= Solution
In a collapsing spacetime, positive-frequency in-modes are defined at past null infinity and outgoing modes at future null infinity. Tracing a late outgoing wave packet backward toward the forming <event horizon> produces an exponential blueshift. Its rapidly varying phase has both positive- and negative-frequency parts relative to the past time coordinate, so the Bogoliubov coefficient $\beta$ is nonzero. The resulting bosonic occupation numbers have the <Bose-Einstein distribution> at
$$
T_H=\frac\kappa{2\pi},
$$
where $\kappa$ is the <surface gravity>. The outgoing quanta form <Hawking radiation>, while their correlated partners cross the horizon.
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