A realization of Mach's Principle: Clocks and rods - or something more fundamental?

Abstract

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 fundamental considerations about the nature of spatial and temporal measurement. In answer to the original two questions, the analysis 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 (typically, Joyce, Montuori, Labini astro-ph/9901290, ApJ Lett 1999 and other references in this paper), which find that such distributions are quasi-fractal with D approx = 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.