I thought it might be useful to write a few blog entries, starting with this one, on common misconceptions about cosmology. Please use the comments sections to ask for clarifications on any points that still aren’t clear.
Today I’d like to address a fairly common idea that if space is expanding, time must be also. On the face of it, this seems pretty reasonable. After all, general relativity describes the evolution of space-time. Why should one expand, while the other stays constant?
It’s because of how we define our coordinates. Let me give you one simple equation to hang your hat on:
This is just the definition of velocity and it’s one of the most important equations imaginable as far as physics, and especially as far as relativity is concerned. The reason is that no matter where or when you are, provided you have a good ruler and a good stopwatch, nothing can travel faster than the speed of light:
If I were to suddenly and simultaneously cut your ruler in half and wind your stopwatch so that it ran at double the normal rate, you wouldn’t notice any difference, at least as far as speed measurements are concerned.
Instead of a tortoise traveling 2m in 10 seconds, it would travel 4m (according to your broken ruler) in 20 seconds (according to your broken stopwatch). The ratio is still the same.
The point is that in many ways distance and time are interchangeable. The definition of a meter, for example, is the distance that light can travel in 3.33564 nanoseconds.
In other words, there’s nothing magical about our units, except that the ratio of distance over time (speed) maxes out at the speed of light.
The measurement that most cosmologists use is called “proper time.” Imagine you had a perfect time piece built at the beginning of the universe, and you handed it to someone at the moment of creation. The time that it now reads is the proper age of the universe. It’s the measure of time that makes most sense.
You could, however, make a watch run on “comoving time.” This would have the freaky property that as the universe expanded, it would appear to tick faster and faster and faster. Most cosmologists don’t find this concept terribly helpful.
So short answer: time doesn’t vary because we’re basically free to fix either the definition of time or of space, since all we ever care about (as physicists) are their ratio. We can’t fix both, however.
The expansion of space is really only relevant on scales of millions of lightyears — distances within a galaxy, for example are determined by the local gravitational field — but time is measurable locally, in systems as small as the radioactive decay of atoms.
As a result, it simply turns out that fixing time is more convenient than fixing space.