Pregeodesic

A pregeodesic is a curve whose tangent obeys ${\nabla }_{U}U=\kappa U$ for some scalar $\kappa$. A change of parameter makes it a geodesic with an affine parameter. A varying speed along a geodesic changes its parameter, not its image.

A null pregeodesic has a null tangent and can be parametrized as a null geodesic. In a two-dimensional Lorentzian metric, the acceleration of a regular null curve is orthogonal to its tangent and therefore proportional to it.