Varying Speed of Light?

There have been theories going around in the undergrowth of theoretical physics for some years that propose that c, the speed of light in a vacuum, may have varied over the course of cosmological evolution. Jakob Bekenstein, John Moffat, Joao Magueijo, John Barrow and Andreas Albrecht have all contributed to "VSL", which is sometimes touted as an alternative to inflation to solve the horizon problem and other puzzles of cosmology. The basic idea is that light was much faster in the early Universe - many orders of magnitude faster. Then at some point it jumped down to the current value. Thus, faraway parts of the observed Universe would have time to communicate with each other, which could explain why they appear so uniform (homogeneous and isotropic).

Recently the measurements of varying alpha (alpha being the fine structure constant) by John Webb and the team at UNSW have revived speculation about varying c, since alpha is defined as e2/(4h-bar c). A veritable storm of media coverage resulted, with headlines like "Was Einstein Wrong?", "Light is Slowing Down", "Is Light strolling along at hot-summer-day-pace?", etc.

Problems

However, both VSL cosmologies and the interpretation of varying alpha as "varying c" suffer from a big problem - that of units. As you will have learnt by reading the other writeups in this node, the metre is defined as the distance travelled by light in 1/299,792,458 seconds. Thus, using this definition, c always takes the same value. What does "varying c" mean now?

Also, suppose you now take a different set of units: the standard second, and the length of a piece of metal containing a certain number of atoms laid end-to-end. The length of the piece of metal in metres will depend on the fine structure constant because of the effect of electromagnetic interactions. So, if alpha varies over time, the speed of light in bits-of-metal per second will also vary. You could also take a different standard of time: a pendulum clock, say. But for every different set of units, the apparent variation in c will be different! Clearly the variation in c is not a physically well-defined thing.

The underlying principle is that only the variation in dimensionless numbers can be measured unambiguously. This is exactly what we do when we use units: we can say that the dimensionless ratio of the length of a table to the standard metre is 1.43. If the number turns out to be 1.45 tomorrow, something is clearly a bit odd, but you can't say definitely whether the table is bigger or the metre is smaller.

As I indicated, variations in dimensionless numbers such as alpha are all that is needed to relate the measurements in one system to those in another. This interconvertibility extends to the so-called VSL theories. Indeed John Barrow has a paper in which he tells us that any theory of varying alpha that includes electromagnetism and general relativity can be written either as "varying c" or "varying e".