On Discrete Models of Space-Time

Abstract

Analyzing the Einstein radiolocation method we come to the conclusion that results of any measurement of space-time coordinates should be expressed in terms of rational numbers. We show that this property is Lorentz invariant and may be used in the construction of discrete models of space-time different from the models of the lattice type constructed in the process of discretization of continuous models. § 1. Discrete models of space-time without elementary length There is a common belief 1 ) that discrete models of space-time must suffer from at least three chronic diseases: i) They introduce the experimentally not found elementary length. ii) They violate experimentally verified relativistic invariance. iii) They legate their mathematics from the continuous models. In the present paper we shall show that this belief is not justified and in order to present our approach we should like to start from the very beginning, i.e., from the analysis of general properties of the experimental methods used to measure space­ time coordinates. The notion of space-time is a physical concept and if we want to be able to understand all its features we should take into account and understand the measure­ ment methods of any quantity used in the mathematical description of space-time. In particular, we must precisely define the method of measuring space-time coordinates the results of which, due to the principle of relativity, have a subjective character for each observer. We consider Einstein radiolocation method 2 ) as a starting point of our discussion. For the completeness of the paper we shall remind here its basic idea in spite of the fact that it is well-known. In this method in any inertial reference frame stiffly connected to a given observer we measure two quantities: