synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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Tangency
The magic algebraic facts
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differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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 of (flat) Minkowski space 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 is in fact , for a specific embedding of in , and the action by scalar multiplication.
Let be the standard indefinite quadratic form of signature on and define the following map :
This is a diffeomorphism on its image, which can be described as the intersection of the hyperplane and the quadric , where are coordinates on with the quadratic form . In particular, the image of this map avoids the origin in , and rearranging the defining equation for the quadric we get , say, where . We can then scale to so that , which means . [TODO: calculate this normalisation] Then the final quotient by gives the desired dense embedding .
One could also skip the normalisation step if desired, and pass directly to the quotient by , treating 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) is , but then clarifies that this is a ‘double cover’.
A conformal compactification, of complexified Minkowski spacetime , 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 – link appears to be dead)
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 July 18, 2019 at 23:39:51. See the history of this page for a list of all contributions to it.