Leibnizian, Galilean and Newtonian structures of spacetime

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 $\mathcal{G}={\nabla }_{Z}Z$ and vorticity $\omega =rotZ/2$ 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.

Minor errata corrected, to appear in J. Math. Phys.; 22 pages including a table, Latex

References in corpus (2)

• Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory
• Fundamental units of length and time

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