In Appendix A of Davis & Lineweaver (2003) proper distance to a faraway galaxy is defined as the distance along a curve of constant time in the RW metric. I was wondering whether that line of constant time is a geodesic of spacetime. If not then there will be a shorter-distance path from here to the faraway galaxy, that starts and finishes in cosmic 'now' but in between deviates into other cosmic times. The RW coordinate system is, I think, orthogonal. But that is not sufficient for its coordinate lines to be geodesics. For example, polar coordinates in E2 are orthogonal but the geodesic from (1,0) to (0,1) is not along the line of constant r=1. Is the line of RW constant time a spacetime geodesic? If so, is there a proof I can look at? If not, is there an easy counterexample like the above one for polar coordinates?