A photon's geodesic equation is defined by re-parameterizing the geodesic equation to some parameter other than proper time. This is done because $ds=0$ for the photon.
Again if we use affine parameter (say $\lambda$) for the path of the null geodesic, we get that the new parameter used is some linear function of proper time.
My question is: If $ds=0$, if the new parameter is linear with proper time, $d\lambda=0$. Then how can this parameter be used to write the null geodesic.
