= Solution
{c}
A regular <hypersurface> $h=0$, with $dh\ne0$, is a <null hypersurface> if its normal is null:
$$
\boxed{g^{\mu\nu}\partial_\mu h\,\partial_\nu h=0\quad\text{on }h=0.}
$$
The raised normal is then also tangent to the <hypersurface>, and its <induced metric> is degenerate. This is a local geometric property.
In an <asymptotically flat spacetime>, choose its <future null infinity> $\mathcal I^+$. The <black hole> region and future <event horizon> are
$$
\boxed{\mathcal B=\mathcal M\setminus J^-(\mathcal I^+),\qquad \mathcal H^+=\partial J^-(\mathcal I^+).}
$$
Here $J^-(\mathcal I^+)$ is the <causal past> of <future null infinity>: the region from which a future-directed <causal curve> can escape to arbitrarily distant observers. An <event horizon> therefore depends on the global future <causal structure>, and an arbitrary <null hypersurface> need not be an <event horizon>.
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