Boundary Construction for B.H. & B.B. Singularities

[bcrowell]

There is a general topic of boundary constructions, which means how to adjoin idealized points in a sensible way to a given spacetime. There is a menagerie of these methods, including the g-boundary (Geroch), b-boundary (Schmidt), c-boundary (Geroch, Kronheimer, and Penrose) and a-boundary (Scott and Szekeres). (Some of these are nonunique, and the a-boundary in particular seems to be more like a general framework than a specific prescription.) Surveys are given in these references:

Sanchez, "Causal boundaries and holography on wave type spacetimes," http://arxiv.org/abs/0812.0243

Ashley, "Singularity theorems and the abstract boundary construction," https://digitalcollections.anu.edu.au/handle/1885/46055

My understanding is based on a fairly casual reading of the introductory material in Ashley's thesis.

Do these methods only disagree on pathological examples, while agreeing on the common examples of interest such as Minkowski space, the Schwarzschild spacetime, and Friedmann spacetimes? In particular, what dimensionality do they give for the boundaries corresponding to the singularities? I think most people intuitively think of the Schwarzschild singularity as being one-dimensional and of cosmological singularities as three-dimensional, but I would be curious to know whether the various boundary constructions agree with these intuitions or not.

 

[martinbn]

I think (and I might be wrong, I can try to find where I have seen it) but the b-boundary is problematic, in a way, for the Friedmann space-times as well. They are completed with one boundary point and the resulting space is not Hausdorff, that point cannot be separated from the rest. It is a single point even for the models with an initial and final singularity.

 

[bcrowell]

@martinbn : Oh, right -- I read right through Ashley's description of that, but I missed that point (get it? heh heh). It's zero-dimensional in that approach. Interesting!

 

[bcrowell]

I found some discussion here: http://physics.stackexchange.com/a/170893 . After noting the problems with the b-boundary as applied to Friedmann spacetimes, they say that the c-boundary is three-dimensional, by invoking the Penrose diagram. I don't know if it's rigorously true that you can always characterize the c-boundary in this way just by looking at the Penrose diagram, but it would certainly make sense, because the c-boundary is clearly designed to harmonize with that approach.

Since people don't like the fact that the b-boundary of a Friedmann spacetime is a single point, I assume they intuitively expect it to be a 3-dimensional surface. But I wonder what they expect the boundary of the Schwarzschild metric to be? It seems equally plausible to me that it would be 1-dimensional or 3-dimensional. On a Penrose diagram it looks 3-dimensional, so I assume the c-boundary is 3-dimensional.

 

[bcrowell]

After some further digging around, it seems to me that the answer must be that there is no clear answer.

For both the Schwarzschild spacetime and the Friedmann spacetimes, the b-boundary gives a topology that's non-Hausdorff, which seems obviously unacceptable; the singularities are in some sense arbitrarily close to every point in spacetime. (In the closed FRW universe, it also identifies the initial and final singularities, which is also silly.)

For anti-de Sitter space, it appears that the GKP-style c-boundary represents the initial and final singularities as points ( http://arxiv.org/abs/gr-qc/0501069 , p. 72). The original GKP paper apparently also discussed the Schwarzschild spacetime, but it's paywalled, so I can't see it. Anyway, the state of the art in constructions in the c-boundary style seems to have moved on. Flores http://arxiv.org/abs/gr-qc/0608063 http://arxiv.org/abs/1001.3270 has a new version that he claims is in some sense "optimal," but he never seems to explicitly talk about examples of physical interest, just artificial examples like Minkowski space with pieces cut out.