# What is a black hole?

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