Ask a Physicist

Which would be considered more fundamental - the constant "c" (speed of light in vacuum), or the constant "epsilon_0" (permittivity of free space), or the constant "alpha" (fine structure)? Are they equally fundamental?
Submitted by Chipper Q from the USA

These constants all include the same information - they can be expressed in terms of each other (and some other constants). But they have different dimensions: "c" is a speed or distance per unit time, "epsilon_0" is some funny electrical thing, and "alpha" has none - it's just a number.

Here we need to distinguish between dimensions (like speed) and units, the specific things we measure with (like miles per hour or meters per second).

Most physicists would say that alpha is a more fundamental constant than c or something else because dimensionless numbers are more fundamental than anything with dimensions. Really what we measure are numbers. When I say it's fifty yards to the street corner, what I mean is that the ratio of that distance to the stride of some long-dead English king is fifty. That ratio would not change if I measured both lengths in meters. It's only forty-five meters to the corner, but the king's stride is a little less than a meter, so it's still fifty of them.

What good are dimensions then, if we really measure dimensionless numbers?

Dimensions let us distinguish between things that cannot be compared without reference to some other factor. For instance, I can't define a distance as a time without using some outside information: I can say that my house is four hours from Philadelphia if I use my average driving speed as a conversion. But if I used light speed as a conversion, I would say that Philadelphia is a millisecond away. Contrast this to saying Philly is two hundred miles away. At first that looks just as arbitrary as quoting a time, but really I am saying the ratio of that distance to that English king's stride is 350,000. And that ratio doesn't change no matter what units I use, and I don't need to bring anyone or anything into it except me and the king.