Snyder's Model -- de Sitter Special Relativity Duality and de Sitter Gravity

TL;DR

This paper explores a duality between Snyder's quantized space-time in de Sitter space and de Sitter special relativity, proposing a fundamental constant linking quantum and cosmological scales, with implications for a new model of de Sitter gravity.

Contribution

It introduces a duality framework connecting Snyder's model and de Sitter relativity, and proposes a gauge-like de Sitter gravity model consistent with observations.

Findings

Identification of a fundamental constant g linking quantum and cosmological scales.

Proposal of a gauge-like de Sitter gravity model on umbilical manifolds.

Support for the duality through observational consistency.

Abstract

Between Snyder's quantized space-time model in de Sitter space of momenta and the \dS special relativity on \dS-spacetime of radius $R$ with Beltrami coordinates, there is a one-to-one dual correspondence supported by a minimum uncertainty-like argument. Together with Planck length $\ell_P$, $R\simeq (3/\Lambda)^{1/2}$ should be a fundamental constant. They lead to a dimensionless constant $g{\sim\ell_PR^{-1}}=(G\hbar c^{-3}\Lambda/3)^{1/2}\sim 10^{-61}$. These indicate that physics at these two scales should be dual to each other and there is in-between gravity of local \dS-invariance characterized by $g$. A simple model of \dS-gravity with a gauge-like action on umbilical manifolds may show these characters. It can pass the observation tests and support the duality.