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
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?
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?
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 $i^\pm$?
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