Conformally Compactified Minkowski Space: A Re-Examination with Emphasis on the Double Cover and Conformal Infinity

This paper presents a detailed re-examination of the conformalcompactification M¯ of Minkowski space M, constructed as the projective null cone of the six-dimensional space R4,2. We provide an explicit and basis-independent formulation, emphasizing geometric clarity. A central result is the explicit identification of M¯ with the unitary group U(2) via a diffeomorphism, offering a clear matrix representation for points in the compactified space. We then systematically construct and analyze the action of the full conformal group O(4,2) and its connected component SO0(4,2) on this manifold. A key contribution is the detailed study of the double cover, M˜, which is shown to be diffeomorphic to S3×S1. This construction resolves the non-effectiveness of the SO(4,2) action on M¯, yielding an effective group action on the covering space. A significant portion of our analysis is devoted to a precise and novel geometric characterization of the conformal infinity. Moving beyond the often-misrepresented “double cone” description, we demonstrate that the infinity of the double cover, M˜∞, is a squeezed torus (specifically, a horn cyclide), while the simple infinity, M¯∞, is a needle cyclide. We provide explicit parametrizations and graphical representations of these structures. Finally, we explore the embedding of five-dimensional constant-curvature spaces, whose boundary is the compactified Minkowski space. The paper aims to clarify long-standing misconceptions in the literature and provides a robust, coordinate-free geometric foundation for conformal compactification, with potential implications for cosmology and conformal field theory.