Leibniz–Mach Foundations for GR and Fundamental Physics

Consider the configuration space Q for some physical system, and a continuous group of transformations G whose action on the configurations is declared to be physically irrelevant. G is to be implemented indirectly by adjoining 1 auxiliary g per independent generator to Q, by writing the action for the system in an arbitrary G-frame, and then passing to the quotient Q/G thanks to the constraints encoded by variation w.r.t the g’s. I show that this arbitrary G-frame principle supercedes (and indeed leads to a derivation of) the Barbour–Bertotti best matching principle. I furthermore consider absolute external time to be meaningless for the universe as a whole. For various choices of Q and G, these Leibniz–Mach considerations lead to Barbour–Bertotti’s proposed absolute-structure-free replacement of Newtonian Mechanics, to Gauge Theory, and to the 3-space approach (TSA) formulation of General Relativity (GR). For the TSA formulation of GR with matter fields, I clarify how the Special Relativity postulates emerge, discuss whether the Principle of Equivalence emerges, and study which additional simplicity postulates are required. I further develop my explanation of how a full enough set of fundamental matter fields to describe nature can be accommodated in the TSA formulation, and further compare the TSA with the ‘split spacetime formulation’ of Kuchař. I explain the emergence of broken and unbroken Gauge Theories as a consequence of the Principle of Equivalence. Whereas for GR one would usually quotient out 3-diffeomorphisms, I also consider as further examples of the arbitrary G-frame principle the further quotienting out of conformal transformations or volume-preserving conformal transformations. Depending on which choices are made, this leads to York’s initial value formulation (IVF) of GR, new alternative foundations for the GR IVF, or alternative theories of gravity which are built out of similar conformal mathematics to the GR IVF and yet admit no GR-like spacetime interpretation.