Hello all, In Carroll's on page 109 it is pointed out that for derivation of the geodesic equation, 3.44, a "hidden" assumption is that we have used an affine parameter. Some few lines below we see that "any other parametrization" could be used, called alpha, but in that case the general form of geodesic equation will be 3.58 and not 3.44. However, when plugging 3.59 into 3.58 and rearranging I can only recover the geodesic equation, 3.44, provided that alpha is itself a linear function of λ, and not "some general parameter alpha." So unlike what is said below 3.59, we cannot "always" find an affine parameter λ. What am I missing here or getting wrong? Could someone please help? Also, related to this confusion is what we call "a null geodesic." To make it clear for myself I need to see an example of a null path which is NOT a geodesic. Can someone please introduce or describe such a null path? Thanks a lot in advance.