[Paper Review] Newton-Hooke Limit of Beltrami-de Sitter Spacetime, Principles of Galilei-Hooke's Relativity and Infinite Signal-velocity

Han-Ying Guo, Chao-Guang Huang+3 more|arXiv (Cornell University)|Feb 29, 2004

Relativity and Gravitational Theory1 citations

TL;DR

This paper introduces the Newton-Hooke spacetime as a contraction of Beltrami-de Sitter spacetime under the Newton-Hooke group, establishing inertial-type coordinate systems and observers moving uniformly along straight lines. It proposes the Galilei-Hooke relativity principle and the principle of infinite signal velocity to uniquely define the limit, extending the model 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 principle of infinite velocity of signal. All results are readily extended to the Newton-Hooke model as a contraction of Beltrami-anti-de Sitter spacetime with negative cosmological constant.

Motivation & Objective

• To derive a consistent Newton-Hooke model from the Beltrami-de Sitter spacetime through group-theoretic contraction.
• To define inertial-type coordinate systems and observers that move uniformly along straight lines in Newton-Hooke spacetime.
• To establish the Galilei-Hooke relativity principle as a foundational postulate for uniquely determining the Newton-Hooke limit.
• To extend the Newton-Hooke model to the Beltrami-anti-de Sitter spacetime with negative cosmological constant.

Proposed method

• Perform a group contraction of the Beltrami-de Sitter spacetime's isometry group to obtain the Newton-Hooke group.
• Analyze the resulting algebraic and geometric structure to identify inertial-type observers and coordinate systems.
• Introduce the Galilei-Hooke relativity principle as a symmetry principle for the Newton-Hooke spacetime.
• Incorporate the principle of infinite signal velocity to uniquely fix the Newton-Hooke limit.
• Extend the contraction procedure to the Beltrami-anti-de Sitter spacetime with negative cosmological constant.

Experimental results

Research questions

• RQ1 How can the Newton-Hooke spacetime be derived as a limit of the Beltrami-de Sitter spacetime through group contraction?
• RQ2 What defines inertial-type observers and coordinate systems in the Newton-Hooke spacetime?
• RQ3 How does the Galilei-Hooke relativity principle constrain the Newton-Hooke limit?
• RQ4 What role does the principle of infinite signal velocity play in uniquely determining the Newton-Hooke model?
• RQ5 Can the Newton-Hooke model be consistently extended to spacetimes with negative cosmological constant?

Key findings

• The Newton-Hooke spacetime admits inertial-type coordinate systems and observers that move uniformly along straight lines, invariant under the Newton-Hooke group.
• The Galilei-Hooke relativity principle is proposed as a fundamental symmetry principle for the Newton-Hooke model.
• The principle of infinite signal velocity is introduced to uniquely determine the Newton-Hooke limit of the Beltrami-de Sitter spacetime.
• The Newton-Hooke model is successfully extended to the Beltrami-anti-de Sitter spacetime with negative cosmological constant.
• The contraction procedure preserves the geometric and algebraic structure of the original spacetime in the Newton-Hooke limit.