Editing World line (section)

===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.