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I have recently become interested in special relativity. I would like to stress that I am not a physicist, but just a curious person. I read that a piecewise twice continuously differentiable curve $\gamma$ in the Minkowski spacetime can be used to represent any massive particle, and every vector tangent to $\gamma$ is timelike. Is this an assumption, or is it possible to prove that massive particles follow timelike worldlines only?

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The tangent vector to the curve $\gamma$ is interpreted as the four-velocity $U$ $$\frac{dX}{d\tau}=U $$ where I denoted the four-position as $X$ and proper time as $\tau$.

Directly from axioms of special relativity we can see that velocity of any massive particle is always less than the speed of light, from this we directly get that every vector tangent to some world-line of a massive particle is always time-like.

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