nLab conformal compactification

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 2004 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.

Related entries

• conformal geometry

• compactification

• asymptotically flat spacetime

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)

• 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:

• Daniel Platt: Conformal Geodesics in Cartan Calculus, MSc thesis (2017) [pdf, pdf]

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]