Length of ruler in expanding Universe

[johne1618]

Actually I still have a query about all this. Here is another way of putting my argument.

Light travels on a null geodesic, ds=0, so that its path obeys the relationship
$$a \ dx = dt$$
So at the present time with $a=1$ light travels 1 light-second in 1 second of cosmological time.

But at a future time with $a=2$ light travels 2 light-seconds in 1 second of cosmological time.

If a future observer is going to measure a constant speed of light then his time scale must change so that light travels the 2 light-seconds in 2 of his seconds.

Therefore the future observer's clock must run twice as fast as our present clock (which by definition runs on cosmological time).

Actually I now accept my argument is wrong after all for the reason that Chalnoth pointed out!

The null geodesic for light gives
$$a(t) \ dx = c \ dt$$
where I am retaining $c$ for clarity.

$dx$ is an interval of comoving distance and $dt$ is an interval of cosmological time.

Now an interval of proper distance $ds$ at any time $t$ is given by
$$ds=a(t) \ dx.$$
Therefore the above geodesic can be written simply as
$$ds = c \ dt$$
which is true for any cosmological time $t$.

Thus light travels at a constant velocity $c$ for all observers who simply use cosmological time.

To illustrate the situation as I now see it:

For $a=1$
$$dx = c \ dt \\<br /> ds = dx = c \ dt$$
For $a=2$
$$2 \ dx = c \ dt \\<br /> dx = c \ dt / 2 \\<br /> ds = 2 \ dx = 2 \cdot (c \ dt / 2) = c \ dt$$