- #1

- 560

- 2

## Main Question or Discussion Point

We have a general spacetime interval ##ds^2 = g_{\mu \nu} dx^\mu dx^\nu##.

One way to define an affine parameter is to define it to be any parameter ##u## which is related to the path length ##s## by ##u = as + b## for two constants ##a,b##. One can show that for the tangent vector ##u^\alpha = \frac{dx^\alpha}{du}## to a geodesic this implies a geodesic equation

$$\nabla_{u} u^\alpha = 0$$

in comparison to the more general

$$\nabla_{u} u^\alpha = f(x) u^\alpha.$$

However, if the geodesic is a null-geodesic, we can not use parameters ##u## which is related to the pathlength by ##u=as + b## since ##s = 0## along any null curve. However, it is still claimed that one can find parameters for which

$$\nabla_{u} u^\alpha = 0$$

holds also for null-geodesics. This thus defines a more general class of "affine parameters".

Now, my question is -- how do we know that there exists such parameters also for null-geodesics?

One way to define an affine parameter is to define it to be any parameter ##u## which is related to the path length ##s## by ##u = as + b## for two constants ##a,b##. One can show that for the tangent vector ##u^\alpha = \frac{dx^\alpha}{du}## to a geodesic this implies a geodesic equation

$$\nabla_{u} u^\alpha = 0$$

in comparison to the more general

$$\nabla_{u} u^\alpha = f(x) u^\alpha.$$

However, if the geodesic is a null-geodesic, we can not use parameters ##u## which is related to the pathlength by ##u=as + b## since ##s = 0## along any null curve. However, it is still claimed that one can find parameters for which

$$\nabla_{u} u^\alpha = 0$$

holds also for null-geodesics. This thus defines a more general class of "affine parameters".

Now, my question is -- how do we know that there exists such parameters also for null-geodesics?