Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime

Abstract

We give a causal version of Eisenhart's geodesic characterization of classical mechanics. We emphasize the geometric, coordinate independent properties needed to express Eisenhart's theorem in light of modern studies on the Bargmann structures (lightlike dimensional reduction, pp-waves). The construction of the space metric, Coriolis 1-form and scalar potential through which the theorem is formulated is shown in detail, and in particular it is proved a one-to-one correspondence between Newtonian frames and Abelian connections on suitable lightlike principal bundles. The relation of Eisenhart's theorem in the lightlike case with a Fermat type principle is pointed out. The operation of lightlike lift is introduced and the existence of minimizers for the classical action is related to the causal simplicity of Eisenhart's spacetime.

arXiv commentLatex2e, 29 pages, 2 figures, uses psfrag package. v2: a problem with theorem 5.5, which was erroneously stated twice, and some misprints have been fixed, v3: change of definition of function I in theorem 5.5

References in corpus (7)

• Celestial Mechanics, Conformal Structures, and Gravitational Waves
• Null Killing Vector Dimensional Reduction and Galilean Geometrodynamics
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Leibnizian, Galilean and Newtonian structures of spacetime
• Causal inheritance in plane wave quotients
• Classical aspects of lightlike dimensional reduction
• On the existence of maximizing curves for the charged-particle action

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