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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.

The conformal group of a metric manifold consists of diffeomorphisms $f$ with $f^{\ast }g={\Omega }^{2}g$ for a positive function $\Omega$. Its infinitesimal generators are Conformal Killing vector fields. The choice of manifold matters: in flat Euclidean dimension greater than two, special conformal transformations can be regular on the conformal compactification of Euclidean space while having finite poles in the uncompactified space.

Choose an antilinear quaternionic structure $J$ on ${\mathbb{C}}^4$. It induces a fixed-point-free involution on projective twistor space. A real Euclidean spacetime point is a $J$-invariant complex two-plane, equivalently a quaternionic line in ${\mathbb{H}}^2$. These form ${\mathbb{HP}}^1\cong S^4$, the conformal compactification of Euclidean space. Every projective twistor lies on the unique invariant line $\mathbb{P}{\mathrm{span}}_{\mathbb{C}}(Z,JZ)$, giving a fibration ${\mathbb{CP}}^3\to S^4$ with fiber ${\mathbb{CP}}^1$.

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.

The pullback of the flat Euclidean metric under $x^{\prime }{}^{\mu }=x^{\mu }+ϵ{\omega }^{\mu }(x)$ isFor a conformal transformation it must be a scalar multiple of ${\delta }_{\mu \nu }$. Taking the trace determines that multiple. With $\rho ={\partial }_{\alpha }{\omega }^{\alpha }$, the necessary and sufficient infinitesimal condition is the Conformal Killing equationDifferentiate its $(\alpha ,\beta )$ instance with respect to $x^{\gamma }$, its $(\alpha ,\gamma )$ instance with respect to $x^{\beta }$, and its $(\beta ,\gamma )$ instance with respect to $x^{\alpha }$. Adding the first two and subtracting the last, commuting partial derivatives, leaves $2{\partial }_{\beta }{\partial }_{\gamma }{\omega }_{\alpha }$. Hence flat conformal Killing integrability givesContracting $\beta$ with $\gamma$ gives $\Delta {\omega }_{\alpha }=(2-d){\partial }_{\alpha }\rho /d$. Its divergence is $\Delta \rho =(2-d)\Delta \rho /d$, so $2(d-1)\Delta \rho =0$: thus $\boxed{\Delta \rho =0\text{ for }d>1}$. Taking instead the $\alpha$ divergence of the displayed second-derivative identity givesConsequently $\boxed{{\partial }_{\beta }{\partial }_{\gamma }\rho =0\text{ for }d>2}$.

On a connected flat domain, write $\rho =d\lambda +2db\cdot x$, with constant $\lambda$ and $b$. Integrating the second-derivative identity, and imposing the Conformal Killing equation on its remaining affine part, gives the complete Conformal Killing vector fieldIndeed the quadratic part has precisely the required Hessian matrix, while the symmetric part of the remaining linear term must be $\lambda {\delta }_{\mu \nu }$. The constants generate translations, rotations, dilations and special conformal transformations, respectively. There are $d$, $d(d-1)/2$, $1$ and $d$ independent parameters, so the full conformal group has dimensionHere “full” means local transformations, or global transformations of the conformal compactification of Euclidean space; its connected group is locally $SO(d+1,1)$. Requiring every transformation to be nonsingular on all of uncompactified ${\mathbb{R}}^d$ would exclude the special conformal transformations with finite poles.

The starting point is the Grassmannian of complex two-planes $W\subset \mathbb{T}\cong {\mathbb{C}}^4$. Choose a basis $u,v$ and form the simple bivector $P=u\wedge v\in {\Lambda }^2\mathbb{T}$. A change of basis multiplies $P$ by a nonzero determinant, so its projective class is independent of that choice. The Plücker embedding is thereforeIt is injective because $W=\{z:z\wedge P=0\}$. In a basis of $\mathbb{T}$, its six homogeneous coordinates $p_{ij}=u_i{v}_j-u_j{v}_i$ satisfyThis is the condition $P\wedge P=0$. Conversely, a nonzero antisymmetric $4\times 4$ matrix satisfying it has Pfaffian zero, hence rank at most two; its nonzero rank is even, so it has rank two and is a simple bivector. Therefore the equation characterizes the image, the Klein quadric $Q\subset {\mathbb{CP}}^5$. Its gradient cannot vanish at a projective point, since its six partial derivatives are the six coordinates up to permutation and sign. Thus $Q$ is smooth and has complex dimension four, agreeing with $\mathrm{Gr}(2,4)$.

The Klein quadric is the complexified compactified Minkowski space of the twistor correspondence. A volume form on $\mathbb{T}$ turns $P\wedge P^{\prime }$ into a scalar symmetric bilinear form $B(P,P^{\prime })$. It is zero exactly when the two underlying two-planes have nontrivial intersection: the four spanning vectors are then linearly dependent. In projective twistor space this says that the two twistor lines meet. In the affine graph chart supplied by the twistor incidence relation, it is $\det (x-y)=0$, so it is precisely null separation.

This also determines a conformal geometry on $Q$. At $[P]$, the tangent space is $P^{\perp }/⟨P⟩$ for $B$. The restriction of $B$ descends to a nondegenerate quadratic form on this four-dimensional quotient. Rescaling the representative $P$ changes the identification by a scalar, so its invariant content is a conformal class. In a graph chart its null tangent vectors are the rank-one matrices in $\mathrm{Hom}(W,\mathbb{T}/W)$, equivalently $d{x}^{A{A}^{\prime }}=a^A{b}^{A^{\prime }}$.

The compact complex geometry does not yet specify a real spacetime. The Lorentzian slice comes from the twistor Hermitian form $h$ of signature $(2,2)$: real points are maximal totally isotropic two-planes. In the incidence chart, isotropy of $W_x=\{(ix\pi ,\pi )\}$ is equivalent to $x=x^{†}$, the usual real Minkowski condition. Its twistor lines lie wholly in projective null twistor space. Diagonalizing $h$ as positive and negative two-dimensional blocks also gives a useful global description: an isotropic two-plane projects isomorphically to the positive block, so it is a graph $(z,Uz)$, and isotropy is $U^{†}U=1$. The real compact space is therefore $U(2)$, equivalently $(S^1\times S^3)/{\mathbb{Z}}_2$.

The same result is especially transparent in coordinates of the Plücker embedding after a complex linear change of coordinates on ${\Lambda }^2\mathbb{T}$. The real Lorentzian cone has equationIts nonzero real generators represent points. Normalize the lengths of its positive and negative parts to one; real projectivization identifies their simultaneous antipodes. ConsequentlyThis is the Projective Minkowski compactification. Its universal cover is $\mathbb{R}\times S^3$, with deck generator $(s,v)\mapsto (s+\pi ,-v)$, so its fundamental group is $\mathbb{Z}$. The projective version includes periodic identifications; an unwrapped Einstein-cylinder or Penrose diagram description uses the appropriate cover and affine domain instead.

For example, use homogeneous cone coordinates $[U,V,t,\mathbf{x}]$ with quadratic equation $2UV+t^2-|\mathbf{x}{|}^2=0$. The affine patch $U\ne 0$ sets $U=1$ and then $V=-(t^2-|\mathbf{x}{|}^2)/2$. The complement $U=0$ is the null cone $t^2-|\mathbf{x}{|}^2=0$ at infinity. It is much larger than a single point: different null directions retain different limits there.

For the Euclidean slice, choose an antilinear quaternionic structure on a complex vector space $J$ on twistor space, for exampleThe Euclidean reality structure on twistor space selects $J$-invariant complex two-planes. Such a plane is a quaternionic line in ${\mathbb{H}}^2$, givingThere are no fixed projective twistors for $J$: if $JZ=cZ$, applying $J$ again would give $-Z=|c{|}^{2}Z$. Every projective twistor instead lies on the unique invariant line $\mathbb{P}\mathrm{span}(Z,JZ)$. Thus the Euclidean correspondence becomes a fibration ${\mathbb{CP}}^3\to S^4$ with fiber ${\mathbb{CP}}^1$, in contrast to the Lorentzian correspondence where only null twistors represent real null geodesics. Distinct Euclidean real points have disjoint twistor lines, since the line through any $Z$ and $JZ$ is unique. This reflects the absence of nonzero real null vectors for a positive-definite metric.

On the Klein quadric the Euclidean real cone has signature $(1,5)$; normalize its timelike coordinate to one and obtain the unit sphere $S^4$. Its affine Euclidean chart omits one point. Inverse stereographic projection explicitly gives the round conformal metricwhich extends smoothly across that point. This proves the one-point conformal compactification of Euclidean space. The distinction between the Lorentzian Hermitian and Euclidean quaternionic reality structures is also described in Adamo's lectures on twistor theory; the global statements here follow from the explicit models just derived.

Finally, Wick rotation $t\mapsto it$ compares affine real slices through their complexification. Locally it changes $d{t}^2-|d\mathbf{x}{|}^2$ to $-d{\tau }^2-|d\mathbf{x}{|}^2$, the negative of the Euclidean metric; the overall sign does not change its null cone in the complexified setting. It does not identify the two real conformal compactifications globally. The Lorentzian projective space has fundamental group $\mathbb{Z}$, while $S^4$ is simply connected. Its causal universal cover $\mathbb{R}\times S^3$ is noncompact, so passing to that cover does not repair the mismatch with the compact Euclidean sphere.

There is also a direct algebraic obstruction. In the homogeneous coordinates above the affine Wick substitution extends only as the rational mapSubstitution verifies that the image satisfies the same complex quadric equation. For $U=0$ and $t\ne 0$ it collapses the point to $[0,1,0,0]$, while at some infinity points every output coordinate vanishes. In particular it is indeterminate at $[0,1,0,0]$. This is a genuine failure of holomorphic extension: the affine paths $(t,\mathbf{x})=(0,(s,0,0))$ and $(s,(is,0,0))$, as $s\to \infty$, both tend to that input point. Their Wick images tend respectively to $[0,1,0,0]$ and $[0,0,1,(1,0,0)]$, which are distinct points of the complex quadric.

Thus the answer to the global compatibility question is no: ordinary affine Wick rotation is a local analytic continuation, not a globally regular invertible map of the conformally compactified real spacetimes. The global obstruction to Wick rotation of conformal compactifications does not prevent both real slices from being studied inside the same smooth complex Klein quadric.