http://arxiv.org/PS_cache/astro-ph/pdf/0310/0310808.pdf [Broken] uses the term "proper distance", but doesn't define it. Presumably this must be a standard defintion. So far, though, I have not been able to track down a definitive defintion (I'm still looking). My guess is that this distance is integrated along a spatial geodesic (as contrasted to comoving distance, also used in the above paper, which is integrated along a curve of constant cosmological time). However, I'd much rather not guess :-). Also, if it turns out that the distance measured along a geodesic is NOT proper distance, does this sort of distance have a name? Alternate names and the most common usage for "proper distance" are also of interest. I've seen a sourced claim that Weinberg uses "proper distance" as a synonym for "comoving" distance, for instance - unfortunatley I don't have that textbook to check the source. This seems to conflict with the Lineweaver & Davis usage, as well. [add] In SR there isn't any confusion as to what proper distance means. It's a lot less clear what that term means in GR or cosmology. Being literal minded, it seems to me that any distance measured by integrating the Lorentz interval along a curve could be called "proper".