Conformal compactification

A conformal compactification realizes a noncompact metric space as a dense open subset of a compact space whose metric on that subset is conformally related to the original one. Its smoothness and boundary interpretation depend on the example. Conformal compactification of Euclidean space adds one point to four-space; Projective Minkowski compactification adds a null cone. A regular null-boundary conformal completion and a projective compactification need not have the same global topology or causal covering convention.

A boundary obtained by extending the rescaled metric $\tilde{g}={\Omega }^{2}g$ smoothly and nondegenerately to a hypersurface on which $\Omega$ vanishes with nonzero derivative. Its causal character is measured using the extended Lorentzian metric, not the divergent physical metric. In signature $(+---)$, a timelike normal makes the boundary spacelike, a spacelike normal makes it timelike, and a null normal makes it null. These cases lead to different causal and initial-value properties. The conformal cylinder of two-dimensional de Sitter spacetime has spacelike past/future boundaries; the conformal strip of two-dimensional anti-de Sitter spacetime has two timelike boundaries.

Inverse stereographic projection identifies Euclidean four-space with a sphere minus one point. Multiplying its flat metric by $4/(1+|x{|}^2{)}^2$ gives the round metric, which extends smoothly over the added point. In the Klein quadric description, the compact space is the projectivized real null cone of signature $(1,5)$. It differs topologically from Minkowski conformal compactification.

The Projective Minkowski compactification has infinite cyclic fundamental group, while the conformal compactification of Euclidean space is $S^4$ and simply connected. Thus the affine substitution $t\mapsto it$ cannot extend to an invertible smooth identification of these real compactifications. Both are real slices of the same complex Klein quadric, which permits local analytic continuation without equating their global real topologies.

In affine cone coordinates $[U,V,t,\mathbf{x}]$ with $2UV+t^2-|\mathbf{x}{|}^2=0$ and $U=1$, the substitution is represented on the complex quadric by the rational map $[U,V,t,\mathbf{x}]\mapsto [U^2,UV+t^2,iUt,U\mathbf{x}]$. It preserves the quadric equation but is indeterminate at some infinity points, including $[0,1,0,0]$. This is a direct algebraic manifestation of the obstruction to a global regular Wick map.