Interpretation of the Cosmological Metric

The cosmological Robertson-Walker metric of general relativity is often said to have the consequences that (1) the recessional velocity \(v\) of a galaxy at proper distance \(\ell\) obeys the Hubble law \(v=H\ell\), and therefore galaxies at sufficiently great distance \(\ell\) are receding faster than the speed of light \(c\); (2) faster than light recession does not violate special relativity theory because the latter is not applicable to the cosmological problem, and because ``space itself is receding'' faster than \(c\) at great distance, and it is velocity relative to local space that is limited by \(c\), not the velocity of distant objects relative to nearby ones; (3) we can see galaxies receding faster than the speed of light; and (4) the cosmological redshift is not a Doppler shift, but is due to a stretching of photon wavelength during propagation in an expanding universe. We present a particular Robertson-Walker metric (an empty universe metric) for which a coordinate transformation shows that none of these interpretation necessarily holds. The resulting paradoxes of interpretation lead to a deeper understanding of the meaning of the cosmological metric.