Poincar\'e-De Sitter Flow and Cosmological Meaning

Abstract

We introduce the Poincar\'e-de Sitter flow with real numbers $\{r,s\}$ to parameterize the relativistic quadruple ${\frak Q}_{PoR}=[{\cal P}, {\cal P}_2, {\cal D}_+,{\cal D}_-]_{M/M_\pm/D_\pm}$ for the triple of Poincar\'e/\dS/\AdS\ group ${\cal P}/{\cal D}_+/{\cal D}_-$ invariant special relativity. The dual Poincar\'e group ${\cal P}_2$-invariant degenerated Einstein manifold $M_\pm$ of $\Lambda_\pm=\pm3l^{-2}$ is for the space/time-like domain $\dot R_\pm$ of the compact lightcone $\bar C_O$ associated to the common space/time-like region $R_\pm$ of the lightcone $C_O$ at common origin on Minkowski/\dS/\AdS\ spacetime $M/D_+/D_-$. Based on the principle of relativity with two universal constants $(c, l)$, there are the law of inertia, coordinate time simultaneity and so on for the flow on a Poincar\'e-\dS\ symmetric Einstein manifold of $\Lambda_{s}=3{s}l^{-2}$. Further, there is Robertson-Walker-like cosmos of the flow for the propertime simultaneity. The \dS\ special relativity with double $[{\cal D}_+,{\cal P}_2]_{D_+/M_+}$ can provide a consistent kinematics for the cosmic scale physics with an upper entropy bound $S_R=k_B\pi g^{-2}, g^2:=(\ell_P/R)^2 \simeq 10^{-122}$, for $R\simeq (3/\Lambda)^{1/2}\sim 13.7 Gly$.