synthetic differential geometry
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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 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 2004 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]
Nikolay M. Nikolov, Ivan T. Todorov, §4.2 in: Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory, in: Modern Mathematical Physocs, Proceedings of: 3rd Summer School in Modern Mathematical Physics (MPHYS3) (2004) 1-93 [arXiv:math-ph/0412039, InSpire:674206]
Rita Fioresi, Maria A. Lledo, Veeravalli Varadarajan, The Minkowski and conformal superspaces, J. Math. Phys. 48 (2007) 113505 [arXiv:math/0609813, doi:10.1142/8972]
Willie Wong, Conformal compactification of space-time (2009) [web]
Juan A. Valiente Kroon: Conformal Methods in General Relativity, Cambridge University Press (2017, 2022) [doi:10.1017/9781009291309, oapen:20.500.12657/59217, pdf]
On conformal compactification via Cartan geometry:
On conformal boundaries, mostly in the context of the AdS/CFT correspondence:
Edward Witten, Shing-Tung Yau, Connectedness Of The Boundary In The AdS/CFT Correspondence, Adv. Theor. Math. Phys. 3 (1999) 1635-1655 [arXiv:hep-th/9910245, doi:10.4310/ATMP.1999.v3.n6.a1]
Henrique Boschi-Filho, Nelson R. F. Braga: Compact AdS space, brane geometry, and the AdS/CFT correspondence, Phys. Rev. D 66 025005 (2002) [doi:10.1103/PhysRevD.66.025005, arXiv:hep-th/0112196]
Jörg Frauendiener, Conformal Infinity, Living Rev. Relativ. 7 1 (2004) [doi:10.12942/lrr-2004-1]
C. A. Ballon Bayona, Nelson R. F. Braga, Anti-de Sitter boundary in Poincaré coordinates, Gen. Rel. Grav. 39 (2007) 1367-1379 [arXiv:hep-th/0512182, doi:10.1007/s10714-007-0446-y]
Charles Frances, The conformal boundary of anti-de Sitter space-times, in: AdS/CFTCorrespondence: Einstein Metrics and Their Conformal Boundaries, IRMA Lectures in Mathematical and Theoretical Physics, EMS (2011) 205-216 [ems:irma/9/206, hal:03195056, pdf]
Last revised on July 2, 2024 at 09:41:10. See the history of this page for a list of all contributions to it.