View source for World line

A line in [[Space-time|space-time]] which is the space-time trajectory of a material point. Introduce a local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981501.png" />, in some domain of space-time, and let the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981502.png" /> lie on a world line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981503.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981504.png" /> is called a world point; it describes the event that at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981505.png" /> the material point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981506.png" /> has space coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981507.png" />. The concept of an event, and the related concepts of a world point and 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|geodesic hypothesis]], a space-time geodesic. The unit tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981508.png" /> to a world line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w0981509.png" /> is a four-dimensional velocity vector; in local coordinates it has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w09815010.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098150/w09815011.png" /></td> </tr></table>

See also [[Minkowski space|Minkowski space]].



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Taylor,   J.A. Wheeler,   "Space-time physics" , Freeman  (1963)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.S. Eddington,   "The mathematical theory of relativity" , Cambridge Univ. Press  (1960)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.G. Bergmann,   "Introduction to the theory of relativity" , Dover, reprint  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.F. Lawden,   "Tensor calculus and relativity" , Methuen  (1962)</TD></TR></table>