# Questions on Penrose's paper - Conformal Treatment of Infinity

I have several questions. Perhaps it would be better to separate them into different posts. However, given their relative closeness to each other, I think putting it all in one place would be better. On suggestion, I will modify this post.

I am reading Penrose's paper on the conformal treatment of infinity (find here). The basic concept that underlies this is the fact that in order to study asymptotic behaviour of a space-time with metric ${\tilde g}_{\mu\nu}$, we may instead study a space-time that is conformally related to it by defining an unphysical metric on a compact manifold $g_{\mu\nu} = \Omega^2 {\tilde g}_{\mu\nu}$. He then goes on to say that the asymptotic properties of fields can then be investigated by studying local behaviour of fields at infinity on this unphysical manifold provided that the relevant concepts can be put into a conformally invariant form

1. What are the relevant conformally invariance concepts that one can study? What I can think of is the causal structure of space-time, gravitational waves (since they are described by a conformally invariant Weyl tensor). What else is there?

2. I have also often heard that massless fields satisfy conformally invariant equations in curved spacetimes. (see this question). Indeed Penrose claims that this can be done if ""interpreted suitably". What does he mean by this? Further, since massless particles can only reach $\mathscr{I}^\pm$, does his formalism only apply to null infinity?

3. What about massive particles? Surely, the equations for such particles are not going to be conformally invariant. Further, these particles would reach $i^\pm$. One can't apply the above formalism to massive particles right? Is there an alternative way of constructing the asymptotic structures of $i^\pm$?

• Ask yourself, are your different questions really asking one thing? If so, they're fine in one post, but otherwise they should be split up. A good rule of thumb is that the one thing you're asking should form the title of the post. If you can't condense the essence of your post down to one main title question, then it's a strong indication the post needs to be split. – David Z Aug 23 '13 at 3:32
• I am reading Penrose's paper on the conformal treatment of infinity. Please provide the actual reference. In general, conformal infinity is just an ideal set of points at infinity that we adjoin onto our spacetime as a matter of convenience. It can be convenient regardless of whether there is any conformal invariance. Cf. the convenience of being able to write $\int_1^\infty x^{-3}dx$. – user4552 Aug 23 '13 at 4:17
• Link added. I understand that while defining infinity for spacetimes is important in the sense that you have mentioned. However, one often studies the asymptotic structure of spacetimes near the conformal infinity. I presume that the "structures that are studied here are necessarily conformally invariant. I am then asking, what precisely are these structures that are studied in the various contexts as mentioned in the problem. – Prahar Aug 23 '13 at 4:22
• " I am then asking, what precisely are these structures that are studied in the various contexts as mentioned in the problem. " Things like the asymptotic behaviours of conformally invariant quantities like the Weyl tensor - as in the peeling theorem – twistor59 Aug 23 '13 at 7:13
• Right. I noted that in the "gravitational waves/Weyl tensor" bit. Are there any other quantities of interest? – Prahar Aug 23 '13 at 7:35