Compactified Minkowski Space: Myths and Facts

TL;DR

This paper clarifies the mathematical structure of compactified Minkowski space, emphasizing its conformal symmetry, and introduces Clifford algebra tools and the Hodge star operator to resolve misconceptions.

Contribution

It presents two clear constructions of compactified Minkowski space based on Penrose's ideas, highlighting subtle mathematical points and connecting to Clifford algebra techniques.

Findings

Clarifies the structure of compactified Minkowski space.

Introduces the antilinear Hodge star operator in this context.

Relates the constructions to Clifford algebra and conformal symmetry.

Abstract

Minkowski space, conformal group, compactification, conformal infinity, conformal inversion, light cone at infinity, SU(2,2), SO(4,2), Hodge star operator, Clifford algebra, spinors, twistors, antilinear operators, exterior algebra, bivectors, isotropic subspaces, null geodesics, Lie spheres, Dupin cyclides, gravitation} \abstract{Maxwell's equations are invariant not only under the Lorentz group but also under the conformal group. Irving E. Segal has shown that while the Galilei group is a deformation of the Poincar\'e group, and the Poincar\'e group is a deformation of the conformal group, the conformal group ends the road, it is {\em rigid\}. There are thus compelling mathematical and physical reasons for promoting the conformal group to the role of the fundamental symmetry of space--time, more important than the Poincar\'e group that formed the group-theoretical basis of special and general theories of relativity. While the action of the conformal group on Minkowski space is singular, it naturally extends to a nonsingular action on the compactified Minkowski space, often referred to in the literature as "Minkowski space plus light-cone at infinity". Unfortunately in some textbooks the true structure of the compactified Minkowski space is sometimes misrepresented, including false proofs and statements that are simply wrong. In this paper we present in, a simple way, two different constructions of the compactified Minkowski space, both stemming from the original idea of Roger Penrose, but putting stress on the mathematically subtle points and relating the constructions to the Clifford algebra tools. In particular the little-known antilinear Hodge star operator is introduced in order to connect real and complex structures of the algebra. A possible relation to Waldyr Rodrigues' idea of gravity as a plastic deformation of Minkowski's vacuum is also indicated.