Scale-invariant gravity: particle dynamics

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 Machʹs 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 ʹisland universeʹ 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 that due to Einsteinʹs 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 a scale-invariant law, steadily increasing inhomogeneity, not expansion of the universe, causes the Hubble redshift. The application of best matching to geometrodynamics is treated in a companion paper.