Publications

Abstract

The history and meaning of Mach's principle are reviewed. In rotating and expanding nearly spherical distributions, the potential $\omega(r, t)$ that governs the rotations of the local inertial frames is instantaneously related to the angular momentum distribution $J(\leq r,t)$ by $\omega(r,t)=2Gc^{-2}(WJr^{-3}+\int\limits_{r}^{\infty}Wr^{-3}\frac{\partial J}/{\partial r}{\mathrm d}r)$, where $W(r, t)$ is a weight function related to the unperturbed metric, which is unity for flat space. In closed universes where r cannot reach infinity, only relative rotations are meaningful, but the relative angular velocities of inertial frames at different points and of inertial frames relative to matter can be found in terms of the angular momentum and the distribution and motion of the matter. A general non-linear discussion of conservation laws yields a general proof that the total angular momentum (and the total of any conserved quantity the current of which derives from a superpotential) must be zero in any closed universe. By looking at conservation laws for angular momentum and quasi-linear momentum in perturbed Robertson-Walker universes, it is shown that Mach's principle follows from the constraint equations of General Relativity, provided that the universe is closed. Closed spaces with little mass are not almost flat, but exist for only short times and are always small; thus the conundrum of inertia in almost empty nearly flat spaces disappears if space is closed.