Quote by robphy
While geodesics are nicelybehaved curves, there are physically important nongeodesic curves....like worldlines of piecewiseinertial particles and worldlines of accelerated particles.
In the proofs of some theorems involving causal structure, there are "causal curves" whose tangents can be timelike or null.

A causal curve might play a role in a proof. So might any curve. But no physical entity can follow curve that is part timelike and part null.