Standard and Generalized Newtonian Gravities as ``Gauge'' Theories of the Extended Galilei Group - I: The Standard Theory

TL;DR

This work recasts Newtonian gravity as a gauge theory of the extended Galilei group, linking a manifestly Galilean-covariant formulation to a nonrelativistic limit of general relativity via an ADM–DeWitt approach. It demonstrates that a 27-field Galilei-invariant action can reproduce Newtonian gravity, showing that most fields are auxiliary and that the Newtonian potential is carried by the field $\tilde{A}$ which obeys a covariant Poisson equation $\Delta \tilde{A} = -4\pi G m \delta^3(\mathbf{z}-\mathbf{x}(t))$. The paper then develops Galilean-covariant Newtonian gravity in arbitrary reference frames, deriving the standard Poisson equation in covariant form and explicit inertial forces in rotating frames, thereby clarifying how Newtonian dynamics remains form-invariant under Galilean transformations. These results lay a foundation for a generalized gauge-theoretic treatment of non-relativistic gravitation and its possible extensions.

Abstract

Newton's standard theory of gravitation is reformulated as a {\it gauge} theory of the {\it extended} Galilei Group. The Action principle is obtained by matching the {\it gauge} technique and a suitable limiting procedure from the ADM-De Witt action of general relativity coupled to a relativistic mass-point.

Figures (1)

• Figure 1: What is being fixed by the constraints' chains for the 27-fields theory.