# Speed of light in an expanding universe

The path of a light ray satisfies, in terms of the scale factor:

$c_m(t)=\frac{dx}{dt}=\frac{c}{a(t)}$

Is $c_m(t)$ the locally measured speed of light? Why not? (I assume $c_m(t)$ cannot the speed of light, as there is no evidence that it has changed in the past)

The FLRW metric for a flat universe is beguilingly simple:

$$ds^2 = -c^2dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$

and using the fact that light follows null geodesics, i.e. $ds = 0$, allows us to immediately write the equation for the speed of light (assuming it propagates in the $x$ direction):

$$\frac{dx}{dt} = \frac{c}{a(t)}$$

But there is a pitfall waiting to trap the unwary. The spatial coordinates $x$, $y$ and $z$ are not the distances measured by observers. Instead these are comoving cordinates i.e. coordinates in which all comoving observers are stationary and remain at a constant distance from each other.

The distance measured by observers using their rulers, i.e. the proper distance $d\ell$, can be obtained by taking a spatial hyspersurface and using the resulting metric:

$$d\ell^2 = a^2(t)(dx^2 + dy^2 + dz^2)$$

so in this case we get:

$$d\ell = a(t)dx$$

and the speed of light is:

$$v_\text{light} = \frac{d\ell}{dt} = a(t)\frac{dx}{dt} = c$$