Unit and dimension of angles

Question

usually in physics (at least in the SI) angles are regarded as dimensionless. Is it possible to give a dimension (and a unit) also to angles and still have a system of units of measure as coherent as the previous one? Is there something in the concept of dimensional analysis that prevents physicians from doing that? Please note that I'm not saying it is convenient! I'm just asking if it is possible.

Read below only if you find this question stupid.

The dimensionless argument is often justified saying that the definition of "angle" is given by a length divided by a length.

I think this is just false since the definition of an angle is "two rays sharing a common endpoint", it is not even in the definition of "measure of an angle" since it is the comparison of the angle I want to measure with my fixed unit angle, something that is not related at all with length. Apparently this is only due to the operation we want to do with angles: multiply them with the radius of a circle to get the length of an arc. Here we can face the problem from two different points of view:

1. Forget about multiplying angles and length together. A bit drastic but we might be able to construct our physics anyway. Honestly I don't remember doing this multiplication of angles and radius much often, especially in important theorems.
2. In the same way in which we jump magically from charge to force in Coulomb's law we can invent a constant that like Coulomb's constant is not dimensionless and allows me to go from angles to lengths.

Another thing I was considering was to measure the radius with the same unit we have for length associating to each angle the measure of the corresponding arc on the unit circle, but I soon realized that this leads to problem when we do conversions since the unit circle in one unit is different from the unit circle in another unit...

Still if we can fix a certain length we can use it to fix the length of the circle we use, is much complicated since is like introducing a measure for angles by firstly introducing a second measure on length but what's the problem? My question is not if this method is efficient, it is about coherence!

Apparently on physics.stackexchange the difference between "one good method" and "the only way of doing it" is not well understood since the answers I got were embarassing: https://physics.stackexchange.com/questions/226332/dimension-of-an-angle

I hope to find better answers here. At least I would find a better definitions of dimension.

Answer 1

You are conflating two different ideas here: an angle (which is a geometric construct that can be defined in numerous ways; the definition you give is just one possibility), and an angle measurement (which is a numerical object; more precisely, a mapping $\theta \mapsto m(\theta) \in \mathbb{R}$).

Of course one can measure angles using a unit if one wants to: "degrees" is the conventional choice for this. You decide on a reference angle, declare that it is equal to "1 degree", and then every other angle is measured relative to the reference angle. "Gradians" is another possible unit of measure for angles, less commonly used but still conventional in certain applications.

The downside of such a choice is that it complicates your system of measurement by introducing an additional, arbitrary unit. It also requires you to always specify how functions like $\sin(\theta)$ and its cousins are meant to be interpreted, what their period is, and so forth.

The advantage of measuring angles in radians is that they are, at a fundamental level, intrinsic to the geometry of the plane: regardless of the unit one chooses to measure length, a semicircle is always $\pi$ times as long in arc length as it is in diameter. For this reason radians have a universality to them that makes them much more useful.

Note that "more useful" means only what it says. You can measure angles in degrees, gradians, or any other unit if you really want to; it will just make all of your formulas more complicated.

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Edited: As one of my professors once said to me: "You can define things any way you want to in mathematics, as long as you are prepared to live with the consequences. But why would you want to?"

If for some reason you were dead set on measuring angles in units of mass, you would first have to decide on what the conversion would be: how many "grams" are in a right angle? Note that there is no way at all to decide this either mathematically or experimentally, so you would have to just make up some conversion factor. Let's say that a right angle is 50 grams, so that a straight angle is 100 grams.

Now having done that, consider a function like $$f(\theta)=\sin\theta$$ The $\sin$ function is usually understood to take in a unitless number and output a unitless number. If you want to measure angles in grams, what do you want to measure sines in? Also grams? Or seconds? Or meters? Or would you like that to be dimensionless? You can make whatever choice you want, as long as you are prepared to live with the consequences.

Let's say that sines are dimensionless. So now we need (based on the choices already made) $\sin(50 \text{ grams})=1$, and the period of $f(\theta)$ would have to be $200$. But the sine function we know doesn't work that way, so you need to (essentially) define a new sine function (maybe distinguish it with an overbar, so it would be called $\overline{\sin}\theta$) with the property that

$$f(\theta)=\overline{\sin}\theta = \sin \left( \frac{\pi}{50 \text{ grams}}\theta \right)$$ where the sine function on the right is the "normal" sine function, with a period of $2 \pi$.

Now consider the derivative, $$f'(\theta)=\frac{\pi}{50 \text{ grams}} \cos\left( \frac{\pi}{50 \text{ grams}}\theta \right)$$ This would have to be be understood as having units of $1/\text{mass}$. Notice that we have replaced the simple and elegant formula $$\frac{\rm{d}}{\rm{d}\theta}\sin\theta = \cos\theta$$ with the formula $$\frac{\rm{d}}{\rm{d}\theta}\overline{\sin}\theta =\frac{\pi}{50 \text{ grams}} \overline{\cos}\theta$$ and whereas the usual sine function is a solution to the differential equation $$\frac{\rm{d}^2}{\rm{d}\theta^2}f(\theta) + f(\theta) = 0$$ the new function $\overline{\sin}\theta$ is not. You would need to insert all kinds of constants to get the dimension to work out and to get the functions to be equal.

And all of this is dealing with purely mathematical relationships, before you start talking about physical constants like $c$ and $G$ and $h$, which would all have to be redefined to keep things consistent. And in the background, lurking behind it all, the old "dimensionless" way of measuring angles is still there, buried inside the definitions of $\overline{\sin}$ and $\overline{\cos}$. So what have we really accomplished?

I am not saying that it is impossible to do this in a coherent way -- but it would be a mess. So why would you want to?