The Beltrami -- de Sitter model

TL;DR

The paper builds a unified Beltrami -- de Sitter framework on R^n by gluing a Riemannian Beltrami metric inside the unit ball to a Lorentzian de Sitter metric outside, with a boundary where the signature changes and the conformal structure remains continuous. It develops a Radon-like transform between the two regions, interprets these transforms in a twistorial Penrose framework on a correspondence space, and uncovers hidden G2-symmetry in low dimensions via a circle twistor bundle and a (2,3,5) distribution. The constructions are then generalized to arbitrary dimensions, including n = 4, and extended via the dancing metric to a conformal, split-signature geometry with a rich symmetry algebra such as sl(n+1,R) and, in 2D and 3D, exceptional G2 structures. The work connects to Penrose's CCC by offering a purely geometric, signature-changing model where light cones are tangent to the boundary, and it highlights potential higher-dimensional hidden symmetries and twistor-theoretic interpretations that could illuminate conformal and cosmological applications.

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.

Figures (21)

• Figure 1: The Beltrami-Klein distance between points $p$ and $q$ is determined by the cross-ratio $\frac{d_E(q,a)}{d_E(q,b)}:\frac{d_E(p,a)}{d_E(p,b)}$ of the Euclidean distances between the four points $(a,p,q,b)$ lying on circle's cord passing through $p$ and $q$, for which points $a$ and $b$ are the end points.
• Figure 2: The $point\leftrightarrow line$ correspondence in the Beltrami -- de Sitter model. On the left:Every point inside the Beltrami-Klein disk, as for example the green point in this figure, defines a line outside the disk - the green line on the figure. The green line is obtained as the locus of points at which pairs of red lines intersect. Each pair of red lines consists of lines tangent to the circle $x^2+y^2=1$ at the end points of some cord passing through the green point. The (green) lines outside the disk, corresponding to the (green) points inside the disk, are spacelike geodesics in the Lorentzian Beltrami metric (\ref{['beme']}) outside the disk. On the right:Every point outside the disk, i.e. an event in the Lorentzian Beltrami spacetime (a green point), defines a (green) cord in the Beltrami-Klein disk with the end points belonging to the two tangent lines to the circle outgoing from the green event. The corresponding (green) cord is a geodesic in the Riemannian Beltrami metric (\ref{['beme']}) inside the disk. Compare this picture with the last picture in Section 2.4 in Thurston.
• Figure 3: The conformal structure as defined by cones tangent to the unit disk. The larger brown point outside the disk defines a cone in the region $x^2+y^2>1$. This cone is interpreted as a light cone of a spacetime event marked by the larger brown point. We interpret the white area in the cone between the point $p$ and the part of the circle $x^2+y^2=1$ between $r$ and $s$ as the possible future of the larger brown point. This is the region of spacetime which the larger brown point can achieve by moving along timelike curves. Every point outside the Beltrami-Klein disk defines its own cone in the same manner. This defines a conformal structure in the region $x^2+y^2>1$. In this conformal structure the null geodesics are just the straight lines tangent to the circle $x^2+y^2=1$, and the time of observers in $x^2+y^2>1$ region has an arrow from a particular event towards the circle $x^2+y^2=1$. In particular, the vector $\stackrel{\longrightarrow}{pO}$ shows the time arrow within the cone of the larger brown point.
• Figure 4: Two types of hyperboloids in the 3-dimensional Minkowski space with the metric $-{\rm d} T^2+{\rm d} X^2+{\rm d} Y^2$. The points $(T,X,Y)$ of the cyan colored hyperboloid satisfy $-T^2+X^2+Y^2=-1$ and $T>0$, whereas the points of the white hyperboloid satisfy $-T^2+X^2+Y^2=1$. The purple cone is the future light cone of the observer at the origin $(0,0,0)$. In the top figure it is visible that there is a one--to--one correspondence between the points of the cyan hyperboloid and the points on the disk of points $(T,X,Y)$ such that $X^2+Y^2<1$ and $T=1$. The correspondence is achieved by the projection of a point $(T,X,Y)$ from the hyperboloid, along a black line passing through this point and the origin, to the point of the intersection of this line with the disk. In the bottom figure we show that there is also a one--to--one correspondence between the points of the white hyperboloid and the points on the complement to the cyan disk in the plane $T=1$. Again, the correspondence is achieved by connecting a point $(T,X,Y)$ on a white hyperboloid by a black line passing through the origin, and getting point of intersection of this line with the plane $T=1$ as the point corresponding to $(T,X,Y)$.
• Figure 5: Four types of geodesics in the Beltrami--de Sitter model

Theorems & Definitions (33)

• Corollary 2.1
• Remark 3.1
• Theorem 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6
• Remark 3.7
• Proposition 3.8
• Theorem 4.1