Black hole solutions I have a question, that may sound a little silly. Suppose that I have an $n$-dimensional metric given by
$$dS^2=e^{2 A(r)}[-f(r)dt^2+\frac{dr^2}{f(r)}+\eta_{m n}\,dx^m\,dx^n]$$
with $A(r)$ is a warp factor and $f(r)$ is blackening factor. Besides the condition that $f(r_h)=0$ for the event horizon, what other conditions should obey the metric above to be considered as a black hole solution.
 A: Informally, black hole is a region   of spacetime which cannot be seen from far away, more technically from the conformal spacetime boundary.
Let us start with the definition of black hole region from Wald's  textbook:
$$ B=M - J^{-}(\mathscr{I}^{+}),$$
where $M$ is our manifold (which is assumed to be asymptotically flat), $\mathscr{I}^{+}$ is the future null infinity, $J^{-}(\mathscr{I}^{+})$ is the causal past of future null infinity. So the black hole is the region of manifold that lie outside of causal past of future null infinity, in other words no signal (that propagates causally) can reach $\mathscr{I}^{+}$.
Can this definition be applied to OP's metric? Not directly, even with the OP's ansatz, because we do not know what is the conformal boundary for this metric and what are its causal properties. Instead one has to look at the explicit form of metric, identify asymptotic region(s) (let us denote it $\mathscr{I}$) and determine whether $r=r_h$ represents the boundary of the causal past ($J^{-}(\mathscr{I})$) of this region.
One thing to keep in mind, is that while metric does not have to satisfy any specific set of field equations to be a black hole, knowing the theory for which the metric is a solution of can give insight into the structure of asymptotic region.
