nLab conformal compactification

Contents

Context

Riemannian geometry

Riemannian geometry

Applications

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A conformal compactification is an embedding of a non-compact Lorentzian manifold into a compact Lorentzian manifold as a dense open subspace, such that the embedding is a conformal map .

Example

The (or rather, any) conformal compactification $\overline{M}$ of (flat) Minkowski space $M = \mathbb{R}^{n,1}$ is important in various treatments of behaviour at infinity of physics thereon (see e.g. Penrose-Hawking theorem).

Section 4.2 of (Nikolov-Todorov) gives a thorough geometric discussion of the construction and gives explicit expressions in coordinates, and shows that (the underlying manifold of) this $\overline{M}$ is in fact $(S^n\times S^1)/\{\pm 1\}$, for a specific embedding of $S^n\times S^1$ in $\mathbb{R}^{n+1,2}$, and the action by scalar multiplication.

Let $q(x,t)$ be the standard indefinite quadratic form of signature $(n,1)$ on $\mathbb{R}^{n,1}$ and define the following map $\mathbb{R}^{n,1} \to \mathbb{R}^{n+1,2} \simeq \mathbb{R}^{n,1}\times \mathbb{R}^{1,1}$:

$\tilde{C}\colon (x,t) \mapsto (x,t,\frac{1-q(x,t)}{2},\frac{-1-q(x,t)}{2})$

This is a diffeomorphism on its image, which can be described as the intersection of the hyperplane $v-w=1$ and the quadric $q(x,t) + q'(v,w)=0$, where $(v,w)$ are coordinates on $\mathbb{R}^{1,1}$ with the quadratic form $q'(v,w) = v^2-w^2$. In particular, the image of this map avoids the origin in $\mathbb{R}^{n+1,2}$, and rearranging the defining equation for the quadric we get $|x|^2 + v^2 = t^2 + w^2$ $=K$, say, where $K\neq 0$. We can then scale $\tilde{C}(x,t)$ to $C(x,t)$ so that $K=1$, which means $C(x,t) \in S^n\times S^1$. [TODO: calculate this normalisation] Then the final quotient by $\{\pm 1\}$ gives the desired dense embedding $\mathbb{R}^{n,1} \to (S^n\times S^1)/\{\pm 1\}$.

One could also skip the normalisation step if desired, and pass directly to the quotient by $\mathbb{R}^\times \simeq \mathbb{R}_{\gt0}^* \times \{\pm 1\}$, treating $\tilde{C}(x,t)$ as homogeneous coordinates.

The blog post (Wong 2009)) gives a discussion of comformal compactification in general, with Minkowski space as an example. It describes the underlying manifold of (the ‘usual’ construction of) $\overline{M}$ is $S^n\times S^1$, but then clarifies that this is a ‘double cover’.

A conformal compactification, of complexified Minkowski spacetime $\mathbb{R}^{3,1}$, is given by the Klein quadric. (eg, Fioresi-Lledo-Varadarajan 07, section 2). This plays a key role in the twistor correspondence.

References

• Roger Penrose, Relativistic Symmetry Groups, in A.O.Barut (ed.), Group Theory in Non-Linear Problems: Lectures Presented at the NATO Advanced Study Institute on Mathematical Physics (1974)