Kinematics of a Spacetime with an Infinite Cosmological Constant

TL;DR

This paper explores the properties of spacetime solutions with an infinite cosmological constant, revealing a conical spacetime structure and a modified Galilei group relevant to early Universe cosmology.

Contribution

It introduces a novel spacetime geometry for infinite  with a duality to Minkowski space and analyzes the associated kinematical group in the non-relativistic limit.

Findings

Spacetime becomes a four-dimensional cone as    .

The non-relativistic limit yields a conformal Galilei group with the same Lie algebra as the Galilei group.

Results have potential implications for early Universe cosmology.

Abstract

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant \Lambda is discussed by using Inonu-Wigner contractions of the de Sitter groups and spaces. When \Lambda --> infinity, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c --> infinity is further considered, the kinematical group in this case being a modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology.