Mach's Principle in General Relativity

Abstract

Mach's Principle is taken as a criterion for selecting cosmological solutions of the Einstein field equations, in which, in a well-defined manner, the metric arises from material sources alone. In such model universes inertial forces are due to the gravitational interaction of matter, and there is a relativity of accelerated motion. The problem of stating such a selection rule in general relativity divides into two parts: an analysis of the relation of the metric to the Riemann curvature, and of the curvature to the stress tensor, with associated Machian criteria. From the first criterion we show that Mach's Principle is not satisfied in Minkowski space. It seems that asymptotically fiat space-times are also non-Machian, as required by the Machian philosophy. The second criterion rules out vacuum solutions and spatially homogeneous cosmological models containing perfect fluids in which there is anisotropic expansion or rotation. Mach's Principle is found to be satisfied in Robertson–Walker models and in a simple class of inhomogeneous solutions. These results lead us to suggest that Mach's Principle may play a role in explaining the observed gross features of the Universe.