Steven Weinberg derived a corollary to Birkhoff's theorem stating that inertial frames cannot rotate or accelerate relative to the rest of the universe. For this he assumes a relatively small vacuum spherical cavity in a spherically symmetric universe, just so that Birkhoff's theorem can be applied. The bubble is assumed to be small enough so that the surrounding metric is not disturbed. He provides mathematical arguments that inside the bubble the metric is flat, which proves the corollary. It does not matter if the entire universe is a massive thin spherical shell around the vacuum bubble, or if it is stretched out uniformly and homogeneously even beyond the cosmological event horizon; by Birkhoff's theorem, this does not affect the metric.
To me, this sounds like a proof of Mach0 in Bondi's list of 11 versions of Mach's principle. Bondi defines Mach0 as follows: "The universe, as represented by the average motion of distant galaxies does not appear to rotate relative to local inertial frames." He further writes: "Mach0 is an experimental observation and not a principle". I presume this means that there is no model-based proof.
Amazingly enough, more than a century after general relativity, people still seem to be puzzled about this. There was an article in American Scientist, "The forgotten Mystery of Inertia", that claims that after all these years we still don't understand how gyros (and by extension, Foucault pendulums and satellite orbits) work in the sense of how they orientate themselves to the universe.
My question is: Has Mach's principle (the Mach0 version) really not been solved? Does Weinberg's corollary not provide a sufficient proof? True, he assumes a vacuum bubble small enough that the metric of the universe is not perturbed, so the isotropy and homogeneity is more like that off a Swiss cheese than a dustbowl of galaxies but that does not strike me as an essential difference. Please let me know if I missed something.
The corollary that I am talking about is in Weinberg's book "Gravitation and Cosmology" in the conclusion of section 11.7. He elaborates in section 15.1 page 474: "The only reference frames in which the universe appears spherically symmetric, so that Birkhoff's theorem applies, are the frames at the center of our cavity, which do not rotate with respect to the exapnding cloud of 'typical' galaxies. The inertial frames are reference frames that move at constant velocity, and without rotation, relative to the frames in which the universe appears spherically symmetric.". For those who don't have the book, a PDF of Weinberg's book can be found online.
American Scientist, "The Forgotten Mystery of Inertia"
Steven Weinberg, "Gravitation and Cosmology"
Herman Bondi and Joseph Samuel, "The Lense–Thirring Effect and Mach’s Principle"
