# 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). |

**How to Cite This Entry:**

World line.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=World_line&oldid=40155