= Solution
A gravitational-collapse spacetime is stationary and approximately empty in the remote past and settles to a stationary black-hole exterior in the future, with a dynamical region between them. The same comparison of in- and out-positive-frequency modes therefore applies. Tracing an outgoing late-time mode backward through the collapse produces an exponentially blueshifted mixture of early positive and negative frequencies. Its nonzero Bogoliubov beta coefficient yields the thermal occupation numbers of <Hawking radiation>, while partner modes pass through the <event horizon>.
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