Cosmological implications of the Machian principle

Abstract

The famous idea of Ernst Mach concerning the non-absolute but relational character of particle inertia is taken up in this paper and is reinvestigated with respect to its cosmological implications. From Thirring’s general relativistic study of the old Newtonian problem of the relativity of rotations in different reference systems, it appears that the equivalence principle with respect to rotating reference systems, if at all, can only be extended to the system of the whole universe, if the mass of the universe scales with the effective radius or extent of the universe. A reanalysis of Thirring’s derivations still reveals this astonishing result, and thus the general question must be posed: how serious this result has to be taken with respect to cosmological implications. As we will show, the equivalence principle is, in fact, fulfilled by a universe with vanishing curvature, i.e. with a curvature parameter$k = 0$, which just has the critical density $\rho _{{crit}} = {3H^{2} } \mathord{\left/ {\vphantom {{3H^{2} } {8\pi G}}} \right. \kern-\nulldelimiterspace} {8\pi G}$, where $H$ is the Hubble constant. It turns out, however, that this principle can only permanently be fulfilled in an evolving cosmos, if the cosmic mass density, different from its conventional behaviour, varies with the reciprocal of the squared cosmic scale. This, in fact, would automatically be realized, if the mass of each cosmic particle scales with the scale of the universe. The latter fact, on one hand, is a field-theoretical request from a general relativistic field theory which fulfills H. Weyl’s requirement of a conformal scale invariance. On the other hand, it can perhaps also be concluded on purely physical grounds, when taking into account that as source of the cosmic metrics only an effective mass density can be taken. This mass density represents the bare mass density reduced by its mass equivalent of gravitational self-binding energy. Some interesting cosmological conclusions connected with this fact are pointed out in this paper.