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

TL;DR

This paper develops a geometrical framework based on Symmetrical Special Relativity (SSR) with an invariant minimum speed, linking Lorentz violation to the uncertainty principle and addressing the cosmic coincidence problem through modified spacetime structures in Minkowski, dS, and AdS spaces.

Contribution

It introduces SSR as a new causal spacetime structure incorporating a minimum speed, providing a geometric basis for the uncertainty principle and explaining the cosmological constant's effects.

Findings

SSR predicts different spacetime geometries depending on speed regimes.

The approach relates vacuum energy density to scalar curvature in Einstein's equations.

Current universe's flatness and vacuum energy are explained via SSR's effective curvature.

Abstract

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 ($\Phi<0$) connected to the cosmological parameter $\Lambda>0$. For $v=v_0$, Minkowski (pseudo-Euclidian) space is recovered for representing the flat space ($\Lambda=0$), and for $v>v_0$ ($\Phi>0$), Anti-de Sitter (AdS) scenario prevails ($\Lambda<0$). 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 ($\rho_{\Lambda}$ given for a slightly positive curvature $\Lambda$), 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 ($p=-\rho_{\Lambda}$). Einstein equation for vacuum given by such SSR approach allows us to obtain $\rho_{\Lambda}$ associated with a scalar curvature $\Lambda$, 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 $\Lambda$ opposes the Riemannian curvature $R$, thus leading to a current effective curvature $R_{eff}=R+\Lambda\approx 0$ according to observations.