The following three geometrical structures on a manifold are studied in detail: Leibnizian: a nonvanishing one-form Ω plus a Riemannian metric 〈⋅,⋅〉 on its annhilator vector bundle. In particular, the possible dimensions of the automorphism group of a Leibnizian -structure are characterized. Galilean: Leibnizian structure endowed with an affine connection ∇ (gauge field) which parallelizes Ω and 〈⋅,⋅〉. For any fixed vector field of observers an explicit Koszul-type formula which reconstructs bijectively all the possible ∇’s from the gravitational and vorticity ω≔ rot fields (plus eventually the torsion) is provided. Newtonian: Galilean structure with 〈⋅,⋅〉 flat and a field of observers which is inertial (its flow preserves the Leibnizian structure and ω≡0). Classical concepts in Newtonian theory are revisited and discussed.
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March 2003
Research Article|
March 01 2003
Leibnizian, Galilean and Newtonian structures of space–time Available to Purchase
Antonio N. Bernal;
Antonio N. Bernal
Dpto. de Geometrı́a y Topologı́a, Universidad de Granada, Facultad de Ciencias, Fuentenueva s/n, E-18071 Granada, Spain
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Miguel Sánchez
Miguel Sánchez
Dpto. de Geometrı́a y Topologı́a, Universidad de Granada, Facultad de Ciencias, Fuentenueva s/n, E-18071 Granada, Spain
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Antonio N. Bernal
Dpto. de Geometrı́a y Topologı́a, Universidad de Granada, Facultad de Ciencias, Fuentenueva s/n, E-18071 Granada, Spain
Miguel Sánchez
Dpto. de Geometrı́a y Topologı́a, Universidad de Granada, Facultad de Ciencias, Fuentenueva s/n, E-18071 Granada, Spain
J. Math. Phys. 44, 1129–1149 (2003)
Article history
Received:
October 28 2002
Accepted:
December 02 2002
Citation
Antonio N. Bernal, Miguel Sánchez; Leibnizian, Galilean and Newtonian structures of space–time. J. Math. Phys. 1 March 2003; 44 (3): 1129–1149. https://doi.org/10.1063/1.1541120
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