The Redshift Paradox Resolved
I'm currently taking a course on general relativity, and I have found some course notes by S. Carroll to be rather useful. While reading them I ran across a passage which quite nicely resolves the redshift paradox by pointing out the ultimate falsehood of a common and convenient assumption. It speaks for itself:
Note: "curved manifold" here means space as Einstein saw it.
We simply must learn to live with the fact that two vectors can only be compared in a natural way if they are elements of the same tangent space... For example, two particles passing by each other have a well-defined relative velocity (which cannot be greater than the speed of light). But two particles at different points on a curved manifold do not have any well-defined notion of relative velocity --- the concept simply makes no sense. Of course, in certain special situations it is still useful to talk as if it did make sense, but it is necessary to understand that occasional usefulness is not a substitute for rigorous definition. In cosmology, for example, the light from distant galaxies is redshifted with respect to the frequencies we would observe from a nearby stationary source. Since this phenomenon bears such a close resemblance to the conventional Doppler effect due to relative motion, it is very tempting to say that the galaxies are "receding away from us" at a speed defined by their redshift. At a rigorous level this is nonsense, what Wittgenstein would call a "grammatical mistake" --- the galaxies are not receding, since the notion of their velocity with respect to us is not well-defined. What is actually happening is that the metric of spacetime between us and the galaxies has changed (the universe has expanded) along the path of the photon from here to there, leading to an increase in the wavelength of the light. As an example of how you can go wrong, naive application of the Doppler formula to the redshift of galaxies implies that some of them are receding faster than light, in apparent contradiction with relativity. The resolution of this apparent paradox is simply that the very notion of their recession should not be taken literally.
---S. Carroll, in Lecture Notes on General Relativity
Originally published on Quasiphysics.