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
<br /> a \ dx = dt<br />
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
<br /> a(t) \ dx = c \ dt<br />
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
<br /> ds=a(t) \ dx.<br />
Therefore the above geodesic can be written simply as
<br /> ds = c \ dt<br />
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
<br /> dx = c \ dt \\<br /> ds = dx = c \ dt<br />
For a=2
<br /> 2 \ dx = c \ dt \\<br /> dx = c \ dt / 2 \\<br /> ds = 2 \ dx = 2 \cdot (c \ dt / 2) = c \ dt<br />