I Affine parameter and non-geodesic null curves

JimWhoKnew
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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?)
 
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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).
 
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