# Basic question about units of velocity and speed of a curve on a smooth manifold

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?

• 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 – Ben Crowell Sep 30 '19 at 16:05

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.