# Principle of Relativity, Dual Poincar\'e Group and Relativistic Quadruple

## TL;DR

This paper introduces a new framework extending special relativity using dual Poincaré groups, leading to a relativistic quadruple that unifies different spacetime geometries and offers new insights for cosmic scale physics.

## Contribution

It proposes a relativistic quadruple structure based on dual Poincaré groups, unifying Minkowski, de Sitter, and anti-de Sitter spacetimes within a new kinematic framework.

## Key findings

- Dual Poincaré group preserves the origin lightcone.
- Existence of a Poincaré double and dS double for different spacetime geometries.
- Introduction of a relativistic quadruple for three types of special relativity.

## Abstract

Based on the principle of relativity with two universal constants (c, l) and in the inertial motion group IM(1,3)\sim PGL(5,R), with Lorentz isotropy, in addition to Poincar\'e group of Einstein's SR the dual Poincar\'e group preserves the origin lightcone and its space/time-like region R_\pm appeared at common origin of intersected Minkowski/dS/AdS space. The dual Poincare kinematics is on a pair of degenerate Einstein manifolds with \Lambda_\pm=\pm3l^{-2} for R_\pm, respectively. Thus, there is a Poincar\'e double and the dS double for dS/AdS SR. Further, with other four doubles they form a relativistic quadruple for three kinds of SR on M/D_\pm, respectively. The dS SR with the dS-dual Poincare double provides new kinematics for cosmic scale physics.