![]() It turns out, although I don't know the details of a proof, that there is a unique affine parameter for any geodesic, up to transformations of the form $t \to at b$. Another way is to say that the acceleration is perpendicular to the velocity given an affine parameter, as Ron did. You have to use an affine parameter.) Another way is to say that iff the parametrization is affine, parallel transport preserves the tangent vector, as Wikipedia does. (Note: the geodesic equation does not work for just any arbitrary parametrization of a geodesic. In particular, one way to define an affine parameter is that it satisfies the geodesic equation. However, it's possible to pick a way to parametrize the null geodesic in a way that is "sensible" in the same way that proper time is "sensible" for a timelike geodesic. In principle, again, it can be any monotonic function that maps points on the geodesic to unique values of the parameter. ![]() So you have to pick some other parametrization. With null geodesics, you don't have the proper time as an option because the proper time mapping assigns the same value to all points on the geodesic. But for timelike geodesics, you almost always use the proper time because it's a nice, sensible physical quantity that also happens to work as a parameter. Actually, you can parametrize any geodesic (heck, even any curve) in any way you want all you need is a monotonic function that maps points on the geodesic to unique values of the parameter. So, I would like to put some examples in front of you before discussing the zero vector in depth. Many people have misconceptions about this zero vector or null vector. ![]() Which you call zero vector or null vector. If you forget about the affine-ness for a moment: you can parametrize a null geodesic in any way you want. What is the Zero Vector (Null Vector) by Jidan / SeptemIn this tutorial, we will discuss an important topic of vectors. ![]()
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