# How unique is a conformal compactification?

I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, a conformal compactification of a manifold $M$ is a compactification $\tilde{M}$ in which all conformal transformations defined locally extend globally. Firstly, is this correct?

Now for my main question. Obviously not all compactifications are conformal compactifications (take the Alexandroff compactification of Minkowski space for example). But what about the other direction? Are all conformal completions compact? I can't think of a counterexample, but then my intuition about conformal completions is pretty shaky. I hoped that some combination of Liouville's theorem and perhaps the Hopf-Rinow theorem would help, but I'm not sure it does.

Finally how unique are conformal compactifications? I'd like to think that for any manifold there's only one conformal compactification of the same dimension, given by the usual Penrose construction. But I can't find any references to help me start to get some intuition.

If anyone has any hints or suggestions for good literature I'd be very pleased to hear them! Many thanks!

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You should be aware that physicists often use math words in a context where they do not strictly apply. For example, a "conformal compactification" of $M$ might not be a compactification of $M$, as the inclusion $M\hookrightarrow \overline M$ might not be dense. – André Henriques Apr 13 '13 at 18:02
@Andre - thanks for this. Have you got an explicit example, because it seems that all the ones I know are compact! – Edward Hughes Apr 13 '13 at 19:13
I was thinking specifically of manifolds with Minkowskian metric, so maybe that's not what you care about most. – André Henriques Apr 14 '13 at 23:14
@André Henriques: “manifolds with Minkowskian metric” are specially named “lorentzian”. – Incnis Mrsi Apr 1 at 9:30

## 1 Answer

For Lorentzian manifolds, the conformal completion need not be compact. A typical example is the universal covering of the $d$-dimensional anti-de Sitter space-time (the maximally symmetric solution of the vacuum Einstein equations with negative cosmological constant) - its conformal boundary is diffeomorphic to $\mathbb{R}\times\mathbb{S}^{d-2}$.

Several examples of conformal completions of different space-times (i.e. time oriented Lorentzian manifolds) such as above can be found in Chapter 5 of S. W. Hawking and G. F. R. Ellis, "The Large Scale Structure of Space-Time" (Cambridge, 1973). As for the uniqueness of the procedure of conformal completion, there is of course (at least) the freedom to multiply the metric in the conformal completion by a positive smooth function thereof. There is an extensive discussion on the ambiguities in the definition of a conformal completion by Robert Geroch in F. P. Esposito and L. Witten (eds.), "The Asymptotic Structure of Space-Time" (Plenum, 1977), pp. 1-105. The book "General Relativity" by R. M. Wald (Chicago University Press, 1984) also discusses conformal completions to some length in Chapter 11.

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Thanks - that's extremely useful! – Edward Hughes Apr 13 '13 at 20:43
Just out of interest - do you know who invented the term conformal compactification, and why they didn't just call it conformal completion? Many thanks! – Edward Hughes Apr 14 '13 at 8:57
@Edward, for that you probably need to check out some early papers by Roger Penrose. – Igor Khavkine Apr 14 '13 at 9:59
As mentioned by Igor, the terminology was introduced by Penrose in the particular case of Minkowski space-time ("Asymptotic Properties of Fields and Space-Times", Phys. Rev. Lett. 10 (1963) 66-68). When developing the procedure in detail ("Zero Rest-Mass Field Including Gravitation: Asymptotic Behavior", Proc. Roy. Soc. London A284 (1965) 159-203), he was already aware that compactness of the conformal completion was a rather special property of Minkowski space-time. However, the name stuck to many physicists because of the convenience of the conformal representation of Minkowski space-time. – Pedro Lauridsen Ribeiro Apr 15 '13 at 3:42