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Is there a definition of a black hole in a generic spacetime? In some books, for example Wald's, black holes are defined for asymptotically flat spacetime with strong asymptotic predictability, although the definition makes sense without the second condition. Is there a notion of a black hole in general spacetime, not necessarily asymptotically flat? Or is it the case that there is not a "natural" or agreed upon definition?

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Asymptotic flatness is not necessary: if you can define a notion of $\mathscr{I}^\pm$ you can apply the usual definition. You can consider the Schwarzschild-(A)dS spacetimes there. If you accept the teleological definition that a black hole is the set of space-time events from which light cannot "escape to infinity", the black hole $B$ is just the manifold $M$ with the causal past $J^-(A)$ of some set removed. The question then becomes what do you mean by $A$, the set of "infinities." –  Willie Wong Oct 18 '11 at 14:12
    
Yes, so the question is, is there a definition of plus null infinity (how did you make that symbol) for a general spacetime? –  MBN Oct 18 '11 at 14:20
    
$\mathscr{I}$ is \mathscr{I} enclosed in dollar signs. In the most generality there is no definition I am aware of. But you can (probably) define it for asymptotically AdS/hyperbolic space-times and space-times admitting some sort of conformal compactification, modulo some technical issues on decay rates of the metric. –  Willie Wong Oct 18 '11 at 21:32
    
Is it possible to define it for a closed universe? Sorry for the many questions. –  MBN Oct 19 '11 at 8:56
    
I suggest you go look up the Schwarzschild--anti-de Sitter space-times that I mentioned before. For example, see Section 9.2 of this Living Reviews article and references therein. The important thing to note is that there is no absolute reference frame in GR, and when people say "closed" universe they generally mean that it has closed space-like sections for a particular fixed time foliation. For technical reasons in GR we do not entertain the idea that the space-time itself can be a closed four-dimensional manifold (tbc) –  Willie Wong Oct 19 '11 at 9:09
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What the definition needs to capture is that a black hole is not (1) a naked singularity, or (2) a big bang (or big crunch) singularity. We also want the definition to be convenient to work with so that, for example, it's possible to prove no-hair theorems.

Since we want to exclude naked singularities, it's natural that we require an event horizon. Event horizons are by their nature observer-dependent things. For example, if we have a naked singularity, we can always hide its nakedness by picking an observer who is far away from it and accelerating continuously away from it. Such an accelerated observer always has an event horizon, even in Minkowski space. This example shows that it makes a difference what observer we pick.

Actually, we can't have a material observer at null infinity, since timelike infinity, not null infinity, is the elephants' graveyard for material observers. However, the choice of null infinity is the appropriate one because a black hole is supposed to be something that light can't escape from.

Of course the actual universe isn't asymptotically flat, but that doesn't matter. In practice, all we care about is that the black hole is surrounded by enough empty space so that the notion of light escaping from it is well defined for all practical purposes.

There are other possible ways of defining a black hole, e.g., http://arxiv.org/abs/gr-qc/0508107 .

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I think that there has to be e distinction between event horizons and other types of horizons. The event horizon is a property of the geometry of spacetime and not observer dependent. –  MBN Jun 29 '13 at 12:46
    
@MBN: Most observer-dependent statements can be made observer-independent. You can use quantifiers, e.g., "there exists an observer such that ..." or you can put an observer in some special place, e.g., at infinity, which is what's being done here. –  Ben Crowell Jul 1 '13 at 1:03
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