The standard definition of a black hole, which includes all the classic "stellar" black holes, is the complement of the past of future null infinity

$$\mathcal M \setminus I^-(\mathscr I^+)$$

That is, the set of points that cannot reach infinity.

This definition is fairly narrow, though. Is there a definition that would include such black holes as the Schwarzschild-de Sitter black hole (not asymptotically flat), or the elliptic Schwarzschild solution (not causal or time orientable)?

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    $\begingroup$ In practical terms, we're using a black hole metric to approximate some quite small region of space, within a galaxy, so it really doesn't matter if there are other things happening at the cosmological scale. There has been some work on quasilocal definitions. See, e.g., Booth, arxiv.org/abs/gr-qc/0508107 . $\endgroup$ – user4552 Jun 29 '17 at 17:16
  • $\begingroup$ Probably relevant: physics.stackexchange.com/questions/62175 $\endgroup$ – Kyle Kanos Jun 29 '17 at 17:24

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