Kinematics of a Spacetime with an Infinite Cosmological Constant

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant is discussed by using Inönü–Wigner contractions of the de Sitter groups and spaces. When , 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 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.