### Dimensional Analysis and Coordinate Systems

#### Posted by John Baez

We had a nice conversation on dimensional analysis. Here are some things I learned.

As Terry Tao pointed out, systems of units are like coordinates. They’re both systems of arbitrary choices that let us describe reality using numbers. In both cases, we’re ultimately interested in learning things that don’t depend on these arbitrary choices. But to do that, we need to study how things change depending on these choices! We seek *invariant* quantities. But to get them, we must study *covariant* quantities.

Systems of units and coordinate transformations both cause a lot of debates in physics, but these debates tend to be separate. We should try to unify them. A system of units gives a coordinate system on some space of quantities we’re trying to measure. So, if we understand coordinate systems thoroughly, we should understand systems of units.

As Terry and Urs Schreiber noted, we can define a **coordinate system** to be an isomorphism

$f: X \stackrel{\sim}{\to} Y$

where $X$ is an object we’re trying to study and $Y$ is a “known” object. For example, $X$ could be a 4-dimensional spacetime and $Y$ could be $\mathbb{R}^4$. $X$ is like the landscape; $Y$ is like the map. Ideally, each point on the landscape corresponds to a unique point on the map.

If we pick different coordinates

$f': X \stackrel{\sim}{\to} Y$

we can think of these as related to the original ones in either of two ways: by an “active” coordinate transformation, or by a “passive” one. This distinction causes a lot of grief and confusion, so let’s clarify it.

An **active coordinate transformation** is an isomorphism

$g: X \stackrel{\sim}{\to} X.$

It actually moves points around in the landscape. For example, if $X$ were the Earth, $g$ could describe how the Earth rotates. Given an active coordinate transformation $g: X \stackrel{\sim}{\to} X$ and a coordinate system $f: X \stackrel{\sim}{\to} Y$, we get a new coordinate system

$f' = f g.$

A **passive coordinate transformation** is an isomorphism

$h: Y \stackrel{\sim}{\to} Y.$

It moves points around on our map. For example, if $Y$ were a globe depicting the Earth, $h$ could describe how the globe rotates. Given a passive coordinate transformation $h: Y \stackrel{\sim}{\to} Y$ and a coordinate system $f: X \stackrel{\sim}{\to} Y$, we get a new coordinate system

$f' = h f.$

The confusion begins when someone has two coordinate systems $f,f'$ and they ask Tweedledum and Tweedledee how these coordinate systems are related.

Tweedledum says “by an active coordinate transformation, of course!” and writes

$f' = f g.$

Tweedledee says “by a passive coordinate transformation, of course!” and writes

$f' = h f.$

Either one could be right, but Tweedledum insists that we talk about

$g = f^{-1} f'$

while Tweedledee insists that we talk about

$h = f' f^{-1}.$

This allows them to fight endlessly.

For example, when Daylight Savings Time starts, Tweedledum insists that the Sun is rising one hour later, while Tweedledee insists that we’ve just set our watch one hour forwards.

Or, when they’re riding in a train, Tweedledum insists that the scenery is rushing past him, while Tweedledee insists that he’s zipping through the scenery.

Or, when they’re doing quantum mechanics, Tweedledum insists on the Schrödinger picture, where the state of the universe is evolving in time. Tweedledee insists on the Heisenberg picture, where the state remains fixed and it’s observables that change with time.

There’s really no reason to get confused about this stuff - but when you have two equally good conventions for doing something, it’s easy to get stuck in endless arguments about which one to use.

(This is obvious when you have two factions involved, as Swift noted in his parable about the Big-Endians and Little-Endians: the Big-Endians insisted on cracking the big end of the hardboiled egg, while the Little-Endians insisted on cracking the little end. In fact this is not the best example, since the two alternatives are not symmetrical: isn’t it obviously easier to crack the big end? A better example is the dispute between Big-Endians and Little-Endians in computer science.

But, having two equally good choices can be a problem even if there’s just one of you, as Buridan’s Ass noted. This perfectly rational ass got stuck between two equidistant, equally good bales of hay and starved to death. Personally my problem with equally good choices is not making them but remembering which one I made. For example, I can never decide whether to denote the composite of two morphisms $f: X \to Y$ and $g: Y \to Z$ as $f g$ or $g f$. This adds an extra layer of confusion to the above dispute between Tweedledum and Tweedledee - but in case you’re wondering, I’m using the second convention here.)

Anyway, when it comes to systems of units, we are using active coordinate transformations when we imagine light going twice as fast, and passive coordinate transformations when we imagine using rulers half as long.

Everything I’ve said so far applies whenever $X$ and $Y$ are objects in any category. As Terry noted, there’s a lot of specially nice stuff that happens when this category is the groupoid of torsors for some group $G$. Then the group of active coordinate transformations is isomorphic to $G$, though not canonically so unless $G$ is abelian - and similarly for the group of passive coordinate transformations. But, since I’ve discussed torsors and physics elsewhere, I won’t go in that direction here.

Instead, I want to point out that so far our category might as well be a groupoid: I’ve only been talking about *isomorphisms*. My “coordinate systems”, “active coordinate transformations” and “passive coordinate transformations” were all isomorphisms.

But, sometimes coordinate systems aren’t isomorphisms.

For example, we often study an object $X$ using **coordinate charts**

$f: Y \to X$

where $Y$ is some “known” object. For example, $X$ could be an $n$-dimensional manifold and $Y$ could be $\mathbb{R}^n$. There are two new features here.

First, $f$ is pointing the opposite way: our “known” object is getting mapped into the object we’re trying to study. I believe this is not such a big deal, since we can switch the direction of arrows in a category using a purely formal trick, the “opposite category”. The opposite category of a category of “spaces” is often some sort of category of “rings of coordinate functions”, so when we map $\mathbb{R}^n$ into a manifold, we are turning functions on the manifold into functions on $\mathbb{R}^n$.

Second, $f$ is not an isomorphism anymore. I believe this more fundamental. Now we are considering situations where the map is not isomorphic to the landscape! That’s important, because most maps are indeed incomplete in some way. The *unavoidability* of this phenomenon may have first become clear when navigators sought to devise planar maps of the spherical Earth: since the plane and the sphere are not isomorphic, no such map can be perfect.

But, the importance of this phenomenon to physics became clear upon moving from special relativity to general relativity. Not only is the Earth not flat: the Universe is also not flat!

If we allow coordinate charts that are not isomorphisms, we should probably allows coordinate transformations that aren’t isomorphisms. What are these like?

Here I will consider **passive coordinate transformations**. In our new setup, with the arrows reversed, these are morphisms

$g: Y' \to Y$

and they act on any coordinate chart

$f: Y \to X$

to give a new coordinate chart

$f' : Y' \to X$

defined by

$f' = f g.$

We see this happening whenever we’re using a map and then we “zoom in” to a smaller map.

We also use them when we switch from a system of units to a simplified system of units. For example, the SI system has seven units, which measure length, mass, time, current, temperature, amount of substance, and luminous intensity. So, any physical quantity gives a point in $\mathbb{R}^7$ - the “dimensions” of this quantity in the SI system. But, we may choose to work with fewer units. For example, we may decide not to treat “amount of substance” as a dimensionful quantity. SI measures amount of substance in moles, but we can say a mole is just $6.0221415(10) \times 10^{23}$ - a dimensionless number, Avogadro’s number. Our new system of units assigns to each physical quantity a point in $\mathbb{R}^6$, and we have a “change of units”

$g: \mathbb{R}^7 \to \mathbb{R}^6$

which is not an isomorphism. The ultimate extreme is to work in a system where all physical quantities are treated as dimensionless, so any physical quantity has dimensions living in $\mathbb{R}^0$. This is actually very popular in fundamental theoretical physics.

Anyway: we’ve seen that in any category, we can study an object $X$ by mapping “known” objects into it. The ultimate way to do this is to consider *all possible* morphisms

$f : Y \to X$

for *all possible* objects $Y$. This gives a very thorough description of $X$.

For example, we can study an $n$-dimensional manifold $X$ by looking at all possible coordinate charts

$f: Y \to X$

in this manifold, where $Y = \mathbb{R}^n$. People like to use this trick; it’s called a **maximal atlas**. But, we can go even further and let $Y$ range over all possible manifolds!

Suppose we’re trying to understand an object $X$ in some category $C$. If we look at all possible “coordinate charts” on this object:

$f: Y \to X$

and how they are related by “passive coordinate transformations” of the form

$g: Y' \to Y,$

we get a gadget called a presheaf on $C$. This is just a functor

$F: C^{op} \to \mathrm{Set}.$

In the case at hand, $F$ assigns to any object $Y$ the set of all “coordinate charts”$f: Y \to X$. And, it assigns to any “passive coordinate transformation” $g: Y' \to Y$ the function which eats a coordinate chart $f: Y \to X$ and spits out the transformed coordinate chart $f g: Y' \to X$.

If you’re wondering exactly what I mean now by the quoted phrases “coordinate chart” and “passive coordinate transformation”, it’s simple. In both cases, I mean simply *morphism!* We’ve left the warmup case where coordinate systems and coordinate transformations had to be isomorphisms: now we’re letting them be noninvertible.

But here’s the cool part:

The Yoneda Lemma says that everything we might want to know about our object $X$ can be recovered from the presheaf $F$ defined above! For a pretty good explanation of this lemma, try this. It takes a bit of category theory to understand the technical details.

But I don’t want to get into that here. I just want to say what this famous result says about coordinate systems:

**We know an object completely if we know all coordinate charts on that object, together with how these coordinates transform.**

## Re: Dimensional Analysis and Coordinate Systems

I was going to post this comment in the previous discussion about dimensional analysis but comments finished there. I will try and tie it into the latest discussion towards the end.

When I teach about waves, the physics textbooks talk about frequency and the unit “hertz” as being “cycles per second” (or similar). However the textbooks, of course, correctly state that the unit Hz is equivalent to $s^{-1}$ and not actually “cycles per second”. Similarly, of course, wavelength is just measured in meters which is consistent with Hz simply being $s^{-1}$ so that, using dimensional analysis :-), wavelength multiplied by frequency gives us the correct units for a wave’s speed.

However, we could have Hz be “cycles per second” and wavelength as “meters per cycle” and speed would still work out just fine. This all sounds good and consistent to me… and more sensible… in the sense that it would be easier to justify when first taught.

Also, for example, isn’t saying a wavelength is “1m” like saying my speed is “1km”? We actually say 1km/h (or similar). So shouldn’t my wavelength be 1 m/cycle where it is explicit that we have to have knowledge of what a cycle is just as we have to have knowledge of what unit of time we are using?

The name “wavelength” is even misleading. It would be easier to have “meters per cycle”. After all we do not say speed is measured in “timelength” and assume a standard amount of time, but we say “kilometres per hour” (or similar). Yes, a “wavelength” has a canonical choice of “cycle” but we could equally (stealing John’s devil’s advocate role from the last dimensional analysis discussion) have chosen a definition that is equivalent to 2 waves or half a wave for our “cycle”.

Would it be sensible to instigate a campaign to have the “cycle” recognised as an SI base unit?! As, surely it is not a derived unit given its definition is “the distance between two consecutive points in phase” (or something similar). How would you express this in terms of the base units?

Linking in to the current main topic… it would seem like the current situation for SI base units is $\Re^8 \rightarrow \Re^7$ [how do I get a blackboard R in itex?] though I am not sure it is clear to me how this trick has been done for a “cycle” in the same way as using Avogadro’s number for moles.

So maybe there is a difference here and thinking of possible morphisms is part of how the difference can be illustrated? Not sure I quite see this.

On the BIPM website there is a section that talks about dimensionless quantities that are ratios such as refractive index (easy to understand), but it also talks about counting quantities such as a number of molecules (less easy to understand – isn’t the mole one of these?). Perhaps “cycle” comes under this category as there are some in this category (such as the radian) that are given names. But in the case of the cycle we could change the definition (unlike counting the number of molecules for example – although could we talk about half molecules?) as mentioned above so does this really fit in this category and therefore how does it differ from other base units?

I had better stop here and ask… “So what do you all think?” Have I just missed something really obvious?