Scale-invariant gravity: particle dynamics

Abstract

A new and universal method for implementing scale invariance, called best matching, is presented. It extends to scaling the method introduced by Bertotti and the author to create a fully relational dynamics that satisfies Mach's principle. The method is illustrated here in the context of non-relativistic gravitational particle dynamics. It leads to far stronger predictions than general Newtonian dynamics. The energy and angular momentum of an 'island universe' must be exactly zero and its size, measured by its moment of inertia, cannot change. This constancy is enforced because the scale invariance requires all potentials to be homogeneous of degree -2. It is remarkable that one can nevertheless exactly recover the standard observed Newtonian laws and forces, which are merely accompanied by an extremely weak universal force like the one due to Einstein's cosmological constant. In contrast to Newtonian and Einsteinian dynamics, both the gravitational constant G and the strength of the cosmological force are uniquely determined by the matter distribution of the universe. Estimates of their values in agreement with observations are obtained. Best matching implements a dynamics of pure shape for which the action is a dimensionless number. If the universe obeys such scale-invariant law, steadily increasing inhomogeneity, not expansion of the universe, causes the Hubble red shift. The application of best matching to geometrodynamics is treated in a companion paper.

Scale-invariant gravity: geometrodynamics

“…We hope to use a 'top-down' approach: to start from firm classical theory and deduce features of the quantum universe. However, we start from space rather than spacetime for relational reasons [13,47,11,9] and to illustrate the potential naivety of presupposing and extending spacetime structure. The great problems of quantizing gravity are hopelessly interrelated, so that adding to a partial resolution to tackle further problems can spoil that partial resolution [48].…”

“…Superspace is the analogue of the relative configuration space discussed in the particle-dynamics paper [9]. 2 The ADM configuration space is the extension of Riem to Riem × Ξ × P , where Ξ is the space of the ξ i and N ∈ P , the space of (suitably differentiable) positive functions.…”

“…The action must be homogeneous of degree zero in φ. 9 We exhibit this in the first expression for L in (31), in which the two expressions (30) that derive from √ g and √ R in the BSW action have been multiplied by appropriate powers of V . (The second expression is more convenient for calculations.…”

On the recovery of geometrodynamics from two different sets of first principles

“…E.g., given Q n , choosing G to be the trivial group gives Newtonian particle mechanics with its notion of absolute space, while choosing G to be the Euclidean group of translations and rotations gives instead a relational Leibnizian particle mechanics, such as in Barbour and Bertotti's works. The choice of G as the similarity group, which includes dilatations alongside the translations and rotations, has also been investigated (Barbour, 2003). For Q = Riem(Σ), one could again choose G to be the identity, which amounts to the coordinate grids painted on Σ having physical significance, while the persistent relationalist might choose G to be the diffeomorphisms on Σ to ensure that this is not the case (Wheeler, 1968, Barbour & Bertotti, 1982, Barbour, 1994a.…”

“…The scope of this arbitrary G-frame method or best-matching method, 20 is well-illustrated by how one can obtain thus all of Leibnizian particle mechanics models, 1-form gauge theories (Barbour & Bertotti, 1982, Barbour, 2003, Anderson, 2004b, and spatially compact without boundary GR. This last case is demonstrated below.…”