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World line

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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://www.encyclopediaofmath.org/index.php?title=World_line&oldid=40155
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article