Null geodesics of the FRW metric

[center o bass]

When working with light-propagation in the FRW metric
$$ds^2 = - dt^2 + a^2 ( d\chi^2 + S_k(\chi) d\Omega^2)$$
most texts just set $$ds^2 = 0$$ and obtain the equation
$$\frac{d\chi}{dt} = - \frac{1}{a}$$
for a light-ray moving from the emitter to the observer.

Question1: Do we not strictly speaking also have to check that the above equation actually specifies a geodesic?

Setting $ds^2 = 0$ does not automatically guarantee that the obtained relation specifies a geodesic, right?

Question2: Is there a quick way to verify that the above curve indeed is a null-geodesic?

 

[PeterDonis]

Do we not strictly speaking also have to check that the above equation actually specifies a geodesic?

Yes.

Is there a quick way to verify that the above curve indeed is a null-geodesic?

I don't know of any quicker way than finding an affine parametrization of the curve and plugging into the geodesic equation, but someone else might.

 

[PAllen]

Well, the given metric displays spherical symmetry. Then, for a radial null path, there is one solution. Then, if it is not a geodesic, what could choose a direction?

I would thus say, it is an interesting consistency check (which I have done for Kruskal coordinates) to verify satisfaction of the geodesic equation. However, for purposes of doing the least work for a valid conclusion, it is superfluous (in this particular case).