Critical notes of Special Relativity: four-velocity

In this paper, we shall consider - from a critical point of view - the definition of four-velocity. We consider a very precise definition of world line in space time. We emphasize that the space-time of Special Relativity must be considered as a pseudo-Riemaniann manifold, that is, a (topological) space endowed with a chart-atlas and a Riemann metric. Nevertheless, we maintain the explanation at a very elementary level, by using essential basic knowledge coming from Mathematical Analysis. Events and world-lines are elements or subsets of the space-time, therefore we cannot realize them as vectors, or subsets, of any vector space without the adoption of a reference frame. The importance of a reference frame is very well underlined and used explicitly. The definition of reference frame is explicitly emphasized. Moreover, we identify reference frames, observers and charts of the assigned atlas. In other terms, from this point of view, Differential Geometry approach asserts that we cannot interpret or visualize or study the elements (or subsets) of the space-time without the intervention of some observer. As usual, the observers, i.e. reference frames, of our differentiable manifold are smoothly correlated. In some sense, space-time is unequivocally existing but not analyzable without the intervention of some observer. World-lines are defined as smooth curves in space-time, such that, for any reference frame, they can be obtained as the graphics of convenient functions defined - by using the pre-chosen reference frame - on interval of the real line taking values on the Euclidean 3-space. We consider a very elementary example and show how to define world-lines of the space-time by using a reverse approach, starting from one fixed reference frame, hence going from the Minkovskian real (1,3)-space to the Minkovsky space time M4. Then, we explicitly define and consider the natural time parametrization of a world-line induced by a reference frame. So that we can define explicitly the natural time parameter of a particle with respect to a reference frame. Different versions of the same concept will be considered. Then, we consider the analytical definition of proper time of a moving point in the space time, starting from the natural parameter induced by a reference frame. In contrast with the usual exposition and explanation of proper time, we explicitly define the domain of the proper time and it's corresponding clear mathematical definition. We also consider the proper-time differential form, field of linear forms attached with the natural parametrization of a world-line, explicitly noting the independence by reference frame of such differential form. Finally, we define the four velocity of a particle in an unambiguous way and explicitly provide various expressions of it.