Define Black Holes w/o Asymptotic Flatness?

[bcrowell]

I see statements that in order to define a black hole, we need asymptotic flatness, but this only seems to be necessary because we want to define the horizon as the boundary of a region from which light can't escape to null infinity ($\mathscr{I}^+$). It seems like you can have a well defined null infinity without having asymptotic flatness. E.g., Hawking and Ellis describe de Sitter space as having a Penrose diagram that looks like a square, and they do refer to the top edge of the square as $\mathscr{I}^+$. (This means that $\mathscr{I}^+$ in de Sitter space is spacelike rather than timelike.)

So can't we define a black hole simply by saying that light can't escape to $\mathscr{I}^+$? I guess this raises the point of what general definition we would have in mind for $\mathscr{I}^+$. H&E don't actually give an explicit definition, but it seems like the thrust of it is that it's the graveyard of lightlike geodesics on a Penrose diagram, and implicitly I think the intention is that a singularity isn't part of $\mathscr{I}^+$, even though light rays can end up there. So I suppose we want the definition to be that it's the graveyard of complete lightlike geodesics.

If asymptotic flatness really is necessary in order to define black holes, then it seems like we can't, for example, define black holes in de Sitter space. This seems silly to me -- if we lived in de Sitter space, we could certainly tell what was and what wasn't a black hole, couldn't we?

[EDIT] Hmm... possibly this answers my question: http://relativity.livingreviews.org/open?pubNo=lrr-2001-6&page=articlese2.html "Similar definitions of a black hole can be given in other contexts (such as asymptotically anti-deSitter spacetimes) where there is a well defined asymptotic region." Geroch is talking about a-dS, not dS, but the concept seems the same.

 

[PAllen]

Yes, that seems reasonable and consistent with generalization of ADM/Bondi mass to any spacetime with a well defined, open, asymptotic region.

For horizons, even in a closed spacetime, it would seem that some definitions of apparent horizon would work. It also seems like singularity defined as curvature becoming infinite should work.

 

[Haelfix]

There are some subleties with apparent horizons in the context of asymptotic AdS or DS spacetimes. But in principle, you can generalize Reissner Nordstrom and Schwarzschild black holes to the case of a nonvanishing cc without too much fuss, at least naively.

For a treatment more in the spirit of Hawking and Ellis (for instance trying to generalize the singularity theorems and cosmic censorship) is of course far more subtle and I believe still an open mathematics question.