Mach's principle from the relativistic constraint equations

The history and meaning of Mach's principle are reviewed. In rotating and expanding nearly spherical distributions, the potential | (r, t) | that governs the rotations of the local inertial frames is instantaneously related to the angular momentum distribution |J (r, t) | by | (r, t) =2Gc^-2 (WJr^-3+^ᵣ Wr^-3 J/ r dr) |⁠, 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.