Black hole as a solution of Einstein field equation, how about the inverse?

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?

• 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. Jun 30 '17 at 12:58

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.