Leibnizian, Galilean and Newtonian structures of spacetime

TL;DR

This paper explores three interconnected geometric structures of spacetime—Leibnizian, Galilean, and Newtonian—detailing their properties, automorphisms, and connections, and revisiting classical Newtonian concepts within this framework.

Contribution

It provides a detailed analysis of the automorphism groups, explicit formulas for connections, and clarifies the geometric underpinnings of classical Newtonian spacetime structures.

Findings

Characterization of automorphism group dimensions for Leibnizian structures

Explicit Koszul-type formula for Galilean connections

Conditions for Newtonian structures with inertial observers

Abstract

The following three geometrical structures on a manifold are studied in detail: (1) Leibnizian: a non-vanishing 1-form $\Omega$ plus a Riemannian metric $\h$ on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian G-structure are characterized. (2) Galilean: Leibnizian structure endowed with an affine connection $\nabla$ (gauge field) which parallelizes $\Omega$ and $\h$. Fixed any vector field of observers Z ($\Omega (Z) = 1$), an explicit Koszul--type formula which reconstruct bijectively all the possible $\nabla$'s from the gravitational ${\cal G} = \nabla_Z Z$ and vorticity $\omega = rot Z/2$ fields (plus eventually the torsion) is provided. (3) Newtonian: Galilean structure with $\h$ flat and a field of observers Z which is inertial (its flow preserves the Leibnizian structure and $\omega = 0$). Classical concepts in Newtonian theory are revisited and discussed.