Surely proper time expands like proper space?

Question

The FRW metric at the origin $r=0$ is simply given by: $$ds^2 = -dt^2 + a^2(t)\ dr^2$$ Setting $dt=0$ gives us an element of proper distance $ds$ given by: $$ds = a(t)\ dr$$ Thus we get the well known result that space expands with the scale factor.

Setting $dr=0$ gives us the relationship between an element of proper time $d\tau$ and an element of co-ordinate time $dt$ (using $d\tau^2 = - ds^2$): $$d\tau = dt$$

Thus we get the well known result that for a co-moving observer proper time is the same as co-ordinate time.

However there is one more relation one can derive. We can set $ds=0$. We then obtain a relationship between an element of co-ordinate time $dt$ and an element of co-ordinate separation $dr$ : $$dt = a(t)\ dr$$

Surely this relationship implies that elements of co-ordinate time $dt$ (and thus proper time $d\tau$ for a co-moving observer) expand with the scale factor in the same way that elements of proper distance expand with the scale factor?

Answer 1

As Antonio says in his comment, by setting $ds = 0$ all you are doing is writing the equation of a null geodesic, i.e. the trajectory of a ray of light. And you deduce that (again as Antonio has pointed out):

$$ \frac{dr}{dt}=\frac{1}{a(t)} $$

At first glance this looks as if the speed of light falls as the universe expands, but you need to remember that the $r$ coordinate is comoving distance not proper distance. The speed of light is indeed falling in comoving coordinates, but the speed you and I measure with our rulers and clocks is the proper distance per second not the comoving distance per second, and that stays constant at $c$.

Response to comment: if we call the change in proper distance $dr_p$ then we have:

$$ dr_p = a(t) dr $$

and substituting this in the above equation for the speed of light we get:

$$ \frac{\tfrac{1}{a(t)}dr_p}{dt}=\frac{1}{a(t)} $$

or:

$$ \frac{dr_p}{dt}=1 $$

as we expect. We don't need $dt$ to be changing to keep the speed of light at $c$.