 Quote by robphy
While geodesics are nicely-behaved curves, there are physically important non-geodesic curves....like worldlines of piecewise-inertial 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.
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Since the OP was talking about Minkowski space-time, the causal structure is pretty trivial, so I didn't think such complications are necessary.