Leibnizian, Galilean and Newtonian structures of spacetime

TL;DR

This work develops a unified, multi-layer geometric framework for spacetime by treating Leibnizian, Galilean, and Newtonian structures in a progressive hierarchy. It provides a Koszul-type reconstruction formula that uniquely determines the Galilean affine connection from a fixed observer, the gravitational field, vorticity, and torsion, establishing a bijection between connections and invariants. The authors also analyze automorphism structures of Leibnizian spacetimes, derive explicit coordinate formulas for connections, geodesics, and curvature, and integrate Poisson's equation to connect geometry with mass density in Newtonian spacetimes. Overall, the paper clarifies how time-space measurements dictate gravitational and inertial structure and situates Newtonian physics within a rigorous differential-geometric framework.

Abstract

The following three geometrical structures on a manifold are studied in detail: (1) Leibnizian: a non-vanishing 1-form $Ω$ 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 $Ω$ and $\h$. Fixed any vector field of observers Z ($Ω(Z) = 1$), an explicit Koszul--type formula which reconstruct bijectively all the possible $\nabla$'s from the gravitational ${\cal G} = \nabla_Z Z$ and vorticity $ω= rot Z/2$ 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 $ω= 0$). Classical concepts in Newtonian theory are revisited and discussed.

Theorems & Definitions (42)

• Definition 2.1
• Proposition 2.2
• Definition 3.3
• Proposition 3.4
• Remark 3.5
• Proposition 3.6
• Remark 3.7
• Theorem 3.8
• Remark 3.9
• Definition 4.10