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marcus

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There is a misconception floating around PF about the relation of the cosmological redshift to present and past recession velocity.

If a redshift is Doppler in origin then in the context of Special Relativity one has Einstein's correction of the Doppler formula

1+z = sqrt[(1+beta)/(1-beta)] where beta=v/c

But the relation of redshift to speed is different in cosmology and depends on how the expansion of the universe is modeled. One gets comparably simple formulas, but different ones, in some simple cases, but none of the usual models give the SR formula. Here are two formulas which Ned Wright mentions:

v = c ln(1+z) ------empty universe case

v = 2c[1-(1+z)

^{-0.5}] --------critical density, zero cosmological constant

These are obviously not the result of a Lorentz change of coordinates as in Special Relativity! They simply result from space stretching out and in the process lengthening wavelength, the usual explanation of cosmological, as opposed to Doppler, redshift. But for a side-by-side comparison with the Doppler formula we can do a little algebra on the Doppler formula and solve for v.

beta = [(1+z)

^{2}- 1]/[(1+z)

^{2}+1]

v = c [(1+z)

^{2}- 1]/[(1+z)

^{2}+1]

Ned Wright discusses the cosmological redshift in

http://www.astro.ucla.edu/~wright/cosmology_faq.html#FTL

Since it's a short passage containing the two formulas I first mentioned, I will quote the whole thing:

http://www.astro.ucla.edu/~wright/cosmology_faq.html#FTLCan objects move away from us faster than the speed of light?

Again, this is a question that depends on which of the many distance definitions one uses. However, if we assume that the distance of an object at time t is the distance from our position at time t to the object's position at time t measured by a set of observers moving with the expansion of the Universe, and all making their observations when they see the Universe as having age t, then the velocity (change in D per change in t) can definitely be larger than the speed of light. This is not a contradiction of special relativity because this distance is not the same as the spatial distance used in SR, and the age of the Universe is not the same as the time used in SR. In the special case of the empty Universe, where one can show the model in both special relativistic and cosmological coordinates, the velocity defined by change in cosmological distance per unit cosmic time is given by v = c ln(1+z) which clearly goes to infinity as the redshift goes to infinity, and is larger than c for z > 1.718. For the critical density Universe, this velocity is given by v = 2c[1-(1+z)^-0.5] which is larger than c for z > 3 .