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 conformal compactification of Minkowski space, \( \overline{M} \), constructed as the projective null cone of the six-dimensional space \( \mathbb{R}^{4,2} \). We provide an explicit and basis-independent formulation, emphasizing geometric clarity. A central result is the explicit identification of \( \overline{M} \) 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 \( \mathrm{O}(4,2) \) and its connected component \( \mathrm{SO}_0(4,2) \) on this manifold. A key contribution is the detailed study of the double cover, \( \overline{\overline{M}} \), which is shown to be diffeomorphic to \( S^3 \times S^1 \). This construction resolves the non-effectiveness of the \( \mathrm{SO}(4,2) \) action on \( \overline{M} \), 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, \( \overline{\overline{M}}_\infty \), is a squeezed torus (specifically, a horn cyclide), while the simple infinity, \( \overline{M}_\infty \), 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, and discuss the interpretation of geodesics within these domains. 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.