Measuring proper distance using a light beam I wonder if someone can help me with the following problem.
I send a light beam to a distant galaxy which then bounces back to me.
I measure the travel time of the lightbeam using say a light clock of fixed size which I assume measures conformal time $\tau=\int dt/a(t)$.
Is the proper distance of the galaxy at the moment that I receive the lightbeam given simply by:
$$L = \frac{c\ \tau}{2}?$$
 A: In Minkowski spacetime the one way light travel time to a galaxy at proper distance $\chi$ is just:
$$ t = \frac{\chi}{c} $$
so:
$$ \chi = ct $$
As you say. However in an FRW universe the travel time is given by a different equation so the proper distance is not simply $ct$. Let's assume all motion is in the $x$ direction, so the metric simplifies to:
$$ c^2ds^2 = -c^2dt^2 + a^2(t)dx^2 \tag{1} $$
We'll take our position to be $(0, 0)$ and the galaxy to be at $(0, \chi)$, and we'll adopt the usual convention that $a = 1$ at the current time. To get the proper distance we integrate $ds$, and since $dt = 0$ and $a = 1$ the proper distance is just:
$$ \Delta s = \int_0^\chi dx = \chi $$
Now let's calculate the time it takes the light beam to get from the galaxy back to us (i.e. one half of the journey). Light travels on a null geodesic so $ds = 0$ and putting this into the metric (1) and rearranging we get:
$$ \frac{dx}{dt} = \frac{c}{a(t)} $$
If the universe is static $a(t) = 1$ for all $t$, and we get $x = ct$ so you would be correct that the proper distance is equal to half the total travel time times $c$. But then with $a = 1$ we just have Minkowski spacetime so that's hardly surprising. To calculate the trajectory of the light we need to assume a form for $a(t)$ so let's make the approximation:
$$ a(t) = 1 + Ht $$
where $H$ is the current value of the Hubble constant. Then we get:
$$ \frac{dx}{dt} = \frac{c}{1 + Ht} $$
and this integrates to give us:
$$ \chi = \frac{c}{H} \log(H\tau + 1) $$
or rearranging this to get the travel time:
$$ \tau = \frac{exp(\frac{\chi H}{c}) -1}{H} \tag{2} $$
So the proper distance $\chi$ is not simply the travel time times $c$.
Just to reassure ourselves that we get the correct result in the limit of $H \rightarrow 0$, i.e. Minkowski spacetime, note that for small $H$:
$$ exp(\frac{\chi H}{c}) \approx 1 + \frac{\chi H}{c} $$
Put this back into equation (2) and we get:
$$ \tau \approx \frac{\chi}{c} $$
which is where we came in.
