Here is a picture of (I think) Maru the cat playing in a bag. He loves bags.
Here is the same picture of Maru, at half the size:
Now imagine that Maru is a physicist and the pictures are not pictures but instead windows into the universe he occupies, separate from ours with (possibly) its own unique set of physical laws. The only difference between the two universes is that one has the lengths of everything reduced by a factor of 2.
Can the parallel versions of Maru tell which universe they’re in – the smaller or the larger? Or if you want to imagine what you might do, suppose that in some Kafkaesque way you find that you wake up one morning and everything if twice the size it was the day before, including you. Could you tell? After all, all the rulers are twice as big too.
Physicists don’t generally directly worry about questions like this, but they’re actually very similar to questions about whether fundamental constants like the speed of light are truly constant. If the speed of light doubled overnight, what would change? Maybe lots of things – the famous Einstein relation E = mc^2 would seem to imply that the ratio between a fistfull of matter and the corresponding quantity of energy would change by a factor of four, which would certainly have an observable effect in things like the nuclear reactions that power the sun.
Or, maybe simultaneously everything suddenly got less massive by a compensating factor and the change in m balances the change in c. Or, maybe the rate of flow of time (whatever that might be) changed or maybe length scales changed, or who knows?
This isn’t idle dorm-room philosophy. Physics needs to be able to deal with quantities that change in time, be they truly fundamental or not. There’s actually some real subtlety involved, but the essentials revolve around just what we mean by units and dimension.
Units are specifically defined increments of a physical quantity. Meters, feet, angstroms, miles, astronomical units, and parsecs are all measuring the same thing in different increments. Likewise liters, gallons, and barn-megaparsecs. And teslas and gauss, and so forth. You can use whatever units are convenient, and keeping track of units is one of the first things you learn in freshman physics. In theoretical physics it’s very common to pick a system of units where various fundamental constants are equal to 1. For instance, with c = 1, Einstein’s mass-energy relation becomes the rather zen-like E = m.
Dimensions are the thing being measured, independent of units. Length in space, duration in time, volume, magnetic field strength, momentum, and so forth.
The crucial test of measurement in physics is the variation in ratios of numbers with identical dimensions. If everything in the universe doubles in size, the ratio of your height to the height of a ruler will stay constant. If only you double in size, the ratio of your height to the ruler will double also, leading to a possible NBA career. (As a naturalized Texan, I believe I should now proffer support for Dallas. Go Dallas!) Ratios of numbers with identical dimensions are said to be dimensionless, and they’re the interesting quantities.
So we don’t look for variation in the speed of light as such. We look for variation in dimensionless numbers involving the speed of light. The most famous of these numbers is probably the fine structure constant, which is the square of the electron charge divided by the product of Planck’s constant and the speed of light. (Maybe with another constant in there too, depending on the system of units you’re using for electric charge. The value of the constant, like all dimensionless constants, is that its numerical value is identical regardless of the system of units.)
The fine structure constant is approximately α = 0.0729735…, which happens to be a hair under 1/137. Why this particular value? Nobody knows, but as a dimensionless number appearing all over physics it’s a great test subject for investigating the possible change in the laws of physics (including the speed of light) over time. So far there’s no real indication that it has ever changed, which is a nice thing to know.
But if it ever did, we could learn about it along these lines without having to worry much about changes in our lab equipment.