If GR is applied to z’ < 1, the SR speed limit doesn’t apply.
But now, it is possible to distinguish outcomes between using SR and z versus GR and z’ even when z’= 0.01. Only due to experimental limitations in the past was z instead of z’applied when z’< 1. GR puts no limits on the increase in distance from galaxy A (suppose our Milky Way) to unbounded galaxy B in time t as the universe expands. GR says nothing about velocities attached to spatial expansion, and the special relativity (SR) speed limit applies only to Doppler redshifts and objects moving through space. If we denote z as a Doppler redshift (due to velocity of an object moving through space away from us) and z’ as cosmological redshift (due to the expansion of space), then conceptually, ONLY z’ should be applied to spatial expansion throughout the range of z’ and general relativity (GR) solved for distance as a function of z’. If $v \ll c$, then the above expression can be approximated by Which can become arbitrarily large as $v \rightarrow c$. Your statement that "nothing can exceed the speed of light" is more nuanced in General Relativity and has received many questions and answers in these pages.Ī redshift larger than 1 is also possible when relativistic motion is applied to a Doppler shift. It is the expansion of space that allows things to apparently recede at greater than the speed of light. This expansion could be interpreted as a recession speed at small redshifts, but as you have surmised, that interpretation runs into trouble when redshifts become greater than 1. It is caused by the expansion of space between the time when the light is emitted and when it is received by an observer.
The cosmological redshift is not a Doppler shift and should not be interpreted as such except perhaps at very small redshifts. Redshifts greater than 1 are possible if the redshift is caused by relativistic motion or by cosmological expansion. The formula $z \simeq v/c$ is only approximately true when $v \ll c$.
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