Newtonian Cosmology and Mach's Principle: local vs. global integrability

Abstract

I discuss the transformation theory of Newtonian mechanics and gravity developed by Cartan and Heckmann and show via coordinate transformations that Heckmann's Newtonian cosmological model has a center, so that the cosmological principle cannot hold globally in that model. This viewpoint disagrees with a recent claim, and is based on a particle mechanics method never before used in this model. It is possible to confuse relativity principles with a position of relativism. Mach's principle has much to do with the latter and little to do with the former. Both general relativity and nonlinear dynamics inform us that most coordinate systems are defined locally by differential equations that are globally nonintegrable: global extensions of local coordinates usually do not exist. By emphasizing local integrability vs. global nonintegrability of equations of motion, and correspondingly local inertial frames, I short circuit Mach's philosophic objections to Newtonian dynamics. The short-circuit follows from a physico-mathematical discussion of inertial frames based on local integrability.