Han-Ying Guo

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 $R\simeq (3/\Lambda)^{1/2}\sim 13.7 Gly$.

If our universe is asymptotic to a de Sitter space, it should be closed with curvature in $O(\Lambda)$ in view of dS special relativity. Conversely, its evolution can fix on Beltrami systems of inertia in the dS-space without Einstein's `argument in a circle'. Gravity should be local dS-invariant based on localization of the principle of inertia.

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.

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 $ell_P$, $Rsimeq (3/Lambda)^{1/2}$ should be a fundamental constant. They lead to a dimensionless constant $g{simell_PR^{-1}}=(Ghbar c^{-3}Lambda/3)^{1/2}sim 10^{-61}$. 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.