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American Institute of Physics, Journal of Mathematical Physics, 3(44), p. 1129-1149, 2003

DOI: 10.1063/1.1541120

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Leibnizian, Galilean and Newtonian structures of space–time

Journal article published in 2003 by Antonio N. Bernal, Miguel Sánchez ORCID
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This paper is available in a repository.

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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 $∇$ (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 $∇$'s from the gravitational ${\cal G} = ∇_Z Z$ 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.