Black hole

Remarks

Example: the Schwarzschild spacetime of mass $m$ in ingoing Eddington–Finkelstein coordinates, with $r > 0$. From a point with $r > 2m$, an outgoing radial light ray reaches $\mathcal{I}^+$. From a point with $r \le 2m$, every future-directed causal curve has $r$ non-increasing, so it never reaches $\mathcal{I}^+$. Hence $\mathcal{B} = \set{0 < r \le 2m}$, and its boundary is the sphere $r = 2m$.

Non-example: in Minkowski space, from every point a light ray goes out to $\mathcal{I}^+$, so $J^-(\mathcal{I}^+) \supseteq M$ and $\mathcal{B}$ is empty.

The boundary of $\mathcal{B}$ in $M$ is the event horizon. Many texts add a further condition, such as strong asymptotic predictability, to obtain good properties of black holes, for example in the Hawking area theorem. The definition is global in time: deciding whether a point lies in $\mathcal{B}$ requires knowing the whole future of the spacetime.