The four-velocity, also called proper velocity, is defined as the derivative of the spacetime position paravector with respect to proper time τ : u = d x d τ τ = d x 0 d τ τ + d d τ τ ( x 1 e 1 + x 2 e 2 + x 3 e 3 ) = d x 0 d τ τ [ 1 + d d x 0 ( x 1 e 1 + x 2 e 2 + x 3 e 3 ) ] .