Existence of Minkowski space

TL;DR

This paper argues that Minkowski space's existence can be rigorously established through coordinate-free methods, challenging traditional reliance on coordinate transformations and suggesting possible need for its generalization or abandonment.

Contribution

It provides a formal, coordinate-free proof of Minkowski space's existence and questions its fundamental necessity in physical theory.

Findings

Coordinate-free transformation confirms Minkowski space exists.

Traditional coordinate-based methods are cumbersome and less fundamental.

Suggests potential for generalizing or abandoning Minkowski space concept.

Abstract

Minkowski space serves as a framework for the theoretical constructions that deal with manifestations of relativistic effects in physical phenomena. But neither Minkowski himself nor the subsequent developers of the relativity theory have provided a reasonable rationale for this mathematical construct. In physics, such a rationale should show lower-level statements that determine where the proposed mathematical structure is applicable and yield formal premises for proving its existence. The above failure has apparently been due to the features of the adopted formalism based on the unjustifiably exclusive use of coordinates in the theoretical analysis of physical phenomena, which ignores the necessity of having physical grounds for mathematical concepts. In particular, the use of a coordinate transformation between two inertial reference frames makes the consideration so cumbersome that it appears useless for solving the fundamental problems of physical theory, including the question of whether Minkowski space exists. In contrast, a straightforward calculation proves that the transformation of the time and the position vector of a physical event between two physical spaces establishes an equivalence relation between pairs made of these variables. This means the existence of Minkowski space and shows that the premises for its proof are the same as for the coordinate-free derivation of basic effects of the special relativity theory: the use of Einsteinian time variable and motions of particles able to interact with each other and electromagnetic field over a short spatial range only. The high degeneracy of free motions of point particles, together with the intricacy of the above mentioned calculation, suggests that a further generalization of Minkowski space is beyond belief, so that the modification or even the abandonment of the concept of spacetime seems quite natural.