Mach’s Principle in general relativity

Raine, D. J.

Mach’s Principle is used to establish a criterion for selecting cosmological solutions of the Einstein field equations in which the metric arises exclusively from physically real material sources. The problem of stating such a selection rule is divided into two parts: analysis of the relation of the metric to the Riemann curvature and analysis of the relation of the curvature to the stress tensor, with associated Machian criteria. From the first analysis, it is shown that Mach’s Principle is not satisfied in either Minkowski space or in asymptotically flat space-times. The second analysis is shown to rule 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 only in Robertson-Walker models and in a simple class of inhomogeneous solutions. The results suggest that Mach’s Principle may play a role in explaining the observed gross features of the universe.


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