synthetic differential geometry
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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
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$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
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$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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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 .
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 $\mathbb{R}^{n,1} \to \mathbb{R}^{n+1,2} \simeq \mathbb{R}^{n,1}\times \mathbb{R}^{1,1}$:
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.
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)
Valentina, Conformal compactifications (pdf)
Jörg Frauendiener, Conformal Infinity (living reviews)
Nikolay M. Nikolov, Ivan T. Todorov, Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory, arXiv:math-ph/0412039 (inspirehep pdf).
R. Fioresi, M. A. Lledo, Veeravalli Varadarajan, The Minkowski and conformal superspaces, J.Math.Phys.48:113505,2007 (arXiv:math/0609813)
Willie Wong, Conformal compactification of space-time (web) (2009).
Last revised on August 2, 2016 at 02:24:08. See the history of this page for a list of all contributions to it.