I'm looking for a rigorous exposition of Penrose diagrams (also called conformal diagrams in general relativity. By "rigorous" ("careful" is perhaps a more attractive word) I mean that it should address the following questions:

  • Is there a general mathematical definition of a Penrose diagram? If spacetime is not 2-dimensional, what kind of symmetries do we need to construct a diagram?
  • Given a 2-dimensional metric, is the Penrose diagram unique?
  • How are infinities (timelike, null, spacelike) formally defined? Can they be defined without appealing to the conformal diagram?

I'm thinking there should be an article, book or something like that covering this, but if this can be answered right here in this site, by all means.

  • 2
    $\begingroup$ For a nice discussion in the context of a review article on boundary constructions, see arxiv.org/abs/gr-qc/0501069 , sec 6.5.1, p. 70. See also Winitzski, section 3.2.2, sites.google.com/site/winitzki/index/… . Can they be defined without appealing to the conformal diagram? Yes, the diagrams are secondary. The notions of boundary constructions and conformal infinity are primary. The source you really need is Hawking and Ellis. You could also look at Wald, section 11.1, p. 271. $\endgroup$ – user4552 Feb 21 '18 at 6:00
  • $\begingroup$ Related: mathoverflow.net/questions/216298/… $\endgroup$ – user4552 Feb 21 '18 at 6:00
  • $\begingroup$ Quite a good lecture on Penrose diagrams: youtube.com/watch?v=nAT1PDkufso $\endgroup$ – md2perpe Feb 22 '18 at 7:29
  • $\begingroup$ @BenCrowell it would seem from your sources that this is kind of an open problem. Maybe you could post that as an answer. $\endgroup$ – Javier Feb 22 '18 at 16:06
  • $\begingroup$ Came across this a little late to the party, but I wrote a paper on this topic a few years ago after spending a pretty solid chunk of grad school looking into it. If anyone else is wondering the same questions you were here and I was then, they might find Section 2 (general formalism) helpful (also appendix for historical review): arxiv.org/abs/1802.02263 $\endgroup$ – Joe Schindler Mar 10 '19 at 0:45

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