Newton-Hooke Limit of Beltrami-de Sitter Spacetime, Principles of Galilei-Hooke's Relativity and Postulate on Newton-Hooke Universal Time

TL;DR

This paper develops a Newton-Hooke model based on Beltrami-de Sitter spacetime, introducing inertial frames and observers, and proposes principles to define the Newton-Hooke limit, extending to anti-de Sitter spacetime.

Contribution

It introduces a Newton-Hooke model derived from Beltrami-de Sitter spacetime, including principles for the universal time and inertial frames, extending the spacetime contraction concept.

Findings

Inertial-type coordinate systems exist in Newton-Hooke spacetime.

The Newton-Hooke group preserves inertial motions.

The model extends to anti-de Sitter spacetime with negative cosmological constant.

Abstract

Based on the Beltrami-de Sitter spacetime, we present the Newton-Hooke model under the Newton-Hooke contraction of the $BdS$ spacetime with respect to the transformation group, algebra and geometry. It is shown that in Newton-Hooke space-time, there are inertial-type coordinate systems and inertial-type observers, which move along straight lines with uniform velocity. And they are invariant under the Newton-Hooke group. In order to determine uniquely the Newton-Hooke limit, we propose the Galilei-Hooke's relativity principle as well as the postulate on Newton-Hooke universal time. All results are readily extended to the Newton-Hooke model as a contraction of Beltrami-anti-de Sitter spacetime with negative cosmological constant.