= World line
{wiki}
= Worldline
{synonym}
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 accumulates
$$
d\tau=dt\sqrt{1-\frac{|\mathbf v(t)|^2}{c^2}},
\qquad
\Delta\tau=\int_{t_0}^{t_1}\sqrt{1-\frac{|\mathbf v(t)|^2}{c^2}}\,dt.
$$
The <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 $4\sqrt{1-0.75^2}\,\mathrm s=\sqrt7\,\mathrm s$. If the observer changes <velocity> along the path, use the <integral> rather than assigning the straight-segment formula to the whole path.
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