Mechanics and Newton-Cartan-Like Gravity on the Newton-Hooke Space-time

TL;DR

This paper develops a geometric contraction approach to formulate Newton-Hooke space-time mechanics and gravity, establishing a Newton-Cartan-like gravity theory and deriving an invariant Schrödinger equation.

Contribution

It introduces a consistent Newton-Hooke gravity theory via geometric contraction from de Sitter space and derives an invariant Schrödinger equation within this framework.

Findings

Established Newton-Hooke gravity as a contraction of de Sitter spacetime.

Derived a Newton-Hooke invariant Schrödinger equation.

Linked probability conservation to mass density in Newton-Hooke mechanics.

Abstract

We focus on the dynamical aspects of Newton-Hooke space-time ${\cal NH}_+$ mainly from the viewpoint of geometric contraction of the de Sitter spacetime. 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 ${\cal NH}_-$ contracted from anti-de Sitter spacetime.