How do units work for objects in differential geometry?

Question

Consider a manifold $M$, say representing the Earth. It seems to me that it doesn't make sense to talk about units of points in $M$, and that it is the charts that have units. That is, if $\phi$ is a chart that gives coordinates $(x, y)$, then $x$ and $y$ map from points of $M$ to numbers with some unit of length.

Then, somewhat counterintuitively, the basis vector fields $\partial/\partial x$ have units of inverse length. If one then had a vector field $V$ measuring, say, wind speed, the components $V_i$ would have units of length over time, and so $V$ itself would have units of inverse time.

The basis one-forms $dx$ also have units of length. So if $h$ is a manifold function with some set of units, its differential $dh$ has the same set of units, but its components would pick up a unit of inverse length.

Am I on the right track? Do any of you know of any sources talking about this? I had a hard time finding some.

Answer 1

Mathematically, in my opinion, the right way to deal with units is by vector spaces.

• For each type of unit, scalar quantities of that type form a one-dimensional vector space
• Every one dimensional vector space can be considered as corresponding to a new type of unit
• Products of units correspond to tensor products of vector spaces
• Inverting a unit corresponds to taking the dual space

The same is true here. Let $\mathbb{U}$ be the vector space corresponding to some type of unit. Then:

• There is a notion of a $\mathbb{U}$-valued scalar field, which is simply a smooth $\mathbb{U}$-valued function
• Any bundle can be 'twisted' by $\mathbb{U}$. For example, we get "$\mathbb{U}$-valued differential forms" by taking products of ordinary differential forms with a $\mathbb{U}$-valued scalar. This bundle would be notated as $\mathbb{U} \otimes T^*M$.

One thing I think you got wrong is that you surely want the "components" of a $\mathbb{U}$-valued form $\mathrm{d}h$ to be dimensionless quantities. Normally, the "components" of a field are the coordinates relative to a basis: here, you'd have

$$ \mathrm{d} h = \sum_i a_i \omega_i $$

where $\omega_i$ is $\mathbb{U}$-valued, since $\{ \omega_i \}$ form a basis for the space of $\mathbb{U}$-valued forms.

Answer 2

Differences in values of coordinates in charts---and hence the notion of length you've described here---depend the choice of chart and so generally aren't meaningful unless there's (1) a canonical choice of chart (e.g., the identity map on $\Bbb R^n$), or (2) another object that gives a notion of length of vectors, like a Riemannian (or more generally, Finsler) metric.