I Affine parameter and non-geodesic null curves

[JimWhoKnew]

TL;DR

Is there a sensible way to define an affine parameter for non-geodesic null curves?

Consider the curve (thanks to SE) in flat spacetime, given in Cartesian coordinates by$$x^μ(λ)=\left(λ , R\cos\frac{\lambda⁡}{R} , R\sin\frac{\lambda}{⁡R} ,0\right)$$where $~R~$ is a positive constant. At each point$$\dot x^\mu \dot x_\mu=0$$so it is a null curve but not a geodesic (not a straight line). It also satisfies$$\ddot x^\mu \dot x_\mu=0 \quad .$$If I got the calculation right, it turns out that for any reparametrization $~\lambda'~$ , where $~\lambda'(\lambda)~$ is an arbitrary monotonic function, $~\dot x^\mu \dot x_\mu=\ddot x^\mu \dot x_\mu=0~$ holds in this particular case (dots here w.r.t. $~\lambda'~$).

Is there a sensible way in which we can define an affine parameter for non-geodesic null curves like this, such that certain parametrizations are affine while others are not?

Edit: (We have criteria for parameters to "be affine" in the cases of timelike/spacelike curves and null geodesics. Is the non-geodesic null curve an exception?)

 

[JimWhoKnew]

For the particular example in OP, I think that the time coordinate $~t~$ of the specific reference frame can be regarded as an affine parameter. Because of the symmetry (the Euclidean length traced in each uniform interval $~\Delta t~$ is the same).