Standard and Generalized Newtonian Gravities as ``Gauge'' Theories of the Extended Galilei Group - II: Dynamical Three-Space Theories

Abstract

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