumption that the components of the metric should be viewed as having units of length squared. That assumption is not something that everybody agrees on. At most it's a convention that he prefers. Since your argument doesn't make use of that assumption, I don't think it can be a correct argument.

As a way of seeing that this is an arbitrary assumption or convention, note that if we multiply the metric by any constant, there are no observable effects -- this is just a change of units and/or signature convention (if the constant is negative). So if Schuller prefers $ds$ to be a length, that's fine, but someone else could prefer to have it be a time, and someone else could prefer units in which $c=1$, in which case the units of length and time are the same.

I think if you ask relativists their own preferences, almost all of them will say that they prefer to work in units where $c=1$. Therefore velocities are unitless.

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