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Feb24-12, 05:08 PM
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.
Since the OP was talking about Minkowski space-time, the causal structure is pretty trivial, so I didn't think such complications are necessary.