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Consider the following definition:

A Black Hole in an asymptotically flat spacetime $\mathfrak{M} \equiv (\mathcal{M}, \mathrm{\textbf{g}})$ is the set of events that do not belong to the causal past of the future null infinity $J^{-}(\mathfrak{I}^{+})$, namely: $$\mathfrak{B} = \mathfrak{M}-J^{-}(\mathfrak{I}^{+})\neq \varnothing$$

The boundary of $\mathfrak{B}$ is called event horizon.

My question is: this definition is a general one, i.e., in the sense that this definition is both valid for Astrophysical Black Holes (the soluitons from General Relativity e.g. Schwarzschild,Kerr,Reissner-Nordström,Kerr-Newman solutions) and "quantum black holes"?

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    $\begingroup$ What are quantum black holes? $\endgroup$ – Avantgarde Apr 8 at 20:55
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This is a traditional definition, which doesn't take into account BH evaporation. Evaporation makes it possible that stellar collapse doesn't result in an event horizon (there is an "apparent" or "trapping" horizon instead), in which case no BHs by that traditional definition would exist. There is no commonly accepted general definition for evaporating quantum BHs, although many people have their own opinions about it.

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