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I have in trouble for understanding some conceptual terminology about Black Hole.

As far as i known, Black Hole is related with solution of Einstein field equation.

But apparently, all solution of Einstein field equations are not Black Holes. (Considering flat)

Then when does the solution of Einstein field equation becomes BH?

Usually my guess is if BH has singularity?

Can you give me some mathematical definition of Black Holes?

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    $\begingroup$ If you want to ask, given a stress-energy known tensor $T_{\mu\nu}$, can we uniquely determine what is the geometry of our spacetime using the Einstein equations? Surprisingly enough, no. For example, if $T_{\mu\nu}=0$ then we can have all the following solutions: a completely flat Minkowskian spacetime, a spacetime with some gravitational waves, as well as a Schwarzschild Black Hole. $\endgroup$
    – user87745
    Commented Jun 30, 2017 at 12:58

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The formal definition is linked to the causal structure of the spacetime:

Consider a strongly asymptotically predictable spacetime $M$. A black hole B is defined as:

$$ B = [ M -J^-(\mathscr{I}^+)] $$

In words, in a spacetime $M$ with an open region conformally related to a globally hyperbolic spacetime, a black hole region is defined as the closure of the complementary of the spacetime $M$ with respect to the causal past of future null infinity. The black hole horizon is the boundary of this region.

In simple words: a black hole is a region of space in which once entered, nothing can escape to infinity anymore. Nothing includes light, that's why they are black. Notice that once you have a singularity, due to the cosmic censorship conjecture you must have an horizon.

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  • $\begingroup$ Nice! Where can I read about this definition and related topics (textbook if possible, please) $\endgroup$
    – magma
    Commented Jul 4, 2017 at 16:36
  • $\begingroup$ The standard reference is Wald, General Relativity. $\endgroup$
    – Rexcirus
    Commented Jul 4, 2017 at 17:19

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