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  • $\begingroup$ Can you outline the differences between the various boundary constructions, from a physical point of view? The obvious approach is to form an equivalence class of the geodesics that intersect the singularity, based on whether they converge or not, and it's hard to see what other choices would be reasonable. Also, could you expand on why you say that the Penrose diagram approach gives you a 3-surface rather than a line? That seems wrong. $\endgroup$
    – Harry Johnston
    Commented Aug 15, 2017 at 4:08
  • $\begingroup$ @HarryJohnston: My answer gives links to review articles on boundary constructions. There's no way to explain the whole topic in an SE answer. Also, could you expand on why you say that the Penrose diagram approach gives you a 3-surface rather than a line? That seems wrong. I'm not sure what expansion would be helpful. What seems wrong to you? $\endgroup$
    – user4552
    Commented Aug 30, 2017 at 23:53
  • $\begingroup$ Looking at the first reference, most of the constructions don't look all that hard to summarize: the g-boundary uses geodesics (the same as my suggestion) whereas the b-boundary uses parallel transport and the c-boundary uses causality relationships. The a-boundary I don't know. The b-boundary is clearly dysfunctional, and according to the paper it is believed (though unproven) that the g-boundary and c-boundary are equivalent, at least in the simpler cases. So at this point I'm not entirely convinced that classifying the Schwartzchild singularity is as hard as your answer suggests. :-) $\endgroup$
    – Harry Johnston
    Commented Aug 31, 2017 at 2:05
  • $\begingroup$ As for the Penrose diagram, that looks like a spacelike line to me, not a three-surface, for the same reason that the center of a sphere is a point and not a two-surface. The radial coordinates converge. $\endgroup$
    – Harry Johnston
    Commented Aug 31, 2017 at 2:11
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    $\begingroup$ This is a terrific and very informative answer! $\endgroup$
    – tparker
    Commented Mar 19, 2018 at 2:01