I’m confused about what a specific observer, comoving with the Hubble flow, would measure as the distance to another comoving observer (galaxy) “far away”. I would’ve thought that this should be the “proper distance”, defined as the shortest length of a path connecting the two events at a fixed moment in proper time $\tau$ of the observer. But perhaps this is already the issue – is the “cosmic time” the same as the observer’s proper time?
Assuming that proper distance is what the observer would measure, the problem is then that this distance $D$ can increase arbitrarily quickly. This is just the observation that “recession velocities”, $\mathrm{d}D/\mathrm{d}\tau$, can be “superluminal”. But if the proper distance is supposed to be the one actually measured by an observer in their local frame, and this distance is increasing faster than $c$, then light sent in the same direction would not be able to catch up with the distant galaxy and there would be actual superluminal motion. That can’t be right? Please note that I’m not concerned with any particular definition of relative velocities being larger than $c$ or not, but rather the problem that in a specific observer’s local frame, if distances between objects can be measured, they should (surely?) not increase faster than the distance traveled by light. I’ve also read that cosmological recession velocities are actually rapidities, but it is not clear to me how that is the case.
So something must be wrong, but I’m not sure what it is. Is it just not possible to define a “physical” distance for distant objects? Or perhaps it is, but what is usually called the “proper distance” is not the correct one to use?
