Standard and Generalized Newtonian Gravities as “gauge” Theories of the Extended Galilei Group - Ii: Dynamical Three-space Theories

In a preceding paper we developed a reformulation of Newtonian gravitation as a gauge theory of the extended Galilei group. In the present one we derive two true generalizations of Newton’s theory (a ten-fields and an eleven-fields theory), in terms of an explicit Lagrangian realization of the absolute time dynamics of a Riemannian three-space. They turn out to be gauge invariant theories of the extended Galilei group in the same sense in which general relativity is said to be a gauge theory of the Poincaré group. The ten-fields theory provides a dynamical realization of some of the so-called “Newtonian space-time structures” which have been geometrically classified by Künzle and Kuchař. The eleven-fields theory involves a dilaton-like scalar potential in addition to Newton’s potential and, like general relativity, has a three-metric with two dynamical degrees of freedom. It is interesting to find that, within the linear approximation, such degrees of freedom show graviton-like features: they satisfy a wave equation and propagate with a velocity related to the scalar Newtonian potential.