Editing World line (section)

==World lines as a method of describing events==
[[Image:Brane-wlwswv.png|upright=1.2|thumb|World line, worldsheet, and world volume, as they are derived from [[elementary particle|particles]], [[string theory|strings]], and [[Membrane (M-theory)|brane]]s.]]

A one-dimensional ''line'' or ''curve'' can be represented by the coordinates as a function of one parameter. Each value of the parameter corresponds to a point in spacetime and varying the parameter traces out a line. So in mathematical terms a curve is defined by four coordinate functions <math>x^a(\tau),\; a=0,1,2,3</math> (where <math>x^{0}</math> usually denotes the time coordinate) depending on one parameter <math>\tau</math>. A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant.

Sometimes, the term '''world line''' is used informally for ''any'' curve in spacetime. This terminology causes confusions. More properly, a '''world line''' is a curve in spacetime that traces out the ''(time) history'' of a particle, observer or small object. One usually uses the [[proper time]] of an object or an observer as the curve parameter <math>\tau</math> along the world line.

===Trivial examples of spacetime curves===
[[Image:Worldlines1.jpg|thumb|upright=1.2|Three different world lines representing travel at different constant four-velocities. ''t'' is time and ''x'' distance.]]
A curve that consists of a horizontal line segment (a line at constant coordinate time), may represent a rod in spacetime and would not be a world line in the proper sense. The parameter simply traces the length of the rod.

A line at constant space coordinate (a vertical line using the convention adopted above) may represent a particle at rest (or a stationary observer). A tilted line represents a particle with a constant coordinate speed (constant change in space coordinate with increasing time coordinate). The more the line is tilted from the vertical, the larger the speed.

Two world lines that start out separately and then intersect, signify a ''collision'' or "encounter". Two world lines starting at the same event in spacetime, each following its own path afterwards, may represent e.g. the decay of a particle into two others or the emission of one particle by another.

World lines of a particle and an observer may be interconnected with the world line of a photon (the path of light) and form a diagram depicting the emission of a photon by a particle that is subsequently observed by the observer (or absorbed by another particle).

===Tangent vector to a world line: four-velocity===
The four coordinate functions <math>x^a(\tau),\; a = 0, 1, 2, 3</math>
defining a world line, are real number functions of a real variable <math>\tau</math> and can simply be differentiated by the usual calculus. Without the existence of a metric (this is important to realize) one can imagine the difference between a point <math>p</math> on the curve at the parameter value <math>\tau_0</math> and a point on the curve a little (parameter <math>\tau_0 + \Delta\tau</math>) farther away. In the limit <math>\Delta\tau \to 0</math>, this difference divided by <math>\Delta\tau</math> defines a vector, the '''tangent vector''' of the world line at the point <math>p</math>. It is a four-dimensional vector, defined in the point <math>p</math>. It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore termed [[four-velocity]] <math>\vec{v}</math>, or in components:
<math display="block">\vec{v} = \left(v^0, v^1, v^2, v^3\right) = \left( \frac{dx^0}{d\tau}\;,\frac{dx^1}{d\tau}\;, \frac{dx^2}{d\tau}\;, \frac{dx^3}{d\tau} \right)</math>

such that the derivatives are taken at the point <math>p</math>, so at <math>\tau = \tau_0</math>.

All curves through point p have a tangent vector, not only world lines. The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar. Therefore, all tangent vectors for a point p span a [[linear space]], termed the [[tangent space]] at point p. For example, taking a 2-dimensional space, like the (curved) surface of the Earth, its tangent space at a specific point would be the flat approximation of the curved space.