The **cosmological
redshift**
is something different, although we are often sloppy and refer to it in
the same terms of the doppler redshift. The cosmological redshift is
actually
due to the **expansion of space**
itself.

How do we relate the cosmological redshift to the expansion of the Universe? Start with the R-W metric again

and watch what happens to a light ray
moving through
space.
In this case, **ds ^{2}=0** (

Again orient the coordinate system so
that
theta=phi=0,
then integrate along the path length, from the time of emission (t_{e})
to now (t_{0}). Look at **two wavecrests**
in the light ray, separated in time by .

*(sheesh!)*

Okay, now we can pull R(t) out of the
integral and
treat
it as a constant (** why?**).
Then
we get

Now since wavelength is equal to c we have

Or, by our definition of
redshift:

We get:

So redshift is related to the
expansion factor of
the
Universe. *If we measure a redshift of z=2, the Universe is 3x
bigger
now than it was when that photon was emitted.
*

Also, this gives rise to an expression of cosmological time
dilation:

(1+z) = dt(0) / dt(e)

The event as observed takes longer
(is stretched) than as it happens in the rest frame.

(from Goldhaber et al 1996)