World line

A line in space-time that is the space-time trajectory of a material point. Introduce a local coordinate system $(t,x,y,z)$ in some domain of space-time, and let the point $P(t,x,y,z)$ lie on a world line $\gamma$. We call $P$ a world point; it describes the event that at time $t$, the material point $P$ has space coordinates $(x,y,z)$. The concept of an event, and the related concepts of a world point and of a world line are among the basic notions of relativity theory, adding to the concept of a material point borrowed from classical mechanics. Usually, one considers smooth (or piecewise-smooth) world lines. The world line of a material point with positive rest mass is a time-like curve. The world line of a material point with zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line. An arbitrary point of space-time is considered as a world point, that is, a (potential) event, and each time-like or isotropic line as the (possible) world line of some material point. The world line of a material point not under the influence of non-gravitational fields is, according to the geodesic hypothesis, a space-time geodesic. The unit tangent vector $\dot{\gamma}$ to a world line $\gamma$ is a $4$-dimensional velocity vector; in local coordinates, it has the form $$\left( \frac{1}{\sqrt{1 - \dfrac{\mathbf{v} \cdot \mathbf{v}}{c^{2}}}};\frac{\dfrac{\mathbf{v}}{c}}{\sqrt{1 - \dfrac{\mathbf{v} \cdot \mathbf{v}}{c^{2}}}} \right),$$ where $$\mathbf{v} \stackrel{\text{df}}{=} \left( \frac{\mathrm{d}{x}}{\mathrm{d}{t}},\frac{\mathrm{d}{y}}{\mathrm{d}{t}},\frac{\mathrm{d}{z}}{\mathrm{d}{t}} \right).$$

See also Minkowski space.

References

[a1]	E.F. Taylor, J.A. Wheeler, “Space-time physics”, Freeman (1963).
[a2]	A.S. Eddington, “The mathematical theory of relativity”, Cambridge Univ. Press (1960).
[a3]	P.G. Bergmann, “Introduction to the theory of relativity”, Dover, reprint (1976).
[a4]	D.F. Lawden, “Tensor calculus and relativity”, Methuen (1962).