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The black hole information paradox is the apparent conflict between unitary quantum evolution and the semiclassical prediction that complete black-hole evaporation leaves only thermal Hawking radiation. A resolution must explain how information is preserved or why one of the assumptions behind the conflict fails.

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.

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 atwhere $\kappa$ is the surface gravity. The outgoing quanta form Hawking radiation, while their correlated partners cross the horizon.