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.

Since the OP was talking about Minkowski spacetime, the causal structure is pretty trivial, so I didn't think such complications are necessary.