# Null geodesics of the FRW metric

## Main Question or Discussion Point

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?

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PeterDonis
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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 in to the geodesic equation, but someone else might.

PAllen