World line

A world line is an object's history as a curve of spacetime events. In an inertial frame, a particle at rest has constant spatial coordinates, so its world line is vertical in a spacetime diagram with $ct$ upward. A particle at constant velocity $v$ along one axis has $x=x_0+vt$, a straight sloping world line. The coordinate representation changes with the inertial frame, while the physical history is the same.

A massive observer follows a future-directed timelike curve. Along a differentiable world line with velocity $\mathbf{v}(t)$, its clock accumulatesThe Minkowski metric therefore measures clock time along the curve, not just the difference between two coordinate readings. In the straight $O$-to-$A$ example, $v=0.75c$ for $4\mathrm{s}$, giving $4\sqrt{1-0.75^2}\mathrm{s}=\sqrt{7}\mathrm{s}$. If the observer changes velocity along the path, use the integral rather than assigning the straight-segment formula to the whole path.

A timelike curve has a nonzero tangent vector with negative Minkowski norm in metric signature $(-,+,+,+)$ at every regular point. A physical massive observer's world line is also future-directed. Parametrizing it by coordinate time gives $-c^2+|\mathbf{v}{|}^2<0$, or $|\mathbf{v}|<c$: the local tangent condition is exactly the subluminal speed condition. It is stronger than checking only that two endpoints have timelike separation.