Existence of Minkowski space

Abstract Physics textbooks present Minkowski space as an almost pure mathematical construct, without any explicit restriction on a domain where it is applicable in physics. Meanwhile, its physical meaning cannot but follow the same premises as those which underlies the special relativity theory: motion of free point particles and propagation of electromagnetic waves. However, the common formalism of coordinate transformations between any two inertial frames appears too ponderous to infer the existence of Minkowski space. For this reason, the time dilation and retardation, the contraction of the length along and the spatial invariance across the direction of relative motion of two frames are presented in a coordinate-free manner. This results in the transformation between two frames in the form of relationships between the time moments and the components of the position vector of a given event, along and across the directions of the frames’ motion. The obtained transformation rules for the components of the position vector differ from the vector-like relationship known in the literature because the latter actually deals with column vectors made of Cartesian coordinates of true vectors and appears identical to the so called boost coordinate transformation, which is quite irrelevant to the question under consideration. The direct calculations prove that the transformation of true position vectors establishes an equivalence relation between column vectors made of the time moment and the position vector of a given event in each frame. This means the existence of Minkowski space, into which the set of all events is mapped. The unavoidable intricacy of that proof suggests that the extension of the full concept of Minkowski space beyond motions of point particles, capable of short-range interaction only, is hardly possible.