The four coordinate functions x a ( τ τ ) , a = 0 , 1 , 2 , 3 defining a world line, are real number functions of a real variable τ τ 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 p on the curve at the parameter value τ τ 0 and a point on the curve a little (parameter τ τ 0 + Δ Δ τ τ ) farther away. In the limit Δ Δ τ τ → → 0 , this difference divided by Δ Δ τ τ defines a vector, the tangent vector of the world line at the point p . It is a four-dimensional vector, defined in the point p . It is associated with the normal 3-dimensional velocity of the object (but it is not the same) and therefore termed four-velocity v → → , or in components: v → → = ( v 0 , v 1 , v 2 , v 3 ) = ( d x 0 d τ τ , d x 1 d τ τ , d x 2 d τ τ , d x 3 d τ τ )