Lorentz violation with an invariant minimum speed as foundation of the uncertainty principle in Minkowski, dS and AdS spaces

This research aims to provide the geometrical foundation of the uncertainty principle within a new causal structure of spacetime so-called Symmetrical Special Relativity (SSR), where there emerges a Lorentz violation due to the presence of an invariant minimum speed $V$ related to the vacuum energy. SSR predicts that a dS-scenario occurs only for a certain regime of speeds $v$, where $v<v_0=\sqrt{cV}$, which represents the negative gravitational potentials ($\mathrm{\Phi }<0$) connected to the cosmological parameter $\mathrm{\Lambda }>0$. For $v=v_0$, Minkowski (pseudo-Euclidian) space is recovered for representing the flat space ($\mathrm{\Lambda }=0$), and for $v>v_0$ ($\mathrm{\Phi }>0$), Anti-de Sitter (AdS) scenario prevails ($\mathrm{\Lambda }<0$). The fact that the current universe is flat as its average density of matter distribution (${\rho }_m$ given for a slightly negative curvature $R$) coincides with its vacuum energy density (${\rho }_{\mathrm{\Lambda }}$ given for a slightly positive curvature $\mathrm{\Lambda }$), i.e., the {\it cosmic coincidence problem}, is now addressed by SSR. SSR provides its energy-momentum tensor of perfect fluid, leading to the EOS of vacuum ($p=-{\rho }_{\mathrm{\Lambda }}$). Einstein equation for vacuum given by such SSR approach allows us to obtain ${\rho }_{\mathrm{\Lambda }}$ associated with a scalar curvature $\mathrm{\Lambda }$, whereas the solution of Einstein equation only in the presence of a homogeneous distribution of matter ${\rho }_m$ for the whole universe presents a scalar curvature $R$, in such a way that the presence of the background field $\mathrm{\Lambda }$ opposes the Riemannian curvature $R$, thus leading to a current effective curvature $R_{eff}=R+\mathrm{\Lambda }\approx 0$ according to observations.