This expression can be brought to a more compact form by defining the ordinary velocity as v = d d x 0 ( x 1 e 1 + x 2 e 2 + x 3 e 3 ) , and recalling the definition of the gamma factor : γ γ ( v ) = 1 1 − − | v | 2 c 2 , so that the proper velocity is more compactly: u = γ γ ( v ) ( 1 + v ) .