The title pretty much sums it up. I'm learning about general relativity and I've ran across descriptions of coordinate distance not matching with a moving objects proper distance. We know this is explained in curved spacetime, but is it also possible in flat spacetime? Perhaps due to the expansion of the universe according to the spatially flat FRW metric spacetime? Thanks.
Sure! Just set up coordinates such your coordinate is "0 meters"; a point a proper distance of 1 meter away from you is labeled "2 m"; a point a proper distance 2 meters away from you is labeled "5 meters"; a point a proper distance of 3 meters away from you is labeled "10 meters"; and so on, with no rhyme or reason as to how the numbers behave with increasing proper distance (other than the fact that they're increasing.)
The point is that coordinates mean nothing on their own in general relativity. They are arbitrary labels that we put on points to help us keep track of them, but they do not necessarily correspond to the proper time for any observer or the proper distance along any geodesic. Nothing stops us from writing down any coordinates we choose to describe flat spacetime in the same way, other than the fact that some choices make the calculations easier. If I were to write out the metric in terms of my perverse coordinates above, it would not look like the Minkowski metric, but it would still describe flat space-time, just in a perverse set of coordinates.
Yes. One example is a row of accelerated observers. They keep a constant distance between them, what is indicated by the green line in the picture. An inertial observer agrees with that distances (half l.y. between yellow and blue, 1 l.y. between blue and red) only for t = 0, when is momentarily comoving with the ships.
After that, the coordinate distances are horizontal lines, and the ships seems getting closer.
The spacetime is of course flat for the inertial observer, but it is also flat for each of the accelerated ones. While they can use this system of coordinates, their proper distances and proper time are different from x and t, and require calculation.