How does space change as the universe expands?

Consider a metre ruler. Despite the universe – and space itself – constantly expanding, the ruler maintains its size. If this ruler was alone in empty-ish space, other distant objects would appear to recede from it.

What does the universe's expansion mean for measurement and speed in space? More specifically,

• Does this mean that spatial coordinates are expanding, too, such that the ruler is actually becoming smaller?
• What happens to the speed of light? It is constant, but that is relative to the space it is in (as we measure v = s/t). Measuring using the ruler, would it appear that the speed of light increases?

Space doesn't expand. The standard coordinates (FLRW) do.

The way FLRW coordinates cover spacetime is similar to the way polar coordinates cover Earth's surface. If you look at a top-down map of a region of Earth away from the equator, like this one:

you'll see that the lines of constant latitude are curved, and the lines of constant longitude are roughly straight but not parallel. The same thing happens to FLRW coordinates: surfaces of constant cosmological time are curved (extrinsically in the full spacetime) and lines of constant cosmological position are straight (they're geodesics) but they aren't parallel.

If two people at the same latitude but different longitudes in Australia both walk due north, they will get further apart. This is not because Australia expands as you go north, but because their paths aren't parallel. They are literally walking away from each other. The same is true in spacetime.

On the other hand, if you move an east-west pointing ruler north, its length in longitudinal degrees changes, but its length in meters doesn't. Physically, all you've done is move an object of constant length from one place to another. Polar coordinates are great for some purposes, but if you're only interested in the local properties of rulers, they're outright counterproductive: they make "effects" appear to exist that actually don't.

Unfortunately there are no light cones in this otherwise accurate analogy. But the local geometry of spacetime is everywhere the same, like the Earth is locally flat if you look at a small enough piece of it, and the geometry being the same everywhere means in particular that you'll always measure the same speed of light.

• "Space doesn't expand. The standard coordinates (FLRW) do." This seems strange. For k=0, it is only a little strange, but for k=1 or k=-1, it is very strange since for these cases the radius of curvature changes. For a finite (hyper-spherical) universe the total volume of the universe changes. This results seems to be completely inconsistent with "Space doesn't expand."
– Buzz
Jun 29, 2021 at 16:33