Clock and Rods—Or Something More Fundamental?

This paper is essentially a speculation on the realization of Mach's Principle, and we came to the details of the present analysis via the formulation of two questions: (a) Can a globally inertial space &; time be associated with a non-trivial global matter distribution? (b) If so, what are the general properties of such a global distribution? These questions are addressed within the context of an extremely simple model universe consisting of particles possessing only the property of enumerability existing in a formless continuum.Since there are no pre-specified ideas of clocks and rods in this model universe, we are forced into two fundamental considerations, these being: What invariant meanings can be given to the concepts of spatial displacement and elapsed time in this model universe? Briefly, these questions are answered as follows: the spatial displacement of a particle is defined in terms of its changed relationship with the particle ensemble as a whole—this is similar to the man walking down a street who can estimate the length of his walk by reference to his changed view of the street. Once the concept of invariant spatial displacement is established, a corresponding concept of elapsed time then emerges in a natural way as ‘process’ within the system. Thus, unlike for example, general relativity, which can be considered as a theory describing the behaviour of specified clocks and rods in the presence of matter, the present analysis can be considered as a rudimentary—but fundamental—theory of what underlies the concepts of clocks and rods in a material universe. In answer to the original two questions, this theory tells us that a globally inertial space &; time can be associated with a non-trivial global matter distribution, and that this distribution is necessarily fractal with D = 2. This latter result is compared with the results of modern surveys of galaxy distributions which find that such distributions are quasi-fractal with D ? 2 on the small-to-medium scales, with the situation on the medium-to-large scales being a topic of considerable debate. Accordingly, and bearing in mind the extreme simplicity of the model considered, the observational evidence is consistent with the interpretation that the analysed point-of-view captures the cosmic reality to a good first-order approximation. We consider the implications of these results.