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

TL;DR

This paper reformulates Eisenhart's classical mechanics theorem using modern geometric and causal structures, linking Newtonian frames with lightlike principal bundles and exploring the causal simplicity of Eisenhart's spacetime.

Contribution

It provides a causal, coordinate-independent version of Eisenhart's theorem, connecting Newtonian frames with Abelian connections on lightlike bundles and relating classical action minimizers to spacetime causality.

Findings

Established a one-to-one correspondence between Newtonian frames and Abelian connections.

Linked the existence of classical action minimizers to the causal simplicity of Eisenhart's spacetime.

Extended Eisenhart's theorem to lightlike and modern geometric contexts.

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.