The early and late stationary regions define distinct positive-frequency mode bases and hence an in-vacuum and out-vacuum. Write their Bogoliubov transformation as

u_{i}^{out} = \sum_{j} (α_{ij} u_{j}^{in} + β_{ij} u_{j}^{in *}) .

The corresponding operators satisfy

b_{i}^{out} = \sum_{j} (α_{ij}^{*} a_{j}^{in} - β_{ij}^{*} a_{j}^{in †}) .

(2)

Using

a_{j}^{in} ∣ 0_{in} ⟩ = 0

and the canonical commutator gives the particle number from Bogoliubov coefficients:

⟨ 0_{in} ∣ N_{i}^{out} ∣ 0_{in} ⟩ = j \sum ∣ β_{ij} ∣^{2} .

(3)

Time dependence in the sandwich region can make

β_{ij} \neq = 0

, so the in-vacuum contains out-particles.

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.