The Beltrami -- de Sitter model

Abstract

We combine the well-known Beltrami-Klein model of non-Euclidean geometry on a $2-$dimensional disk, where the geodesics are the chords of the disk, with the $2-$dimensional de Sitter space. The geometry of the de Sitter space is defined on the complement of the Beltrami-Klein disk in the plane, with the de Sitter metric being the unique Lorentzian Einstein metric whose light cones form cones tangent to the disk in this complement. This leads to a Beltrami-de Sitter model on the plane ${\bf R}^2$, which is endowed with the Riemannian Beltrami metric on the disk and the Lorentzian de Sitter metric outside the disk in ${\bf R}^2$. We explore the relevance of this model for Penrose's Conformal Cyclic Cosmology, first in the $2-$dimensional setting and subsequently in higher dimensions, including the physically significant case of four dimensions. In this context, we define a Radon-like transform between the de Sitter and Beltrami spaces, facilitating the purely geometric transformation of physical fields from the Lorentzian de Sitter space to the Riemannian Beltrami space. In the $2-$, and $3-$dimensional cases, we also uncover a hidden ${\bf G}_2$ symmetry associated with the de Sitter spaces in these dimensions, which is related to a certain vector distribution naturally defined by the geometry of the model. We suggest the potential for discovering similar hidden symmetries in the $n-$dimensional Beltrami-de Sitter model.