Re: Cosmological redshift and Doppler shift

In article <[EMAIL PROTECTED]>,
Phillip Helbig---remove CLOTHES to reply <[EMAIL PROTECTED]> wrote:

>I've now finished reading the paper I mentioned a couple of posts back
>in this thread, astro-ph/0310808.  The authors mention that the
>cosmological time dilation---(1+z), independent of the cosmological
>parameters---does not agree with the special-relativistic prediction
>EVEN TO FIRST ORDER (Eqs. 3--5).  If so (and if not please point out
>their error), then special-case cosmological models such as the Milne
>model (i.e. lambda=0, Omega_matter=0; obviously a limiting case) cannot 
>be expressed in the language of SR and agree with observations.

Equation (4) is wrong.  Either that, or the authors mean something
quite different by it than what it seems to mean; I haven't read the
paper carefully enough yet to know.  Either way, the description
following it is extremely misleading.  Equation (4) certainly does not
justify the conclusion that special relativity doesn't reproduce the
cosmological time dilation to first order; it does reproduce it.

When you take an FRW spacetime and approximate it as flat (which you
can do to a good approximation at low redshift in any FRW spacetime,
or exactly at arbitrary redshift in the Milne model), the velocity of a
galaxy relative to you is not the peculiar velocity v_{pec}.  It's (to
first order, anyway) the peculiar velocity plus the Hubble velocity.
So the special-relativistic prediction should not be

\gamma_{SR} = (1-v_{pec}^2/c^2)^{-1/2},

as they say, but

\gamma_{SR} = (1-v^2/c^2)^{-1/2}.

This is a good example of the usefulness of the Milne model in
error-checking.  If you ever draw a conclusion that says that the
Milne model isn't consistent with special relativity, then it's 100%
guaranteed that you've made a mistake, because the Milne spacetime is
*precisely* the same as the Minkowski spacetime of special relativity.

-Ted

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