A reader writes in with a question about the speed of light. Since the meter and second are defined in terms of that speed, how would we be able to tell if the speed of light is changing or has changed throughout history?
It’s a good question. According to the official standard, a meter is defined to be “the length of the path travelled by light in vacuum during a time interval of 1⁄299 792 458 of a second”. If the speed of light gets faster, by definition the meter gets longer. It’s additionally complicated by the fact that the second is also defined in terms of the speed of light. The second is “the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.” If you actually calculate that transition from first principles, you’ll encounter equations involving the speed of light. Therefore a changing speed of light would actually change the definitions of both the meter and the second in a complicated way.
So would scientists notice a changing speed of light, given that the units for distance and time are defined in terms of that speed? The answer, as you might guess, is yes. There’s two classes of constantly ongoing observations that come to mind. We’ll call them the practical and the theoretical.
The Practical: In practice physicists don’t directly use the formal definition of the meter and second. We use measuring devices whose connection to those standards is indirect. If the speed of light changed, those devices wouldn’t automatically resize themselves to fit the new definition. Because we routinely make measurements of length with light to ludicrous precision, we’d notice immediately if the speed of light changed. But even if our measuring devices did instantly update themselves to a new speed of light, we’d still notice just as quickly because all the actual distances being measured (our height, the distance to satellites, the length of interferometers, etc) are still the same, just enumerated with different units. You’d notice the difference too – GPS would quit working, because it functions by measuring the light travel time to you given the distances between the satellites. Those distances wouldn’t change just because our measuring unit changed. A new speed of light, even if the change were just a few parts per billion, would seriously throw off the accuracy of GPS.
The Theoretical: It may not even be meaningful to talk about a change in the speed of light in and of itself. If the other constants of nature changed in compensating ways, the change might be no different from watching a movie in slow motion; nothing in the film happens differently. What we’re really interested in is the speed of light changing with respect to other constants of nature. As such we can look at the ratio of the speed of light with other constants , with the constants arranged in a way that cancels out all the units, leaving just a pure number. The most obvious of these dimensionless constants is the fine structure constant:
It’s equal to about 1/137.035999679. As a dimensionless constant, its value is not in the slightest dependent on how we define our units. If we redefined the meter to be twice its current value, we’d have to re-write the speed of light and the gravitational constant and whatnot to fit the new system, but all of that cancels out for the dimensionless constants. The fine structure constant and the other dimensionless constants would stay exactly the same.
This particular constant is extremely important in the quantum theory of atoms and light. If you want to calculate just about anything in AMO physics, you’ll be using it directly or indirectly. It permeates the laws describing all kinds of processes, including the way atoms emit light. If we look at the spectra of distant stars, we can determine the fine structure constant at the time and place that light was emitted. To the greatest precision we can manage (and it’s pretty great – about one part in a million), the fine structure constant seems to be the same at all times and places we’ve looked in the wider universe. On earth over our short history of measuring the constant in the lab, it’s been rock-solid to a much greater precision. It can’t have changed too much a priori anyway, since stars are only possible for a limited range of values of that constant.
We still can’t absolutely exclude the possibility that the speed of light may not be constant, or may have varied very slightly in he past. But we can be assured that if it changes even slightly, we’ll notice immediately even with our units defined the way they are.