Mechanics and Newton-Cartan-like gravity on the Newton-Hooke space-time

Abstract

We focus on the dynamical aspects on Newton-Hooke space-time ${\mathcal{NH}}_{+}$ mainly from the viewpoint of geometric contraction of the de Sitter spacetime with Beltrami metric. (The term spacetime is used to denote a space with non-degenerate metric, while the term space-time is used to denote a space with degenerate metric.) We first discuss the Newton-Hooke classical mechanics, especially the continuous medium mechanics, in this framework. Then, we establish a consistent theory of gravity on the Newton-Hooke space-time as a kind of Newton-Cartan-like theory, parallel to the Newton's gravity in the Galilei space-time. Finally, we give the Newton-Hooke invariant Schr\"odinger equation from the geometric contraction, where we can relate the conservative probability in some sense to the mass density in the Newton-Hooke continuous medium mechanics. Similar consideration may apply to the Newton-Hooke space-time ${\mathcal{NH}}_{\ensuremath{-}}$ contracted from anti-de Sitter spacetime.